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authorlangou <julien.langou@ucdenver.edu>2017-04-01 13:41:49 (GMT)
committerGitHub <noreply@github.com>2017-04-01 13:41:49 (GMT)
commit2a811b8f213b8f08e93c7b64acbb3864efa7a5dd (patch)
tree4d74d0774bbe1b748731cd6e3b511f27b02813fd
parentce2261daabdee2e0b8532d1fabf879e5418db03a (diff)
parent16db9c24ea1da07e27c22cea5a187545085947a4 (diff)
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Merge pull request #137 from echeresh/xlarrx_fix
Add quick return for *LARR* routines
-rw-r--r--SRC/cgejsv.f4470
-rw-r--r--SRC/clarrv.f7
-rw-r--r--SRC/dlarra.f8
-rw-r--r--SRC/dlarrb.f6
-rw-r--r--SRC/dlarrc.f7
-rw-r--r--SRC/dlarrd.f6
-rw-r--r--SRC/dlarre.f7
-rw-r--r--SRC/dlarrf.f7
-rw-r--r--SRC/dlarrj.f6
-rw-r--r--SRC/dlarrk.f7
-rw-r--r--SRC/dlarrr.f7
-rw-r--r--SRC/dlarrv.f7
-rw-r--r--SRC/slarra.f8
-rw-r--r--SRC/slarrb.f6
-rw-r--r--SRC/slarrc.f7
-rw-r--r--SRC/slarrd.f6
-rw-r--r--SRC/slarre.f7
-rw-r--r--SRC/slarrf.f7
-rw-r--r--SRC/slarrj.f6
-rw-r--r--SRC/slarrk.f7
-rw-r--r--SRC/slarrr.f7
-rw-r--r--SRC/slarrv.f7
-rw-r--r--SRC/zgejsv.f4474
-rw-r--r--SRC/zlarrv.f7
24 files changed, 4618 insertions, 4476 deletions
diff --git a/SRC/cgejsv.f b/SRC/cgejsv.f
index 28804e7..06c9f40 100644
--- a/SRC/cgejsv.f
+++ b/SRC/cgejsv.f
@@ -1,2235 +1,2235 @@
-*> \brief \b CGEJSV
-*
-* =========== DOCUMENTATION ===========
-*
-* Online html documentation available at
-* http://www.netlib.org/lapack/explore-html/
-*
-*> \htmlonly
-*> Download CGEJSV + dependencies
-*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgejsv.f">
-*> [TGZ]</a>
-*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgejsv.f">
-*> [ZIP]</a>
-*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgejsv.f">
-*> [TXT]</a>
-*> \endhtmlonly
-*
-* Definition:
-* ===========
-*
-* SUBROUTINE CGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
-* M, N, A, LDA, SVA, U, LDU, V, LDV,
-* CWORK, LWORK, RWORK, LRWORK, IWORK, INFO )
-*
-* .. Scalar Arguments ..
-* IMPLICIT NONE
-* INTEGER INFO, LDA, LDU, LDV, LWORK, M, N
-* ..
-* .. Array Arguments ..
-* COMPLEX A( LDA, * ), U( LDU, * ), V( LDV, * ), CWORK( LWORK )
-* REAL SVA( N ), RWORK( LRWORK )
-* INTEGER IWORK( * )
-* CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
-* ..
-*
-*
-*> \par Purpose:
-* =============
-*>
-*> \verbatim
-*>
-*> CGEJSV computes the singular value decomposition (SVD) of a complex M-by-N
-*> matrix [A], where M >= N. The SVD of [A] is written as
-*>
-*> [A] = [U] * [SIGMA] * [V]^*,
-*>
-*> where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
-*> diagonal elements, [U] is an M-by-N (or M-by-M) unitary matrix, and
-*> [V] is an N-by-N unitary matrix. The diagonal elements of [SIGMA] are
-*> the singular values of [A]. The columns of [U] and [V] are the left and
-*> the right singular vectors of [A], respectively. The matrices [U] and [V]
-*> are computed and stored in the arrays U and V, respectively. The diagonal
-*> of [SIGMA] is computed and stored in the array SVA.
-*> \endverbatim
-*>
-*> Arguments:
-*> ==========
-*>
-*> \param[in] JOBA
-*> \verbatim
-*> JOBA is CHARACTER*1
-*> Specifies the level of accuracy:
-*> = 'C': This option works well (high relative accuracy) if A = B * D,
-*> with well-conditioned B and arbitrary diagonal matrix D.
-*> The accuracy cannot be spoiled by COLUMN scaling. The
-*> accuracy of the computed output depends on the condition of
-*> B, and the procedure aims at the best theoretical accuracy.
-*> The relative error max_{i=1:N}|d sigma_i| / sigma_i is
-*> bounded by f(M,N)*epsilon* cond(B), independent of D.
-*> The input matrix is preprocessed with the QRF with column
-*> pivoting. This initial preprocessing and preconditioning by
-*> a rank revealing QR factorization is common for all values of
-*> JOBA. Additional actions are specified as follows:
-*> = 'E': Computation as with 'C' with an additional estimate of the
-*> condition number of B. It provides a realistic error bound.
-*> = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
-*> D1, D2, and well-conditioned matrix C, this option gives
-*> higher accuracy than the 'C' option. If the structure of the
-*> input matrix is not known, and relative accuracy is
-*> desirable, then this option is advisable. The input matrix A
-*> is preprocessed with QR factorization with FULL (row and
-*> column) pivoting.
-*> = 'G' Computation as with 'F' with an additional estimate of the
-*> condition number of B, where A=B*D. If A has heavily weighted
-*> rows, then using this condition number gives too pessimistic
-*> error bound.
-*> = 'A': Small singular values are not well determined by the data
-*> and are considered as noisy; the matrix is treated as
-*> numerically rank defficient. The error in the computed
-*> singular values is bounded by f(m,n)*epsilon*||A||.
-*> The computed SVD A = U * S * V^* restores A up to
-*> f(m,n)*epsilon*||A||.
-*> This gives the procedure the licence to discard (set to zero)
-*> all singular values below N*epsilon*||A||.
-*> = 'R': Similar as in 'A'. Rank revealing property of the initial
-*> QR factorization is used do reveal (using triangular factor)
-*> a gap sigma_{r+1} < epsilon * sigma_r in which case the
-*> numerical RANK is declared to be r. The SVD is computed with
-*> absolute error bounds, but more accurately than with 'A'.
-*> \endverbatim
-*>
-*> \param[in] JOBU
-*> \verbatim
-*> JOBU is CHARACTER*1
-*> Specifies whether to compute the columns of U:
-*> = 'U': N columns of U are returned in the array U.
-*> = 'F': full set of M left sing. vectors is returned in the array U.
-*> = 'W': U may be used as workspace of length M*N. See the description
-*> of U.
-*> = 'N': U is not computed.
-*> \endverbatim
-*>
-*> \param[in] JOBV
-*> \verbatim
-*> JOBV is CHARACTER*1
-*> Specifies whether to compute the matrix V:
-*> = 'V': N columns of V are returned in the array V; Jacobi rotations
-*> are not explicitly accumulated.
-*> = 'J': N columns of V are returned in the array V, but they are
-*> computed as the product of Jacobi rotations, if JOBT .EQ. 'N'.
-*> = 'W': V may be used as workspace of length N*N. See the description
-*> of V.
-*> = 'N': V is not computed.
-*> \endverbatim
-*>
-*> \param[in] JOBR
-*> \verbatim
-*> JOBR is CHARACTER*1
-*> Specifies the RANGE for the singular values. Issues the licence to
-*> set to zero small positive singular values if they are outside
-*> specified range. If A .NE. 0 is scaled so that the largest singular
-*> value of c*A is around SQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
-*> the licence to kill columns of A whose norm in c*A is less than
-*> SQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,
-*> where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
-*> = 'N': Do not kill small columns of c*A. This option assumes that
-*> BLAS and QR factorizations and triangular solvers are
-*> implemented to work in that range. If the condition of A
-*> is greater than BIG, use CGESVJ.
-*> = 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)]
-*> (roughly, as described above). This option is recommended.
-*> ===========================
-*> For computing the singular values in the FULL range [SFMIN,BIG]
-*> use CGESVJ.
-*> \endverbatim
-*>
-*> \param[in] JOBT
-*> \verbatim
-*> JOBT is CHARACTER*1
-*> If the matrix is square then the procedure may determine to use
-*> transposed A if A^* seems to be better with respect to convergence.
-*> If the matrix is not square, JOBT is ignored.
-*> The decision is based on two values of entropy over the adjoint
-*> orbit of A^* * A. See the descriptions of WORK(6) and WORK(7).
-*> = 'T': transpose if entropy test indicates possibly faster
-*> convergence of Jacobi process if A^* is taken as input. If A is
-*> replaced with A^*, then the row pivoting is included automatically.
-*> = 'N': do not speculate.
-*> The option 'T' can be used to compute only the singular values, or
-*> the full SVD (U, SIGMA and V). For only one set of singular vectors
-*> (U or V), the caller should provide both U and V, as one of the
-*> matrices is used as workspace if the matrix A is transposed.
-*> The implementer can easily remove this constraint and make the
-*> code more complicated. See the descriptions of U and V.
-*> In general, this option is considered experimental, and 'N'; should
-*> be preferred. This is subject to changes in the future.
-*> \endverbatim
-*>
-*> \param[in] JOBP
-*> \verbatim
-*> JOBP is CHARACTER*1
-*> Issues the licence to introduce structured perturbations to drown
-*> denormalized numbers. This licence should be active if the
-*> denormals are poorly implemented, causing slow computation,
-*> especially in cases of fast convergence (!). For details see [1,2].
-*> For the sake of simplicity, this perturbations are included only
-*> when the full SVD or only the singular values are requested. The
-*> implementer/user can easily add the perturbation for the cases of
-*> computing one set of singular vectors.
-*> = 'P': introduce perturbation
-*> = 'N': do not perturb
-*> \endverbatim
-*>
-*> \param[in] M
-*> \verbatim
-*> M is INTEGER
-*> The number of rows of the input matrix A. M >= 0.
-*> \endverbatim
-*>
-*> \param[in] N
-*> \verbatim
-*> N is INTEGER
-*> The number of columns of the input matrix A. M >= N >= 0.
-*> \endverbatim
-*>
-*> \param[in,out] A
-*> \verbatim
-*> A is COMPLEX array, dimension (LDA,N)
-*> On entry, the M-by-N matrix A.
-*> \endverbatim
-*>
-*> \param[in] LDA
-*> \verbatim
-*> LDA is INTEGER
-*> The leading dimension of the array A. LDA >= max(1,M).
-*> \endverbatim
-*>
-*> \param[out] SVA
-*> \verbatim
-*> SVA is REAL array, dimension (N)
-*> On exit,
-*> - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
-*> computation SVA contains Euclidean column norms of the
-*> iterated matrices in the array A.
-*> - For WORK(1) .NE. WORK(2): The singular values of A are
-*> (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
-*> sigma_max(A) overflows or if small singular values have been
-*> saved from underflow by scaling the input matrix A.
-*> - If JOBR='R' then some of the singular values may be returned
-*> as exact zeros obtained by "set to zero" because they are
-*> below the numerical rank threshold or are denormalized numbers.
-*> \endverbatim
-*>
-*> \param[out] U
-*> \verbatim
-*> U is COMPLEX array, dimension ( LDU, N ) or ( LDU, M )
-*> If JOBU = 'U', then U contains on exit the M-by-N matrix of
-*> the left singular vectors.
-*> If JOBU = 'F', then U contains on exit the M-by-M matrix of
-*> the left singular vectors, including an ONB
-*> of the orthogonal complement of the Range(A).
-*> If JOBU = 'W' .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),
-*> then U is used as workspace if the procedure
-*> replaces A with A^*. In that case, [V] is computed
-*> in U as left singular vectors of A^* and then
-*> copied back to the V array. This 'W' option is just
-*> a reminder to the caller that in this case U is
-*> reserved as workspace of length N*N.
-*> If JOBU = 'N' U is not referenced, unless JOBT='T'.
-*> \endverbatim
-*>
-*> \param[in] LDU
-*> \verbatim
-*> LDU is INTEGER
-*> The leading dimension of the array U, LDU >= 1.
-*> IF JOBU = 'U' or 'F' or 'W', then LDU >= M.
-*> \endverbatim
-*>
-*> \param[out] V
-*> \verbatim
-*> V is COMPLEX array, dimension ( LDV, N )
-*> If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
-*> the right singular vectors;
-*> If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),
-*> then V is used as workspace if the pprocedure
-*> replaces A with A^*. In that case, [U] is computed
-*> in V as right singular vectors of A^* and then
-*> copied back to the U array. This 'W' option is just
-*> a reminder to the caller that in this case V is
-*> reserved as workspace of length N*N.
-*> If JOBV = 'N' V is not referenced, unless JOBT='T'.
-*> \endverbatim
-*>
-*> \param[in] LDV
-*> \verbatim
-*> LDV is INTEGER
-*> The leading dimension of the array V, LDV >= 1.
-*> If JOBV = 'V' or 'J' or 'W', then LDV >= N.
-*> \endverbatim
-*>
-*> \param[out] CWORK
-*> \verbatim
-*> CWORK is COMPLEX array, dimension (MAX(2,LWORK))
-*> If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
-*> LRWORK=-1), then on exit CWORK(1) contains the required length of
-*> CWORK for the job parameters used in the call.
-*> \endverbatim
-*>
-*> \param[in] LWORK
-*> \verbatim
-*> LWORK is INTEGER
-*> Length of CWORK to confirm proper allocation of workspace.
-*> LWORK depends on the job:
-*>
-*> 1. If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
-*> 1.1 .. no scaled condition estimate required (JOBA.NE.'E'.AND.JOBA.NE.'G'):
-*> LWORK >= 2*N+1. This is the minimal requirement.
-*> ->> For optimal performance (blocked code) the optimal value
-*> is LWORK >= N + (N+1)*NB. Here NB is the optimal
-*> block size for CGEQP3 and CGEQRF.
-*> In general, optimal LWORK is computed as
-*> LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF), LWORK(CGESVJ)).
-*> 1.2. .. an estimate of the scaled condition number of A is
-*> required (JOBA='E', or 'G'). In this case, LWORK the minimal
-*> requirement is LWORK >= N*N + 2*N.
-*> ->> For optimal performance (blocked code) the optimal value
-*> is LWORK >= max(N+(N+1)*NB, N*N+2*N)=N**2+2*N.
-*> In general, the optimal length LWORK is computed as
-*> LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF), LWORK(CGESVJ),
-*> N*N+LWORK(CPOCON)).
-*> 2. If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
-*> (JOBU.EQ.'N')
-*> 2.1 .. no scaled condition estimate requested (JOBE.EQ.'N'):
-*> -> the minimal requirement is LWORK >= 3*N.
-*> -> For optimal performance,
-*> LWORK >= max(N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
-*> where NB is the optimal block size for CGEQP3, CGEQRF, CGELQ,
-*> CUNMLQ. In general, the optimal length LWORK is computed as
-*> LWORK >= max(N+LWORK(CGEQP3), N+LWORK(CGESVJ),
-*> N+LWORK(CGELQF), 2*N+LWORK(CGEQRF), N+LWORK(CUNMLQ)).
-*> 2.2 .. an estimate of the scaled condition number of A is
-*> required (JOBA='E', or 'G').
-*> -> the minimal requirement is LWORK >= 3*N.
-*> -> For optimal performance,
-*> LWORK >= max(N+(N+1)*NB, 2*N,2*N+N*NB)=2*N+N*NB,
-*> where NB is the optimal block size for CGEQP3, CGEQRF, CGELQ,
-*> CUNMLQ. In general, the optimal length LWORK is computed as
-*> LWORK >= max(N+LWORK(CGEQP3), LWORK(CPOCON), N+LWORK(CGESVJ),
-*> N+LWORK(CGELQF), 2*N+LWORK(CGEQRF), N+LWORK(CUNMLQ)).
-*> 3. If SIGMA and the left singular vectors are needed
-*> 3.1 .. no scaled condition estimate requested (JOBE.EQ.'N'):
-*> -> the minimal requirement is LWORK >= 3*N.
-*> -> For optimal performance:
-*> if JOBU.EQ.'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
-*> where NB is the optimal block size for CGEQP3, CGEQRF, CUNMQR.
-*> In general, the optimal length LWORK is computed as
-*> LWORK >= max(N+LWORK(CGEQP3), 2*N+LWORK(CGEQRF), N+LWORK(CUNMQR)).
-*> 3.2 .. an estimate of the scaled condition number of A is
-*> required (JOBA='E', or 'G').
-*> -> the minimal requirement is LWORK >= 3*N.
-*> -> For optimal performance:
-*> if JOBU.EQ.'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
-*> where NB is the optimal block size for CGEQP3, CGEQRF, CUNMQR.
-*> In general, the optimal length LWORK is computed as
-*> LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CPOCON),
-*> 2*N+LWORK(CGEQRF), N+LWORK(CUNMQR)).
-*>
-*> 4. If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and
-*> 4.1. if JOBV.EQ.'V'
-*> the minimal requirement is LWORK >= 5*N+2*N*N.
-*> 4.2. if JOBV.EQ.'J' the minimal requirement is
-*> LWORK >= 4*N+N*N.
-*> In both cases, the allocated CWORK can accommodate blocked runs
-*> of CGEQP3, CGEQRF, CGELQF, CUNMQR, CUNMLQ.
-*>
-*> If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
-*> LRWORK=-1), then on exit CWORK(1) contains the optimal and CWORK(2) contains the
-*> minimal length of CWORK for the job parameters used in the call.
-*> \endverbatim
-*>
-*> \param[out] RWORK
-*> \verbatim
-*> RWORK is REAL array, dimension (MAX(7,LWORK))
-*> On exit,
-*> RWORK(1) = Determines the scaling factor SCALE = RWORK(2) / RWORK(1)
-*> such that SCALE*SVA(1:N) are the computed singular values
-*> of A. (See the description of SVA().)
-*> RWORK(2) = See the description of RWORK(1).
-*> RWORK(3) = SCONDA is an estimate for the condition number of
-*> column equilibrated A. (If JOBA .EQ. 'E' or 'G')
-*> SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
-*> It is computed using SPOCON. It holds
-*> N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
-*> where R is the triangular factor from the QRF of A.
-*> However, if R is truncated and the numerical rank is
-*> determined to be strictly smaller than N, SCONDA is
-*> returned as -1, thus indicating that the smallest
-*> singular values might be lost.
-*>
-*> If full SVD is needed, the following two condition numbers are
-*> useful for the analysis of the algorithm. They are provied for
-*> a developer/implementer who is familiar with the details of
-*> the method.
-*>
-*> RWORK(4) = an estimate of the scaled condition number of the
-*> triangular factor in the first QR factorization.
-*> RWORK(5) = an estimate of the scaled condition number of the
-*> triangular factor in the second QR factorization.
-*> The following two parameters are computed if JOBT .EQ. 'T'.
-*> They are provided for a developer/implementer who is familiar
-*> with the details of the method.
-*> RWORK(6) = the entropy of A^* * A :: this is the Shannon entropy
-*> of diag(A^* * A) / Trace(A^* * A) taken as point in the
-*> probability simplex.
-*> RWORK(7) = the entropy of A * A^*. (See the description of RWORK(6).)
-*> If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
-*> LRWORK=-1), then on exit RWORK(1) contains the required length of
-*> RWORK for the job parameters used in the call.
-*> \endverbatim
-*>
-*> \param[in] LRWORK
-*> \verbatim
-*> LRWORK is INTEGER
-*> Length of RWORK to confirm proper allocation of workspace.
-*> LRWORK depends on the job:
-*>
-*> 1. If only the singular values are requested i.e. if
-*> LSAME(JOBU,'N') .AND. LSAME(JOBV,'N')
-*> then:
-*> 1.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
-*> then: LRWORK = max( 7, 2 * M ).
-*> 1.2. Otherwise, LRWORK = max( 7, N ).
-*> 2. If singular values with the right singular vectors are requested
-*> i.e. if
-*> (LSAME(JOBV,'V').OR.LSAME(JOBV,'J')) .AND.
-*> .NOT.(LSAME(JOBU,'U').OR.LSAME(JOBU,'F'))
-*> then:
-*> 2.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
-*> then LRWORK = max( 7, 2 * M ).
-*> 2.2. Otherwise, LRWORK = max( 7, N ).
-*> 3. If singular values with the left singular vectors are requested, i.e. if
-*> (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
-*> .NOT.(LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
-*> then:
-*> 3.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
-*> then LRWORK = max( 7, 2 * M ).
-*> 3.2. Otherwise, LRWORK = max( 7, N ).
-*> 4. If singular values with both the left and the right singular vectors
-*> are requested, i.e. if
-*> (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
-*> (LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
-*> then:
-*> 4.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
-*> then LRWORK = max( 7, 2 * M ).
-*> 4.2. Otherwise, LRWORK = max( 7, N ).
-*>
-*> If, on entry, LRWORK = -1 or LWORK=-1, a workspace query is assumed and
-*> the length of RWORK is returned in RWORK(1).
-*> \endverbatim
-*>
-*> \param[out] IWORK
-*> \verbatim
-*> IWORK is INTEGER array, of dimension at least 4, that further depends
-*> on the job:
-*>
-*> 1. If only the singular values are requested then:
-*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
-*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
-*> 2. If the singular values and the right singular vectors are requested then:
-*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
-*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
-*> 3. If the singular values and the left singular vectors are requested then:
-*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
-*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
-*> 4. If the singular values with both the left and the right singular vectors
-*> are requested, then:
-*> 4.1. If LSAME(JOBV,'J') the length of IWORK is determined as follows:
-*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
-*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
-*> 4.2. If LSAME(JOBV,'V') the length of IWORK is determined as follows:
-*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
-*> then the length of IWORK is 2*N+M; otherwise the length of IWORK is 2*N.
-*>
-*> On exit,
-*> IWORK(1) = the numerical rank determined after the initial
-*> QR factorization with pivoting. See the descriptions
-*> of JOBA and JOBR.
-*> IWORK(2) = the number of the computed nonzero singular values
-*> IWORK(3) = if nonzero, a warning message:
-*> If IWORK(3).EQ.1 then some of the column norms of A
-*> were denormalized floats. The requested high accuracy
-*> is not warranted by the data.
-*> IWORK(4) = 1 or -1. If IWORK(4) .EQ. 1, then the procedure used A^* to
-*> do the job as specified by the JOB parameters.
-*> If the call to CGEJSV is a workspace query (indicated by LWORK .EQ. -1 and
-*> LRWORK .EQ. -1), then on exit IWORK(1) contains the required length of
-*> IWORK for the job parameters used in the call.
-*> \endverbatim
-*>
-*> \param[out] INFO
-*> \verbatim
-*> INFO is INTEGER
-*> < 0 : if INFO = -i, then the i-th argument had an illegal value.
-*> = 0 : successful exit;
-*> > 0 : CGEJSV did not converge in the maximal allowed number
-*> of sweeps. The computed values may be inaccurate.
-*> \endverbatim
-*
-* Authors:
-* ========
-*
-*> \author Univ. of Tennessee
-*> \author Univ. of California Berkeley
-*> \author Univ. of Colorado Denver
-*> \author NAG Ltd.
-*
-*> \date June 2016
-*
-*> \ingroup complexGEsing
-*
-*> \par Further Details:
-* =====================
-*>
-*> \verbatim
-*> CGEJSV implements a preconditioned Jacobi SVD algorithm. It uses CGEQP3,
-*> CGEQRF, and CGELQF as preprocessors and preconditioners. Optionally, an
-*> additional row pivoting can be used as a preprocessor, which in some
-*> cases results in much higher accuracy. An example is matrix A with the
-*> structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
-*> diagonal matrices and C is well-conditioned matrix. In that case, complete
-*> pivoting in the first QR factorizations provides accuracy dependent on the
-*> condition number of C, and independent of D1, D2. Such higher accuracy is
-*> not completely understood theoretically, but it works well in practice.
-*> Further, if A can be written as A = B*D, with well-conditioned B and some
-*> diagonal D, then the high accuracy is guaranteed, both theoretically and
-*> in software, independent of D. For more details see [1], [2].
-*> The computational range for the singular values can be the full range
-*> ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
-*> & LAPACK routines called by CGEJSV are implemented to work in that range.
-*> If that is not the case, then the restriction for safe computation with
-*> the singular values in the range of normalized IEEE numbers is that the
-*> spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
-*> overflow. This code (CGEJSV) is best used in this restricted range,
-*> meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are
-*> returned as zeros. See JOBR for details on this.
-*> Further, this implementation is somewhat slower than the one described
-*> in [1,2] due to replacement of some non-LAPACK components, and because
-*> the choice of some tuning parameters in the iterative part (CGESVJ) is
-*> left to the implementer on a particular machine.
-*> The rank revealing QR factorization (in this code: CGEQP3) should be
-*> implemented as in [3]. We have a new version of CGEQP3 under development
-*> that is more robust than the current one in LAPACK, with a cleaner cut in
-*> rank deficient cases. It will be available in the SIGMA library [4].
-*> If M is much larger than N, it is obvious that the initial QRF with
-*> column pivoting can be preprocessed by the QRF without pivoting. That
-*> well known trick is not used in CGEJSV because in some cases heavy row
-*> weighting can be treated with complete pivoting. The overhead in cases
-*> M much larger than N is then only due to pivoting, but the benefits in
-*> terms of accuracy have prevailed. The implementer/user can incorporate
-*> this extra QRF step easily. The implementer can also improve data movement
-*> (matrix transpose, matrix copy, matrix transposed copy) - this
-*> implementation of CGEJSV uses only the simplest, naive data movement.
-*> \endverbatim
-*
-*> \par Contributor:
-* ==================
-*>
-*> Zlatko Drmac (Zagreb, Croatia)
-*
-*> \par References:
-* ================
-*>
-*> \verbatim
-*>
-*> [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
-*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
-*> LAPACK Working note 169.
-*> [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
-*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
-*> LAPACK Working note 170.
-*> [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
-*> factorization software - a case study.
-*> ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
-*> LAPACK Working note 176.
-*> [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
-*> QSVD, (H,K)-SVD computations.
-*> Department of Mathematics, University of Zagreb, 2008, 2016.
-*> \endverbatim
-*
-*> \par Bugs, examples and comments:
-* =================================
-*>
-*> Please report all bugs and send interesting examples and/or comments to
-*> drmac@math.hr. Thank you.
-*>
-* =====================================================================
- SUBROUTINE CGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
- $ M, N, A, LDA, SVA, U, LDU, V, LDV,
- $ CWORK, LWORK, RWORK, LRWORK, IWORK, INFO )
-*
-* -- LAPACK computational routine (version 3.7.0) --
-* -- LAPACK is a software package provided by Univ. of Tennessee, --
-* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* December 2016
-*
-* .. Scalar Arguments ..
- IMPLICIT NONE
- INTEGER INFO, LDA, LDU, LDV, LWORK, LRWORK, M, N
-* ..
-* .. Array Arguments ..
- COMPLEX A( LDA, * ), U( LDU, * ), V( LDV, * ), CWORK( LWORK )
- REAL SVA( N ), RWORK( LRWORK )
- INTEGER IWORK( * )
- CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
-* ..
-*
-* ===========================================================================
-*
-* .. Local Parameters ..
- REAL ZERO, ONE
- PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
- COMPLEX CZERO, CONE
- PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ), CONE = ( 1.0E0, 0.0E0 ) )
-* ..
-* .. Local Scalars ..
- COMPLEX CTEMP
- REAL AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1, COND_OK,
- $ CONDR1, CONDR2, ENTRA, ENTRAT, EPSLN, MAXPRJ, SCALEM,
- $ SCONDA, SFMIN, SMALL, TEMP1, USCAL1, USCAL2, XSC
- INTEGER IERR, N1, NR, NUMRANK, p, q, WARNING
- LOGICAL ALMORT, DEFR, ERREST, GOSCAL, JRACC, KILL, LQUERY,
- $ LSVEC, L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN, NOSCAL,
- $ ROWPIV, RSVEC, TRANSP
-*
- INTEGER OPTWRK, MINWRK, MINRWRK, MINIWRK
- INTEGER LWCON, LWLQF, LWQP3, LWQRF, LWUNMLQ, LWUNMQR, LWUNMQRM,
- $ LWSVDJ, LWSVDJV, LRWQP3, LRWCON, LRWSVDJ, IWOFF
- INTEGER LWRK_CGELQF, LWRK_CGEQP3, LWRK_CGEQP3N, LWRK_CGEQRF,
- $ LWRK_CGESVJ, LWRK_CGESVJV, LWRK_CGESVJU, LWRK_CUNMLQ,
- $ LWRK_CUNMQR, LWRK_CUNMQRM
-* ..
-* .. Local Arrays
- COMPLEX CDUMMY(1)
- REAL RDUMMY(1)
-*
-* .. Intrinsic Functions ..
- INTRINSIC ABS, CMPLX, CONJG, ALOG, MAX, MIN, REAL, NINT, SQRT
-* ..
-* .. External Functions ..
- REAL SLAMCH, SCNRM2
- INTEGER ISAMAX, ICAMAX
- LOGICAL LSAME
- EXTERNAL ISAMAX, ICAMAX, LSAME, SLAMCH, SCNRM2
-* ..
-* .. External Subroutines ..
- EXTERNAL SLASSQ, CCOPY, CGELQF, CGEQP3, CGEQRF, CLACPY, CLAPMR,
- $ CLASCL, SLASCL, CLASET, CLASSQ, CLASWP, CUNGQR, CUNMLQ,
- $ CUNMQR, CPOCON, SSCAL, CSSCAL, CSWAP, CTRSM, CLACGV,
- $ XERBLA
-*
- EXTERNAL CGESVJ
-* ..
-*
-* Test the input arguments
-*
- LSVEC = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' )
- JRACC = LSAME( JOBV, 'J' )
- RSVEC = LSAME( JOBV, 'V' ) .OR. JRACC
- ROWPIV = LSAME( JOBA, 'F' ) .OR. LSAME( JOBA, 'G' )
- L2RANK = LSAME( JOBA, 'R' )
- L2ABER = LSAME( JOBA, 'A' )
- ERREST = LSAME( JOBA, 'E' ) .OR. LSAME( JOBA, 'G' )
- L2TRAN = LSAME( JOBT, 'T' ) .AND. ( M .EQ. N )
- L2KILL = LSAME( JOBR, 'R' )
- DEFR = LSAME( JOBR, 'N' )
- L2PERT = LSAME( JOBP, 'P' )
-*
- LQUERY = ( LWORK .EQ. -1 ) .OR. ( LRWORK .EQ. -1 )
-*
- IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR.
- $ ERREST .OR. LSAME( JOBA, 'C' ) )) THEN
- INFO = - 1
- ELSE IF ( .NOT.( LSVEC .OR. LSAME( JOBU, 'N' ) .OR.
- $ ( LSAME( JOBU, 'W' ) .AND. RSVEC .AND. L2TRAN ) ) ) THEN
- INFO = - 2
- ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.
- $ ( LSAME( JOBV, 'W' ) .AND. LSVEC .AND. L2TRAN ) ) ) THEN
- INFO = - 3
- ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) ) THEN
- INFO = - 4
- ELSE IF ( .NOT. ( LSAME(JOBT,'T') .OR. LSAME(JOBT,'N') ) ) THEN
- INFO = - 5
- ELSE IF ( .NOT. ( L2PERT .OR. LSAME( JOBP, 'N' ) ) ) THEN
- INFO = - 6
- ELSE IF ( M .LT. 0 ) THEN
- INFO = - 7
- ELSE IF ( ( N .LT. 0 ) .OR. ( N .GT. M ) ) THEN
- INFO = - 8
- ELSE IF ( LDA .LT. M ) THEN
- INFO = - 10
- ELSE IF ( LSVEC .AND. ( LDU .LT. M ) ) THEN
- INFO = - 13
- ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN
- INFO = - 15
- ELSE
-* #:)
- INFO = 0
- END IF
-*
- IF ( INFO .EQ. 0 ) THEN
-* .. compute the minimal and the optimal workspace lengths
-* [[The expressions for computing the minimal and the optimal
-* values of LCWORK, LRWORK are written with a lot of redundancy and
-* can be simplified. However, this verbose form is useful for
-* maintenance and modifications of the code.]]
-*
-* .. minimal workspace length for CGEQP3 of an M x N matrix,
-* CGEQRF of an N x N matrix, CGELQF of an N x N matrix,
-* CUNMLQ for computing N x N matrix, CUNMQR for computing N x N
-* matrix, CUNMQR for computing M x N matrix, respectively.
- LWQP3 = N+1
- LWQRF = MAX( 1, N )
- LWLQF = MAX( 1, N )
- LWUNMLQ = MAX( 1, N )
- LWUNMQR = MAX( 1, N )
- LWUNMQRM = MAX( 1, M )
-* .. minimal workspace length for CPOCON of an N x N matrix
- LWCON = 2 * N
-* .. minimal workspace length for CGESVJ of an N x N matrix,
-* without and with explicit accumulation of Jacobi rotations
- LWSVDJ = MAX( 2 * N, 1 )
- LWSVDJV = MAX( 2 * N, 1 )
-* .. minimal REAL workspace length for CGEQP3, CPOCON, CGESVJ
- LRWQP3 = N
- LRWCON = N
- LRWSVDJ = N
- IF ( LQUERY ) THEN
- CALL CGEQP3( M, N, A, LDA, IWORK, CDUMMY, CDUMMY, -1,
- $ RDUMMY, IERR )
- LWRK_CGEQP3 = CDUMMY(1)
- CALL CGEQRF( N, N, A, LDA, CDUMMY, CDUMMY,-1, IERR )
- LWRK_CGEQRF = CDUMMY(1)
- CALL CGELQF( N, N, A, LDA, CDUMMY, CDUMMY,-1, IERR )
- LWRK_CGELQF = CDUMMY(1)
- END IF
- MINWRK = 2
- OPTWRK = 2
- MINIWRK = N
- IF ( .NOT. (LSVEC .OR. RSVEC ) ) THEN
-* .. minimal and optimal sizes of the complex workspace if
-* only the singular values are requested
- IF ( ERREST ) THEN
- MINWRK = MAX( N+LWQP3, N**2+LWCON, N+LWQRF, LWSVDJ )
- ELSE
- MINWRK = MAX( N+LWQP3, N+LWQRF, LWSVDJ )
- END IF
- IF ( LQUERY ) THEN
- CALL CGESVJ( 'L', 'N', 'N', N, N, A, LDA, SVA, N, V,
- $ LDV, CDUMMY, -1, RDUMMY, -1, IERR )
- LWRK_CGESVJ = CDUMMY(1)
- IF ( ERREST ) THEN
- OPTWRK = MAX( N+LWRK_CGEQP3, N**2+LWCON,
- $ N+LWRK_CGEQRF, LWRK_CGESVJ )
- ELSE
- OPTWRK = MAX( N+LWRK_CGEQP3, N+LWRK_CGEQRF,
- $ LWRK_CGESVJ )
- END IF
- END IF
- IF ( L2TRAN .OR. ROWPIV ) THEN
- IF ( ERREST ) THEN
- MINRWRK = MAX( 7, 2*M, LRWQP3, LRWCON, LRWSVDJ )
- ELSE
- MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
- END IF
- ELSE
- IF ( ERREST ) THEN
- MINRWRK = MAX( 7, LRWQP3, LRWCON, LRWSVDJ )
- ELSE
- MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
- END IF
- END IF
- IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
- ELSE IF ( RSVEC .AND. (.NOT.LSVEC) ) THEN
-* .. minimal and optimal sizes of the complex workspace if the
-* singular values and the right singular vectors are requested
- IF ( ERREST ) THEN
- MINWRK = MAX( N+LWQP3, LWCON, LWSVDJ, N+LWLQF,
- $ 2*N+LWQRF, N+LWSVDJ, N+LWUNMLQ )
- ELSE
- MINWRK = MAX( N+LWQP3, LWSVDJ, N+LWLQF, 2*N+LWQRF,
- $ N+LWSVDJ, N+LWUNMLQ )
- END IF
- IF ( LQUERY ) THEN
- CALL CGESVJ( 'L', 'U', 'N', N,N, U, LDU, SVA, N, A,
- $ LDA, CDUMMY, -1, RDUMMY, -1, IERR )
- LWRK_CGESVJ = CDUMMY(1)
- CALL CUNMLQ( 'L', 'C', N, N, N, A, LDA, CDUMMY,
- $ V, LDV, CDUMMY, -1, IERR )
- LWRK_CUNMLQ = CDUMMY(1)
- IF ( ERREST ) THEN
- OPTWRK = MAX( N+LWRK_CGEQP3, LWCON, LWRK_CGESVJ,
- $ N+LWRK_CGELQF, 2*N+LWRK_CGEQRF,
- $ N+LWRK_CGESVJ, N+LWRK_CUNMLQ )
- ELSE
- OPTWRK = MAX( N+LWRK_CGEQP3, LWRK_CGESVJ,N+LWRK_CGELQF,
- $ 2*N+LWRK_CGEQRF, N+LWRK_CGESVJ,
- $ N+LWRK_CUNMLQ )
- END IF
- END IF
- IF ( L2TRAN .OR. ROWPIV ) THEN
- IF ( ERREST ) THEN
- MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
- ELSE
- MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
- END IF
- ELSE
- IF ( ERREST ) THEN
- MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
- ELSE
- MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
- END IF
- END IF
- IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
- ELSE IF ( LSVEC .AND. (.NOT.RSVEC) ) THEN
-* .. minimal and optimal sizes of the complex workspace if the
-* singular values and the left singular vectors are requested
- IF ( ERREST ) THEN
- MINWRK = N + MAX( LWQP3,LWCON,N+LWQRF,LWSVDJ,LWUNMQRM )
- ELSE
- MINWRK = N + MAX( LWQP3, N+LWQRF, LWSVDJ, LWUNMQRM )
- END IF
- IF ( LQUERY ) THEN
- CALL CGESVJ( 'L', 'U', 'N', N,N, U, LDU, SVA, N, A,
- $ LDA, CDUMMY, -1, RDUMMY, -1, IERR )
- LWRK_CGESVJ = CDUMMY(1)
- CALL CUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
- $ LDU, CDUMMY, -1, IERR )
- LWRK_CUNMQRM = CDUMMY(1)
- IF ( ERREST ) THEN
- OPTWRK = N + MAX( LWRK_CGEQP3, LWCON, N+LWRK_CGEQRF,
- $ LWRK_CGESVJ, LWRK_CUNMQRM )
- ELSE
- OPTWRK = N + MAX( LWRK_CGEQP3, N+LWRK_CGEQRF,
- $ LWRK_CGESVJ, LWRK_CUNMQRM )
- END IF
- END IF
- IF ( L2TRAN .OR. ROWPIV ) THEN
- IF ( ERREST ) THEN
- MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
- ELSE
- MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
- END IF
- ELSE
- IF ( ERREST ) THEN
- MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
- ELSE
- MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
- END IF
- END IF
- IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
- ELSE
-* .. minimal and optimal sizes of the complex workspace if the
-* full SVD is requested
- IF ( .NOT. JRACC ) THEN
- IF ( ERREST ) THEN
- MINWRK = MAX( N+LWQP3, N+LWCON, 2*N+N**2+LWCON,
- $ 2*N+LWQRF, 2*N+LWQP3,
- $ 2*N+N**2+N+LWLQF, 2*N+N**2+N+N**2+LWCON,
- $ 2*N+N**2+N+LWSVDJ, 2*N+N**2+N+LWSVDJV,
- $ 2*N+N**2+N+LWUNMQR,2*N+N**2+N+LWUNMLQ,
- $ N+N**2+LWSVDJ, N+LWUNMQRM )
- ELSE
- MINWRK = MAX( N+LWQP3, 2*N+N**2+LWCON,
- $ 2*N+LWQRF, 2*N+LWQP3,
- $ 2*N+N**2+N+LWLQF, 2*N+N**2+N+N**2+LWCON,
- $ 2*N+N**2+N+LWSVDJ, 2*N+N**2+N+LWSVDJV,
- $ 2*N+N**2+N+LWUNMQR,2*N+N**2+N+LWUNMLQ,
- $ N+N**2+LWSVDJ, N+LWUNMQRM )
- END IF
- MINIWRK = MINIWRK + N
- IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
- ELSE
- IF ( ERREST ) THEN
- MINWRK = MAX( N+LWQP3, N+LWCON, 2*N+LWQRF,
- $ 2*N+N**2+LWSVDJV, 2*N+N**2+N+LWUNMQR,
- $ N+LWUNMQRM )
- ELSE
- MINWRK = MAX( N+LWQP3, 2*N+LWQRF,
- $ 2*N+N**2+LWSVDJV, 2*N+N**2+N+LWUNMQR,
- $ N+LWUNMQRM )
- END IF
- IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
- END IF
- IF ( LQUERY ) THEN
- CALL CUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
- $ LDU, CDUMMY, -1, IERR )
- LWRK_CUNMQRM = CDUMMY(1)
- CALL CUNMQR( 'L', 'N', N, N, N, A, LDA, CDUMMY, U,
- $ LDU, CDUMMY, -1, IERR )
- LWRK_CUNMQR = CDUMMY(1)
- IF ( .NOT. JRACC ) THEN
- CALL CGEQP3( N,N, A, LDA, IWORK, CDUMMY,CDUMMY, -1,
- $ RDUMMY, IERR )
- LWRK_CGEQP3N = CDUMMY(1)
- CALL CGESVJ( 'L', 'U', 'N', N, N, U, LDU, SVA,
- $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
- LWRK_CGESVJ = CDUMMY(1)
- CALL CGESVJ( 'U', 'U', 'N', N, N, U, LDU, SVA,
- $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
- LWRK_CGESVJU = CDUMMY(1)
- CALL CGESVJ( 'L', 'U', 'V', N, N, U, LDU, SVA,
- $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
- LWRK_CGESVJV = CDUMMY(1)
- CALL CUNMLQ( 'L', 'C', N, N, N, A, LDA, CDUMMY,
- $ V, LDV, CDUMMY, -1, IERR )
- LWRK_CUNMLQ = CDUMMY(1)
- IF ( ERREST ) THEN
- OPTWRK = MAX( N+LWRK_CGEQP3, N+LWCON,
- $ 2*N+N**2+LWCON, 2*N+LWRK_CGEQRF,
- $ 2*N+LWRK_CGEQP3N,
- $ 2*N+N**2+N+LWRK_CGELQF,
- $ 2*N+N**2+N+N**2+LWCON,
- $ 2*N+N**2+N+LWRK_CGESVJ,
- $ 2*N+N**2+N+LWRK_CGESVJV,
- $ 2*N+N**2+N+LWRK_CUNMQR,
- $ 2*N+N**2+N+LWRK_CUNMLQ,
- $ N+N**2+LWRK_CGESVJU,
- $ N+LWRK_CUNMQRM )
- ELSE
- OPTWRK = MAX( N+LWRK_CGEQP3,
- $ 2*N+N**2+LWCON, 2*N+LWRK_CGEQRF,
- $ 2*N+LWRK_CGEQP3N,
- $ 2*N+N**2+N+LWRK_CGELQF,
- $ 2*N+N**2+N+N**2+LWCON,
- $ 2*N+N**2+N+LWRK_CGESVJ,
- $ 2*N+N**2+N+LWRK_CGESVJV,
- $ 2*N+N**2+N+LWRK_CUNMQR,
- $ 2*N+N**2+N+LWRK_CUNMLQ,
- $ N+N**2+LWRK_CGESVJU,
- $ N+LWRK_CUNMQRM )
- END IF
- ELSE
- CALL CGESVJ( 'L', 'U', 'V', N, N, U, LDU, SVA,
- $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
- LWRK_CGESVJV = CDUMMY(1)
- CALL CUNMQR( 'L', 'N', N, N, N, CDUMMY, N, CDUMMY,
- $ V, LDV, CDUMMY, -1, IERR )
- LWRK_CUNMQR = CDUMMY(1)
- CALL CUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
- $ LDU, CDUMMY, -1, IERR )
- LWRK_CUNMQRM = CDUMMY(1)
- IF ( ERREST ) THEN
- OPTWRK = MAX( N+LWRK_CGEQP3, N+LWCON,
- $ 2*N+LWRK_CGEQRF, 2*N+N**2,
- $ 2*N+N**2+LWRK_CGESVJV,
- $ 2*N+N**2+N+LWRK_CUNMQR,N+LWRK_CUNMQRM )
- ELSE
- OPTWRK = MAX( N+LWRK_CGEQP3, 2*N+LWRK_CGEQRF,
- $ 2*N+N**2, 2*N+N**2+LWRK_CGESVJV,
- $ 2*N+N**2+N+LWRK_CUNMQR,
- $ N+LWRK_CUNMQRM )
- END IF
- END IF
- END IF
- IF ( L2TRAN .OR. ROWPIV ) THEN
- MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
- ELSE
- MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
- END IF
- END IF
- MINWRK = MAX( 2, MINWRK )
- OPTWRK = MAX( 2, OPTWRK )
- IF ( LWORK .LT. MINWRK .AND. (.NOT.LQUERY) ) INFO = - 17
- IF ( LRWORK .LT. MINRWRK .AND. (.NOT.LQUERY) ) INFO = - 19
- END IF
-*
- IF ( INFO .NE. 0 ) THEN
-* #:(
- CALL XERBLA( 'CGEJSV', - INFO )
- RETURN
- ELSE IF ( LQUERY ) THEN
- CWORK(1) = OPTWRK
- CWORK(2) = MINWRK
- RWORK(1) = MINRWRK
- IWORK(1) = MAX( 4, MINIWRK )
- RETURN
- END IF
-*
-* Quick return for void matrix (Y3K safe)
-* #:)
- IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) THEN
- IWORK(1:4) = 0
- RWORK(1:7) = 0
- RETURN
- ENDIF
-*
-* Determine whether the matrix U should be M x N or M x M
-*
- IF ( LSVEC ) THEN
- N1 = N
- IF ( LSAME( JOBU, 'F' ) ) N1 = M
- END IF
-*
-* Set numerical parameters
-*
-*! NOTE: Make sure SLAMCH() does not fail on the target architecture.
-*
- EPSLN = SLAMCH('Epsilon')
- SFMIN = SLAMCH('SafeMinimum')
- SMALL = SFMIN / EPSLN
- BIG = SLAMCH('O')
-* BIG = ONE / SFMIN
-*
-* Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N
-*
-*(!) If necessary, scale SVA() to protect the largest norm from
-* overflow. It is possible that this scaling pushes the smallest
-* column norm left from the underflow threshold (extreme case).
-*
- SCALEM = ONE / SQRT(REAL(M)*REAL(N))
- NOSCAL = .TRUE.
- GOSCAL = .TRUE.
- DO 1874 p = 1, N
- AAPP = ZERO
- AAQQ = ONE
- CALL CLASSQ( M, A(1,p), 1, AAPP, AAQQ )
- IF ( AAPP .GT. BIG ) THEN
- INFO = - 9
- CALL XERBLA( 'CGEJSV', -INFO )
- RETURN
- END IF
- AAQQ = SQRT(AAQQ)
- IF ( ( AAPP .LT. (BIG / AAQQ) ) .AND. NOSCAL ) THEN
- SVA(p) = AAPP * AAQQ
- ELSE
- NOSCAL = .FALSE.
- SVA(p) = AAPP * ( AAQQ * SCALEM )
- IF ( GOSCAL ) THEN
- GOSCAL = .FALSE.
- CALL SSCAL( p-1, SCALEM, SVA, 1 )
- END IF
- END IF
- 1874 CONTINUE
-*
- IF ( NOSCAL ) SCALEM = ONE
-*
- AAPP = ZERO
- AAQQ = BIG
- DO 4781 p = 1, N
- AAPP = MAX( AAPP, SVA(p) )
- IF ( SVA(p) .NE. ZERO ) AAQQ = MIN( AAQQ, SVA(p) )
- 4781 CONTINUE
-*
-* Quick return for zero M x N matrix
-* #:)
- IF ( AAPP .EQ. ZERO ) THEN
- IF ( LSVEC ) CALL CLASET( 'G', M, N1, CZERO, CONE, U, LDU )
- IF ( RSVEC ) CALL CLASET( 'G', N, N, CZERO, CONE, V, LDV )
- RWORK(1) = ONE
- RWORK(2) = ONE
- IF ( ERREST ) RWORK(3) = ONE
- IF ( LSVEC .AND. RSVEC ) THEN
- RWORK(4) = ONE
- RWORK(5) = ONE
- END IF
- IF ( L2TRAN ) THEN
- RWORK(6) = ZERO
- RWORK(7) = ZERO
- END IF
- IWORK(1) = 0
- IWORK(2) = 0
- IWORK(3) = 0
- IWORK(4) = -1
- RETURN
- END IF
-*
-* Issue warning if denormalized column norms detected. Override the
-* high relative accuracy request. Issue licence to kill nonzero columns
-* (set them to zero) whose norm is less than sigma_max / BIG (roughly).
-* #:(
- WARNING = 0
- IF ( AAQQ .LE. SFMIN ) THEN
- L2RANK = .TRUE.
- L2KILL = .TRUE.
- WARNING = 1
- END IF
-*
-* Quick return for one-column matrix
-* #:)
- IF ( N .EQ. 1 ) THEN
-*
- IF ( LSVEC ) THEN
- CALL CLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR )
- CALL CLACPY( 'A', M, 1, A, LDA, U, LDU )
-* computing all M left singular vectors of the M x 1 matrix
- IF ( N1 .NE. N ) THEN
- CALL CGEQRF( M, N, U,LDU, CWORK, CWORK(N+1),LWORK-N,IERR )
- CALL CUNGQR( M,N1,1, U,LDU,CWORK,CWORK(N+1),LWORK-N,IERR )
- CALL CCOPY( M, A(1,1), 1, U(1,1), 1 )
- END IF
- END IF
- IF ( RSVEC ) THEN
- V(1,1) = CONE
- END IF
- IF ( SVA(1) .LT. (BIG*SCALEM) ) THEN
- SVA(1) = SVA(1) / SCALEM
- SCALEM = ONE
- END IF
- RWORK(1) = ONE / SCALEM
- RWORK(2) = ONE
- IF ( SVA(1) .NE. ZERO ) THEN
- IWORK(1) = 1
- IF ( ( SVA(1) / SCALEM) .GE. SFMIN ) THEN
- IWORK(2) = 1
- ELSE
- IWORK(2) = 0
- END IF
- ELSE
- IWORK(1) = 0
- IWORK(2) = 0
- END IF
- IWORK(3) = 0
- IWORK(4) = -1
- IF ( ERREST ) RWORK(3) = ONE
- IF ( LSVEC .AND. RSVEC ) THEN
- RWORK(4) = ONE
- RWORK(5) = ONE
- END IF
- IF ( L2TRAN ) THEN
- RWORK(6) = ZERO
- RWORK(7) = ZERO
- END IF
- RETURN
-*
- END IF
-*
- TRANSP = .FALSE.
-*
- AATMAX = -ONE
- AATMIN = BIG
- IF ( ROWPIV .OR. L2TRAN ) THEN
-*
-* Compute the row norms, needed to determine row pivoting sequence
-* (in the case of heavily row weighted A, row pivoting is strongly
-* advised) and to collect information needed to compare the
-* structures of A * A^* and A^* * A (in the case L2TRAN.EQ..TRUE.).
-*
- IF ( L2TRAN ) THEN
- DO 1950 p = 1, M
- XSC = ZERO
- TEMP1 = ONE
- CALL CLASSQ( N, A(p,1), LDA, XSC, TEMP1 )
-* CLASSQ gets both the ell_2 and the ell_infinity norm
-* in one pass through the vector
- RWORK(M+p) = XSC * SCALEM
- RWORK(p) = XSC * (SCALEM*SQRT(TEMP1))
- AATMAX = MAX( AATMAX, RWORK(p) )
- IF (RWORK(p) .NE. ZERO)
- $ AATMIN = MIN(AATMIN,RWORK(p))
- 1950 CONTINUE
- ELSE
- DO 1904 p = 1, M
- RWORK(M+p) = SCALEM*ABS( A(p,ICAMAX(N,A(p,1),LDA)) )
- AATMAX = MAX( AATMAX, RWORK(M+p) )
- AATMIN = MIN( AATMIN, RWORK(M+p) )
- 1904 CONTINUE
- END IF
-*
- END IF
-*
-* For square matrix A try to determine whether A^* would be better
-* input for the preconditioned Jacobi SVD, with faster convergence.
-* The decision is based on an O(N) function of the vector of column
-* and row norms of A, based on the Shannon entropy. This should give
-* the right choice in most cases when the difference actually matters.
-* It may fail and pick the slower converging side.
-*
- ENTRA = ZERO
- ENTRAT = ZERO
- IF ( L2TRAN ) THEN
-*
- XSC = ZERO
- TEMP1 = ONE
- CALL SLASSQ( N, SVA, 1, XSC, TEMP1 )
- TEMP1 = ONE / TEMP1
-*
- ENTRA = ZERO
- DO 1113 p = 1, N
- BIG1 = ( ( SVA(p) / XSC )**2 ) * TEMP1
- IF ( BIG1 .NE. ZERO ) ENTRA = ENTRA + BIG1 * ALOG(BIG1)
- 1113 CONTINUE
- ENTRA = - ENTRA / ALOG(REAL(N))
-*
-* Now, SVA().^2/Trace(A^* * A) is a point in the probability simplex.
-* It is derived from the diagonal of A^* * A. Do the same with the
-* diagonal of A * A^*, compute the entropy of the corresponding
-* probability distribution. Note that A * A^* and A^* * A have the
-* same trace.
-*
- ENTRAT = ZERO
- DO 1114 p = 1, M
- BIG1 = ( ( RWORK(p) / XSC )**2 ) * TEMP1
- IF ( BIG1 .NE. ZERO ) ENTRAT = ENTRAT + BIG1 * ALOG(BIG1)
- 1114 CONTINUE
- ENTRAT = - ENTRAT / ALOG(REAL(M))
-*
-* Analyze the entropies and decide A or A^*. Smaller entropy
-* usually means better input for the algorithm.
-*
- TRANSP = ( ENTRAT .LT. ENTRA )
-*
-* If A^* is better than A, take the adjoint of A. This is allowed
-* only for square matrices, M=N.
- IF ( TRANSP ) THEN
-* In an optimal implementation, this trivial transpose
-* should be replaced with faster transpose.
- DO 1115 p = 1, N - 1
- A(p,p) = CONJG(A(p,p))
- DO 1116 q = p + 1, N
- CTEMP = CONJG(A(q,p))
- A(q,p) = CONJG(A(p,q))
- A(p,q) = CTEMP
- 1116 CONTINUE
- 1115 CONTINUE
- A(N,N) = CONJG(A(N,N))
- DO 1117 p = 1, N
- RWORK(M+p) = SVA(p)
- SVA(p) = RWORK(p)
-* previously computed row 2-norms are now column 2-norms
-* of the transposed matrix
- 1117 CONTINUE
- TEMP1 = AAPP
- AAPP = AATMAX
- AATMAX = TEMP1
- TEMP1 = AAQQ
- AAQQ = AATMIN
- AATMIN = TEMP1
- KILL = LSVEC
- LSVEC = RSVEC
- RSVEC = KILL
- IF ( LSVEC ) N1 = N
-*
- ROWPIV = .TRUE.
- END IF
-*
- END IF
-* END IF L2TRAN
-*
-* Scale the matrix so that its maximal singular value remains less
-* than SQRT(BIG) -- the matrix is scaled so that its maximal column
-* has Euclidean norm equal to SQRT(BIG/N). The only reason to keep
-* SQRT(BIG) instead of BIG is the fact that CGEJSV uses LAPACK and
-* BLAS routines that, in some implementations, are not capable of
-* working in the full interval [SFMIN,BIG] and that they may provoke
-* overflows in the intermediate results. If the singular values spread
-* from SFMIN to BIG, then CGESVJ will compute them. So, in that case,
-* one should use CGESVJ instead of CGEJSV.
- BIG1 = SQRT( BIG )
- TEMP1 = SQRT( BIG / REAL(N) )
-* >> for future updates: allow bigger range, i.e. the largest column
-* will be allowed up to BIG/N and CGESVJ will do the rest. However, for
-* this all other (LAPACK) components must allow such a range.
-* TEMP1 = BIG/REAL(N)
-* TEMP1 = BIG * EPSLN this should 'almost' work with current LAPACK components
- CALL SLASCL( 'G', 0, 0, AAPP, TEMP1, N, 1, SVA, N, IERR )
- IF ( AAQQ .GT. (AAPP * SFMIN) ) THEN
- AAQQ = ( AAQQ / AAPP ) * TEMP1
- ELSE
- AAQQ = ( AAQQ * TEMP1 ) / AAPP
- END IF
- TEMP1 = TEMP1 * SCALEM
- CALL CLASCL( 'G', 0, 0, AAPP, TEMP1, M, N, A, LDA, IERR )
-*
-* To undo scaling at the end of this procedure, multiply the
-* computed singular values with USCAL2 / USCAL1.
-*
- USCAL1 = TEMP1
- USCAL2 = AAPP
-*
- IF ( L2KILL ) THEN
-* L2KILL enforces computation of nonzero singular values in
-* the restricted range of condition number of the initial A,
-* sigma_max(A) / sigma_min(A) approx. SQRT(BIG)/SQRT(SFMIN).
- XSC = SQRT( SFMIN )
- ELSE
- XSC = SMALL
-*
-* Now, if the condition number of A is too big,
-* sigma_max(A) / sigma_min(A) .GT. SQRT(BIG/N) * EPSLN / SFMIN,
-* as a precaution measure, the full SVD is computed using CGESVJ
-* with accumulated Jacobi rotations. This provides numerically
-* more robust computation, at the cost of slightly increased run
-* time. Depending on the concrete implementation of BLAS and LAPACK
-* (i.e. how they behave in presence of extreme ill-conditioning) the
-* implementor may decide to remove this switch.
- IF ( ( AAQQ.LT.SQRT(SFMIN) ) .AND. LSVEC .AND. RSVEC ) THEN
- JRACC = .TRUE.
- END IF
-*
- END IF
- IF ( AAQQ .LT. XSC ) THEN
- DO 700 p = 1, N
- IF ( SVA(p) .LT. XSC ) THEN
- CALL CLASET( 'A', M, 1, CZERO, CZERO, A(1,p), LDA )
- SVA(p) = ZERO
- END IF
- 700 CONTINUE
- END IF
-*
-* Preconditioning using QR factorization with pivoting
-*
- IF ( ROWPIV ) THEN
-* Optional row permutation (Bjoerck row pivoting):
-* A result by Cox and Higham shows that the Bjoerck's
-* row pivoting combined with standard column pivoting
-* has similar effect as Powell-Reid complete pivoting.
-* The ell-infinity norms of A are made nonincreasing.
- IF ( ( LSVEC .AND. RSVEC ) .AND. .NOT.( JRACC ) ) THEN
- IWOFF = 2*N
- ELSE
- IWOFF = N
- END IF
- DO 1952 p = 1, M - 1
- q = ISAMAX( M-p+1, RWORK(M+p), 1 ) + p - 1
- IWORK(IWOFF+p) = q
- IF ( p .NE. q ) THEN
- TEMP1 = RWORK(M+p)
- RWORK(M+p) = RWORK(M+q)
- RWORK(M+q) = TEMP1
- END IF
- 1952 CONTINUE
- CALL CLASWP( N, A, LDA, 1, M-1, IWORK(IWOFF+1), 1 )
- END IF
-*
-* End of the preparation phase (scaling, optional sorting and
-* transposing, optional flushing of small columns).
-*
-* Preconditioning
-*
-* If the full SVD is needed, the right singular vectors are computed
-* from a matrix equation, and for that we need theoretical analysis
-* of the Businger-Golub pivoting. So we use CGEQP3 as the first RR QRF.
-* In all other cases the first RR QRF can be chosen by other criteria
-* (eg speed by replacing global with restricted window pivoting, such
-* as in xGEQPX from TOMS # 782). Good results will be obtained using
-* xGEQPX with properly (!) chosen numerical parameters.
-* Any improvement of CGEQP3 improves overal performance of CGEJSV.
-*
-* A * P1 = Q1 * [ R1^* 0]^*:
- DO 1963 p = 1, N
-* .. all columns are free columns
- IWORK(p) = 0
- 1963 CONTINUE
- CALL CGEQP3( M, N, A, LDA, IWORK, CWORK, CWORK(N+1), LWORK-N,
- $ RWORK, IERR )
-*
-* The upper triangular matrix R1 from the first QRF is inspected for
-* rank deficiency and possibilities for deflation, or possible
-* ill-conditioning. Depending on the user specified flag L2RANK,
-* the procedure explores possibilities to reduce the numerical
-* rank by inspecting the computed upper triangular factor. If
-* L2RANK or L2ABER are up, then CGEJSV will compute the SVD of
-* A + dA, where ||dA|| <= f(M,N)*EPSLN.
-*
- NR = 1
- IF ( L2ABER ) THEN
-* Standard absolute error bound suffices. All sigma_i with
-* sigma_i < N*EPSLN*||A|| are flushed to zero. This is an
-* agressive enforcement of lower numerical rank by introducing a
-* backward error of the order of N*EPSLN*||A||.
- TEMP1 = SQRT(REAL(N))*EPSLN
- DO 3001 p = 2, N
- IF ( ABS(A(p,p)) .GE. (TEMP1*ABS(A(1,1))) ) THEN
- NR = NR + 1
- ELSE
- GO TO 3002
- END IF
- 3001 CONTINUE
- 3002 CONTINUE
- ELSE IF ( L2RANK ) THEN
-* .. similarly as above, only slightly more gentle (less agressive).
-* Sudden drop on the diagonal of R1 is used as the criterion for
-* close-to-rank-defficient.
- TEMP1 = SQRT(SFMIN)
- DO 3401 p = 2, N
- IF ( ( ABS(A(p,p)) .LT. (EPSLN*ABS(A(p-1,p-1))) ) .OR.
- $ ( ABS(A(p,p)) .LT. SMALL ) .OR.
- $ ( L2KILL .AND. (ABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3402
- NR = NR + 1
- 3401 CONTINUE
- 3402 CONTINUE
-*
- ELSE
-* The goal is high relative accuracy. However, if the matrix
-* has high scaled condition number the relative accuracy is in
-* general not feasible. Later on, a condition number estimator
-* will be deployed to estimate the scaled condition number.
-* Here we just remove the underflowed part of the triangular
-* factor. This prevents the situation in which the code is
-* working hard to get the accuracy not warranted by the data.
- TEMP1 = SQRT(SFMIN)
- DO 3301 p = 2, N
- IF ( ( ABS(A(p,p)) .LT. SMALL ) .OR.
- $ ( L2KILL .AND. (ABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3302
- NR = NR + 1
- 3301 CONTINUE
- 3302 CONTINUE
-*
- END IF
-*
- ALMORT = .FALSE.
- IF ( NR .EQ. N ) THEN
- MAXPRJ = ONE
- DO 3051 p = 2, N
- TEMP1 = ABS(A(p,p)) / SVA(IWORK(p))
- MAXPRJ = MIN( MAXPRJ, TEMP1 )
- 3051 CONTINUE
- IF ( MAXPRJ**2 .GE. ONE - REAL(N)*EPSLN ) ALMORT = .TRUE.
- END IF
-*
-*
- SCONDA = - ONE
- CONDR1 = - ONE
- CONDR2 = - ONE
-*
- IF ( ERREST ) THEN
- IF ( N .EQ. NR ) THEN
- IF ( RSVEC ) THEN
-* .. V is available as workspace
- CALL CLACPY( 'U', N, N, A, LDA, V, LDV )
- DO 3053 p = 1, N
- TEMP1 = SVA(IWORK(p))
- CALL CSSCAL( p, ONE/TEMP1, V(1,p), 1 )
- 3053 CONTINUE
- IF ( LSVEC )THEN
- CALL CPOCON( 'U', N, V, LDV, ONE, TEMP1,
- $ CWORK(N+1), RWORK, IERR )
- ELSE
- CALL CPOCON( 'U', N, V, LDV, ONE, TEMP1,
- $ CWORK, RWORK, IERR )
- END IF
-*
- ELSE IF ( LSVEC ) THEN
-* .. U is available as workspace
- CALL CLACPY( 'U', N, N, A, LDA, U, LDU )
- DO 3054 p = 1, N
- TEMP1 = SVA(IWORK(p))
- CALL CSSCAL( p, ONE/TEMP1, U(1,p), 1 )
- 3054 CONTINUE
- CALL CPOCON( 'U', N, U, LDU, ONE, TEMP1,
- $ CWORK(N+1), RWORK, IERR )
- ELSE
- CALL CLACPY( 'U', N, N, A, LDA, CWORK, N )
-*[] CALL CLACPY( 'U', N, N, A, LDA, CWORK(N+1), N )
-* Change: here index shifted by N to the left, CWORK(1:N)
-* not needed for SIGMA only computation
- DO 3052 p = 1, N
- TEMP1 = SVA(IWORK(p))
-*[] CALL CSSCAL( p, ONE/TEMP1, CWORK(N+(p-1)*N+1), 1 )
- CALL CSSCAL( p, ONE/TEMP1, CWORK((p-1)*N+1), 1 )
- 3052 CONTINUE
-* .. the columns of R are scaled to have unit Euclidean lengths.
-*[] CALL CPOCON( 'U', N, CWORK(N+1), N, ONE, TEMP1,
-*[] $ CWORK(N+N*N+1), RWORK, IERR )
- CALL CPOCON( 'U', N, CWORK, N, ONE, TEMP1,
- $ CWORK(N*N+1), RWORK, IERR )
-*
- END IF
- IF ( TEMP1 .NE. ZERO ) THEN
- SCONDA = ONE / SQRT(TEMP1)
- ELSE
- SCONDA = - ONE
- END IF
-* SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
-* N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
- ELSE
- SCONDA = - ONE
- END IF
- END IF
-*
- L2PERT = L2PERT .AND. ( ABS( A(1,1)/A(NR,NR) ) .GT. SQRT(BIG1) )
-* If there is no violent scaling, artificial perturbation is not needed.
-*
-* Phase 3:
-*
- IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN
-*
-* Singular Values only
-*
-* .. transpose A(1:NR,1:N)
- DO 1946 p = 1, MIN( N-1, NR )
- CALL CCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )
- CALL CLACGV( N-p+1, A(p,p), 1 )
- 1946 CONTINUE
- IF ( NR .EQ. N ) A(N,N) = CONJG(A(N,N))
-*
-* The following two DO-loops introduce small relative perturbation
-* into the strict upper triangle of the lower triangular matrix.
-* Small entries below the main diagonal are also changed.
-* This modification is useful if the computing environment does not
-* provide/allow FLUSH TO ZERO underflow, for it prevents many
-* annoying denormalized numbers in case of strongly scaled matrices.
-* The perturbation is structured so that it does not introduce any
-* new perturbation of the singular values, and it does not destroy
-* the job done by the preconditioner.
-* The licence for this perturbation is in the variable L2PERT, which
-* should be .FALSE. if FLUSH TO ZERO underflow is active.
-*
- IF ( .NOT. ALMORT ) THEN
-*
- IF ( L2PERT ) THEN
-* XSC = SQRT(SMALL)
- XSC = EPSLN / REAL(N)
- DO 4947 q = 1, NR
- CTEMP = CMPLX(XSC*ABS(A(q,q)),ZERO)
- DO 4949 p = 1, N
- IF ( ( (p.GT.q) .AND. (ABS(A(p,q)).LE.TEMP1) )
- $ .OR. ( p .LT. q ) )
-* $ A(p,q) = TEMP1 * ( A(p,q) / ABS(A(p,q)) )
- $ A(p,q) = CTEMP
- 4949 CONTINUE
- 4947 CONTINUE
- ELSE
- CALL CLASET( 'U', NR-1,NR-1, CZERO,CZERO, A(1,2),LDA )
- END IF
-*
-* .. second preconditioning using the QR factorization
-*
- CALL CGEQRF( N,NR, A,LDA, CWORK, CWORK(N+1),LWORK-N, IERR )
-*
-* .. and transpose upper to lower triangular
- DO 1948 p = 1, NR - 1
- CALL CCOPY( NR-p, A(p,p+1), LDA, A(p+1,p), 1 )
- CALL CLACGV( NR-p+1, A(p,p), 1 )
- 1948 CONTINUE
-*
- END IF
-*
-* Row-cyclic Jacobi SVD algorithm with column pivoting
-*
-* .. again some perturbation (a "background noise") is added
-* to drown denormals
- IF ( L2PERT ) THEN
-* XSC = SQRT(SMALL)
- XSC = EPSLN / REAL(N)
- DO 1947 q = 1, NR
- CTEMP = CMPLX(XSC*ABS(A(q,q)),ZERO)
- DO 1949 p = 1, NR
- IF ( ( (p.GT.q) .AND. (ABS(A(p,q)).LE.TEMP1) )
- $ .OR. ( p .LT. q ) )
-* $ A(p,q) = TEMP1 * ( A(p,q) / ABS(A(p,q)) )
- $ A(p,q) = CTEMP
- 1949 CONTINUE
- 1947 CONTINUE
- ELSE
- CALL CLASET( 'U', NR-1, NR-1, CZERO, CZERO, A(1,2), LDA )
- END IF
-*
-* .. and one-sided Jacobi rotations are started on a lower
-* triangular matrix (plus perturbation which is ignored in
-* the part which destroys triangular form (confusing?!))
-*
- CALL CGESVJ( 'L', 'N', 'N', NR, NR, A, LDA, SVA,
- $ N, V, LDV, CWORK, LWORK, RWORK, LRWORK, INFO )
-*
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
-*
-*
- ELSE IF ( ( RSVEC .AND. ( .NOT. LSVEC ) .AND. ( .NOT. JRACC ) )
- $ .OR.
- $ ( JRACC .AND. ( .NOT. LSVEC ) .AND. ( NR .NE. N ) ) ) THEN
-*
-* -> Singular Values and Right Singular Vectors <-
-*
- IF ( ALMORT ) THEN
-*
-* .. in this case NR equals N
- DO 1998 p = 1, NR
- CALL CCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
- CALL CLACGV( N-p+1, V(p,p), 1 )
- 1998 CONTINUE
- CALL CLASET( 'U', NR-1,NR-1, CZERO, CZERO, V(1,2), LDV )
-*
- CALL CGESVJ( 'L','U','N', N, NR, V, LDV, SVA, NR, A, LDA,
- $ CWORK, LWORK, RWORK, LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
-
- ELSE
-*
-* .. two more QR factorizations ( one QRF is not enough, two require
-* accumulated product of Jacobi rotations, three are perfect )
-*
- CALL CLASET( 'L', NR-1,NR-1, CZERO, CZERO, A(2,1), LDA )
- CALL CGELQF( NR,N, A, LDA, CWORK, CWORK(N+1), LWORK-N, IERR)
- CALL CLACPY( 'L', NR, NR, A, LDA, V, LDV )
- CALL CLASET( 'U', NR-1,NR-1, CZERO, CZERO, V(1,2), LDV )
- CALL CGEQRF( NR, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
- $ LWORK-2*N, IERR )
- DO 8998 p = 1, NR
- CALL CCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )
- CALL CLACGV( NR-p+1, V(p,p), 1 )
- 8998 CONTINUE
- CALL CLASET('U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV)
-*
- CALL CGESVJ( 'L', 'U','N', NR, NR, V,LDV, SVA, NR, U,
- $ LDU, CWORK(N+1), LWORK-N, RWORK, LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
- IF ( NR .LT. N ) THEN
- CALL CLASET( 'A',N-NR, NR, CZERO,CZERO, V(NR+1,1), LDV )
- CALL CLASET( 'A',NR, N-NR, CZERO,CZERO, V(1,NR+1), LDV )
- CALL CLASET( 'A',N-NR,N-NR,CZERO,CONE, V(NR+1,NR+1),LDV )
- END IF
-*
- CALL CUNMLQ( 'L', 'C', N, N, NR, A, LDA, CWORK,
- $ V, LDV, CWORK(N+1), LWORK-N, IERR )
-*
- END IF
-* .. permute the rows of V
-* DO 8991 p = 1, N
-* CALL CCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA )
-* 8991 CONTINUE
-* CALL CLACPY( 'All', N, N, A, LDA, V, LDV )
- CALL CLAPMR( .FALSE., N, N, V, LDV, IWORK )
-*
- IF ( TRANSP ) THEN
- CALL CLACPY( 'A', N, N, V, LDV, U, LDU )
- END IF
-*
- ELSE IF ( JRACC .AND. (.NOT. LSVEC) .AND. ( NR.EQ. N ) ) THEN
-*
- CALL CLASET( 'L', N-1,N-1, CZERO, CZERO, A(2,1), LDA )
-*
- CALL CGESVJ( 'U','N','V', N, N, A, LDA, SVA, N, V, LDV,
- $ CWORK, LWORK, RWORK, LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
- CALL CLAPMR( .FALSE., N, N, V, LDV, IWORK )
-*
- ELSE IF ( LSVEC .AND. ( .NOT. RSVEC ) ) THEN
-*
-* .. Singular Values and Left Singular Vectors ..
-*
-* .. second preconditioning step to avoid need to accumulate
-* Jacobi rotations in the Jacobi iterations.
- DO 1965 p = 1, NR
- CALL CCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 )
- CALL CLACGV( N-p+1, U(p,p), 1 )
- 1965 CONTINUE
- CALL CLASET( 'U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
-*
- CALL CGEQRF( N, NR, U, LDU, CWORK(N+1), CWORK(2*N+1),
- $ LWORK-2*N, IERR )
-*
- DO 1967 p = 1, NR - 1
- CALL CCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )
- CALL CLACGV( N-p+1, U(p,p), 1 )
- 1967 CONTINUE
- CALL CLASET( 'U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
-*
- CALL CGESVJ( 'L', 'U', 'N', NR,NR, U, LDU, SVA, NR, A,
- $ LDA, CWORK(N+1), LWORK-N, RWORK, LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
-*
- IF ( NR .LT. M ) THEN
- CALL CLASET( 'A', M-NR, NR,CZERO, CZERO, U(NR+1,1), LDU )
- IF ( NR .LT. N1 ) THEN
- CALL CLASET( 'A',NR, N1-NR, CZERO, CZERO, U(1,NR+1),LDU )
- CALL CLASET( 'A',M-NR,N1-NR,CZERO,CONE,U(NR+1,NR+1),LDU )
- END IF
- END IF
-*
- CALL CUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
- $ LDU, CWORK(N+1), LWORK-N, IERR )
-*
- IF ( ROWPIV )
- $ CALL CLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
-*
- DO 1974 p = 1, N1
- XSC = ONE / SCNRM2( M, U(1,p), 1 )
- CALL CSSCAL( M, XSC, U(1,p), 1 )
- 1974 CONTINUE
-*
- IF ( TRANSP ) THEN
- CALL CLACPY( 'A', N, N, U, LDU, V, LDV )
- END IF
-*
- ELSE
-*
-* .. Full SVD ..
-*
- IF ( .NOT. JRACC ) THEN
-*
- IF ( .NOT. ALMORT ) THEN
-*
-* Second Preconditioning Step (QRF [with pivoting])
-* Note that the composition of TRANSPOSE, QRF and TRANSPOSE is
-* equivalent to an LQF CALL. Since in many libraries the QRF
-* seems to be better optimized than the LQF, we do explicit
-* transpose and use the QRF. This is subject to changes in an
-* optimized implementation of CGEJSV.
-*
- DO 1968 p = 1, NR
- CALL CCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
- CALL CLACGV( N-p+1, V(p,p), 1 )
- 1968 CONTINUE
-*
-* .. the following two loops perturb small entries to avoid
-* denormals in the second QR factorization, where they are
-* as good as zeros. This is done to avoid painfully slow
-* computation with denormals. The relative size of the perturbation
-* is a parameter that can be changed by the implementer.
-* This perturbation device will be obsolete on machines with
-* properly implemented arithmetic.
-* To switch it off, set L2PERT=.FALSE. To remove it from the
-* code, remove the action under L2PERT=.TRUE., leave the ELSE part.
-* The following two loops should be blocked and fused with the
-* transposed copy above.
-*
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL)
- DO 2969 q = 1, NR
- CTEMP = CMPLX(XSC*ABS( V(q,q) ),ZERO)
- DO 2968 p = 1, N
- IF ( ( p .GT. q ) .AND. ( ABS(V(p,q)) .LE. TEMP1 )
- $ .OR. ( p .LT. q ) )
-* $ V(p,q) = TEMP1 * ( V(p,q) / ABS(V(p,q)) )
- $ V(p,q) = CTEMP
- IF ( p .LT. q ) V(p,q) = - V(p,q)
- 2968 CONTINUE
- 2969 CONTINUE
- ELSE
- CALL CLASET( 'U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV )
- END IF
-*
-* Estimate the row scaled condition number of R1
-* (If R1 is rectangular, N > NR, then the condition number
-* of the leading NR x NR submatrix is estimated.)
-*
- CALL CLACPY( 'L', NR, NR, V, LDV, CWORK(2*N+1), NR )
- DO 3950 p = 1, NR
- TEMP1 = SCNRM2(NR-p+1,CWORK(2*N+(p-1)*NR+p),1)
- CALL CSSCAL(NR-p+1,ONE/TEMP1,CWORK(2*N+(p-1)*NR+p),1)
- 3950 CONTINUE
- CALL CPOCON('L',NR,CWORK(2*N+1),NR,ONE,TEMP1,
- $ CWORK(2*N+NR*NR+1),RWORK,IERR)
- CONDR1 = ONE / SQRT(TEMP1)
-* .. here need a second oppinion on the condition number
-* .. then assume worst case scenario
-* R1 is OK for inverse <=> CONDR1 .LT. REAL(N)
-* more conservative <=> CONDR1 .LT. SQRT(REAL(N))
-*
- COND_OK = SQRT(SQRT(REAL(NR)))
-*[TP] COND_OK is a tuning parameter.
-*
- IF ( CONDR1 .LT. COND_OK ) THEN
-* .. the second QRF without pivoting. Note: in an optimized
-* implementation, this QRF should be implemented as the QRF
-* of a lower triangular matrix.
-* R1^* = Q2 * R2
- CALL CGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
- $ LWORK-2*N, IERR )
-*
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL)/EPSLN
- DO 3959 p = 2, NR
- DO 3958 q = 1, p - 1
- CTEMP=CMPLX(XSC*MIN(ABS(V(p,p)),ABS(V(q,q))),
- $ ZERO)
- IF ( ABS(V(q,p)) .LE. TEMP1 )
-* $ V(q,p) = TEMP1 * ( V(q,p) / ABS(V(q,p)) )
- $ V(q,p) = CTEMP
- 3958 CONTINUE
- 3959 CONTINUE
- END IF
-*
- IF ( NR .NE. N )
- $ CALL CLACPY( 'A', N, NR, V, LDV, CWORK(2*N+1), N )
-* .. save ...
-*
-* .. this transposed copy should be better than naive
- DO 1969 p = 1, NR - 1
- CALL CCOPY( NR-p, V(p,p+1), LDV, V(p+1,p), 1 )
- CALL CLACGV(NR-p+1, V(p,p), 1 )
- 1969 CONTINUE
- V(NR,NR)=CONJG(V(NR,NR))
-*
- CONDR2 = CONDR1
-*
- ELSE
-*
-* .. ill-conditioned case: second QRF with pivoting
-* Note that windowed pivoting would be equaly good
-* numerically, and more run-time efficient. So, in
-* an optimal implementation, the next call to CGEQP3
-* should be replaced with eg. CALL CGEQPX (ACM TOMS #782)
-* with properly (carefully) chosen parameters.
-*
-* R1^* * P2 = Q2 * R2
- DO 3003 p = 1, NR
- IWORK(N+p) = 0
- 3003 CONTINUE
- CALL CGEQP3( N, NR, V, LDV, IWORK(N+1), CWORK(N+1),
- $ CWORK(2*N+1), LWORK-2*N, RWORK, IERR )
-** CALL CGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
-** $ LWORK-2*N, IERR )
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL)
- DO 3969 p = 2, NR
- DO 3968 q = 1, p - 1
- CTEMP=CMPLX(XSC*MIN(ABS(V(p,p)),ABS(V(q,q))),
- $ ZERO)
- IF ( ABS(V(q,p)) .LE. TEMP1 )
-* $ V(q,p) = TEMP1 * ( V(q,p) / ABS(V(q,p)) )
- $ V(q,p) = CTEMP
- 3968 CONTINUE
- 3969 CONTINUE
- END IF
-*
- CALL CLACPY( 'A', N, NR, V, LDV, CWORK(2*N+1), N )
-*
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL)
- DO 8970 p = 2, NR
- DO 8971 q = 1, p - 1
- CTEMP=CMPLX(XSC*MIN(ABS(V(p,p)),ABS(V(q,q))),
- $ ZERO)
-* V(p,q) = - TEMP1*( V(q,p) / ABS(V(q,p)) )
- V(p,q) = - CTEMP
- 8971 CONTINUE
- 8970 CONTINUE
- ELSE
- CALL CLASET( 'L',NR-1,NR-1,CZERO,CZERO,V(2,1),LDV )
- END IF
-* Now, compute R2 = L3 * Q3, the LQ factorization.
- CALL CGELQF( NR, NR, V, LDV, CWORK(2*N+N*NR+1),
- $ CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, IERR )
-* .. and estimate the condition number
- CALL CLACPY( 'L',NR,NR,V,LDV,CWORK(2*N+N*NR+NR+1),NR )
- DO 4950 p = 1, NR
- TEMP1 = SCNRM2( p, CWORK(2*N+N*NR+NR+p), NR )
- CALL CSSCAL( p, ONE/TEMP1, CWORK(2*N+N*NR+NR+p), NR )
- 4950 CONTINUE
- CALL CPOCON( 'L',NR,CWORK(2*N+N*NR+NR+1),NR,ONE,TEMP1,
- $ CWORK(2*N+N*NR+NR+NR*NR+1),RWORK,IERR )
- CONDR2 = ONE / SQRT(TEMP1)
-*
-*
- IF ( CONDR2 .GE. COND_OK ) THEN
-* .. save the Householder vectors used for Q3
-* (this overwrittes the copy of R2, as it will not be
-* needed in this branch, but it does not overwritte the
-* Huseholder vectors of Q2.).
- CALL CLACPY( 'U', NR, NR, V, LDV, CWORK(2*N+1), N )
-* .. and the rest of the information on Q3 is in
-* WORK(2*N+N*NR+1:2*N+N*NR+N)
- END IF
-*
- END IF
-*
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL)
- DO 4968 q = 2, NR
- CTEMP = XSC * V(q,q)
- DO 4969 p = 1, q - 1
-* V(p,q) = - TEMP1*( V(p,q) / ABS(V(p,q)) )
- V(p,q) = - CTEMP
- 4969 CONTINUE
- 4968 CONTINUE
- ELSE
- CALL CLASET( 'U', NR-1,NR-1, CZERO,CZERO, V(1,2), LDV )
- END IF
-*
-* Second preconditioning finished; continue with Jacobi SVD
-* The input matrix is lower trinagular.
-*
-* Recover the right singular vectors as solution of a well
-* conditioned triangular matrix equation.
-*
- IF ( CONDR1 .LT. COND_OK ) THEN
-*
- CALL CGESVJ( 'L','U','N',NR,NR,V,LDV,SVA,NR,U, LDU,
- $ CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,RWORK,
- $ LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
- DO 3970 p = 1, NR
- CALL CCOPY( NR, V(1,p), 1, U(1,p), 1 )
- CALL CSSCAL( NR, SVA(p), V(1,p), 1 )
- 3970 CONTINUE
-
-* .. pick the right matrix equation and solve it
-*
- IF ( NR .EQ. N ) THEN
-* :)) .. best case, R1 is inverted. The solution of this matrix
-* equation is Q2*V2 = the product of the Jacobi rotations
-* used in CGESVJ, premultiplied with the orthogonal matrix
-* from the second QR factorization.
- CALL CTRSM('L','U','N','N', NR,NR,CONE, A,LDA, V,LDV)
- ELSE
-* .. R1 is well conditioned, but non-square. Adjoint of R2
-* is inverted to get the product of the Jacobi rotations
-* used in CGESVJ. The Q-factor from the second QR
-* factorization is then built in explicitly.
- CALL CTRSM('L','U','C','N',NR,NR,CONE,CWORK(2*N+1),
- $ N,V,LDV)
- IF ( NR .LT. N ) THEN
- CALL CLASET('A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV)
- CALL CLASET('A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV)
- CALL CLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
- END IF
- CALL CUNMQR('L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
- $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR)
- END IF
-*
- ELSE IF ( CONDR2 .LT. COND_OK ) THEN
-*
-* The matrix R2 is inverted. The solution of the matrix equation
-* is Q3^* * V3 = the product of the Jacobi rotations (appplied to
-* the lower triangular L3 from the LQ factorization of
-* R2=L3*Q3), pre-multiplied with the transposed Q3.
- CALL CGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR, U,
- $ LDU, CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR,
- $ RWORK, LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
- DO 3870 p = 1, NR
- CALL CCOPY( NR, V(1,p), 1, U(1,p), 1 )
- CALL CSSCAL( NR, SVA(p), U(1,p), 1 )
- 3870 CONTINUE
- CALL CTRSM('L','U','N','N',NR,NR,CONE,CWORK(2*N+1),N,
- $ U,LDU)
-* .. apply the permutation from the second QR factorization
- DO 873 q = 1, NR
- DO 872 p = 1, NR
- CWORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
- 872 CONTINUE
- DO 874 p = 1, NR
- U(p,q) = CWORK(2*N+N*NR+NR+p)
- 874 CONTINUE
- 873 CONTINUE
- IF ( NR .LT. N ) THEN
- CALL CLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
- CALL CLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
- CALL CLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
- END IF
- CALL CUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
- $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
- ELSE
-* Last line of defense.
-* #:( This is a rather pathological case: no scaled condition
-* improvement after two pivoted QR factorizations. Other
-* possibility is that the rank revealing QR factorization
-* or the condition estimator has failed, or the COND_OK
-* is set very close to ONE (which is unnecessary). Normally,
-* this branch should never be executed, but in rare cases of
-* failure of the RRQR or condition estimator, the last line of
-* defense ensures that CGEJSV completes the task.
-* Compute the full SVD of L3 using CGESVJ with explicit
-* accumulation of Jacobi rotations.
- CALL CGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR, U,
- $ LDU, CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR,
- $ RWORK, LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
- IF ( NR .LT. N ) THEN
- CALL CLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
- CALL CLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
- CALL CLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
- END IF
- CALL CUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
- $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
-*
- CALL CUNMLQ( 'L', 'C', NR, NR, NR, CWORK(2*N+1), N,
- $ CWORK(2*N+N*NR+1), U, LDU, CWORK(2*N+N*NR+NR+1),
- $ LWORK-2*N-N*NR-NR, IERR )
- DO 773 q = 1, NR
- DO 772 p = 1, NR
- CWORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
- 772 CONTINUE
- DO 774 p = 1, NR
- U(p,q) = CWORK(2*N+N*NR+NR+p)
- 774 CONTINUE
- 773 CONTINUE
-*
- END IF
-*
-* Permute the rows of V using the (column) permutation from the
-* first QRF. Also, scale the columns to make them unit in
-* Euclidean norm. This applies to all cases.
-*
- TEMP1 = SQRT(REAL(N)) * EPSLN
- DO 1972 q = 1, N
- DO 972 p = 1, N
- CWORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
- 972 CONTINUE
- DO 973 p = 1, N
- V(p,q) = CWORK(2*N+N*NR+NR+p)
- 973 CONTINUE
- XSC = ONE / SCNRM2( N, V(1,q), 1 )
- IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
- $ CALL CSSCAL( N, XSC, V(1,q), 1 )
- 1972 CONTINUE
-* At this moment, V contains the right singular vectors of A.
-* Next, assemble the left singular vector matrix U (M x N).
- IF ( NR .LT. M ) THEN
- CALL CLASET('A', M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU)
- IF ( NR .LT. N1 ) THEN
- CALL CLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)
- CALL CLASET('A',M-NR,N1-NR,CZERO,CONE,
- $ U(NR+1,NR+1),LDU)
- END IF
- END IF
-*
-* The Q matrix from the first QRF is built into the left singular
-* matrix U. This applies to all cases.
-*
- CALL CUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
- $ LDU, CWORK(N+1), LWORK-N, IERR )
-
-* The columns of U are normalized. The cost is O(M*N) flops.
- TEMP1 = SQRT(REAL(M)) * EPSLN
- DO 1973 p = 1, NR
- XSC = ONE / SCNRM2( M, U(1,p), 1 )
- IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
- $ CALL CSSCAL( M, XSC, U(1,p), 1 )
- 1973 CONTINUE
-*
-* If the initial QRF is computed with row pivoting, the left
-* singular vectors must be adjusted.
-*
- IF ( ROWPIV )
- $ CALL CLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
-*
- ELSE
-*
-* .. the initial matrix A has almost orthogonal columns and
-* the second QRF is not needed
-*
- CALL CLACPY( 'U', N, N, A, LDA, CWORK(N+1), N )
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL)
- DO 5970 p = 2, N
- CTEMP = XSC * CWORK( N + (p-1)*N + p )
- DO 5971 q = 1, p - 1
-* CWORK(N+(q-1)*N+p)=-TEMP1 * ( CWORK(N+(p-1)*N+q) /
-* $ ABS(CWORK(N+(p-1)*N+q)) )
- CWORK(N+(q-1)*N+p)=-CTEMP
- 5971 CONTINUE
- 5970 CONTINUE
- ELSE
- CALL CLASET( 'L',N-1,N-1,CZERO,CZERO,CWORK(N+2),N )
- END IF
-*
- CALL CGESVJ( 'U', 'U', 'N', N, N, CWORK(N+1), N, SVA,
- $ N, U, LDU, CWORK(N+N*N+1), LWORK-N-N*N, RWORK, LRWORK,
- $ INFO )
-*
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
- DO 6970 p = 1, N
- CALL CCOPY( N, CWORK(N+(p-1)*N+1), 1, U(1,p), 1 )
- CALL CSSCAL( N, SVA(p), CWORK(N+(p-1)*N+1), 1 )
- 6970 CONTINUE
-*
- CALL CTRSM( 'L', 'U', 'N', 'N', N, N,
- $ CONE, A, LDA, CWORK(N+1), N )
- DO 6972 p = 1, N
- CALL CCOPY( N, CWORK(N+p), N, V(IWORK(p),1), LDV )
- 6972 CONTINUE
- TEMP1 = SQRT(REAL(N))*EPSLN
- DO 6971 p = 1, N
- XSC = ONE / SCNRM2( N, V(1,p), 1 )
- IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
- $ CALL CSSCAL( N, XSC, V(1,p), 1 )
- 6971 CONTINUE
-*
-* Assemble the left singular vector matrix U (M x N).
-*
- IF ( N .LT. M ) THEN
- CALL CLASET( 'A', M-N, N, CZERO, CZERO, U(N+1,1), LDU )
- IF ( N .LT. N1 ) THEN
- CALL CLASET('A',N, N1-N, CZERO, CZERO, U(1,N+1),LDU)
- CALL CLASET( 'A',M-N,N1-N, CZERO, CONE,U(N+1,N+1),LDU)
- END IF
- END IF
- CALL CUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
- $ LDU, CWORK(N+1), LWORK-N, IERR )
- TEMP1 = SQRT(REAL(M))*EPSLN
- DO 6973 p = 1, N1
- XSC = ONE / SCNRM2( M, U(1,p), 1 )
- IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
- $ CALL CSSCAL( M, XSC, U(1,p), 1 )
- 6973 CONTINUE
-*
- IF ( ROWPIV )
- $ CALL CLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
-*
- END IF
-*
-* end of the >> almost orthogonal case << in the full SVD
-*
- ELSE
-*
-* This branch deploys a preconditioned Jacobi SVD with explicitly
-* accumulated rotations. It is included as optional, mainly for
-* experimental purposes. It does perfom well, and can also be used.
-* In this implementation, this branch will be automatically activated
-* if the condition number sigma_max(A) / sigma_min(A) is predicted
-* to be greater than the overflow threshold. This is because the
-* a posteriori computation of the singular vectors assumes robust
-* implementation of BLAS and some LAPACK procedures, capable of working
-* in presence of extreme values, e.g. when the singular values spread from
-* the underflow to the overflow threshold.
-*
- DO 7968 p = 1, NR
- CALL CCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
- CALL CLACGV( N-p+1, V(p,p), 1 )
- 7968 CONTINUE
-*
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL/EPSLN)
- DO 5969 q = 1, NR
- CTEMP = CMPLX(XSC*ABS( V(q,q) ),ZERO)
- DO 5968 p = 1, N
- IF ( ( p .GT. q ) .AND. ( ABS(V(p,q)) .LE. TEMP1 )
- $ .OR. ( p .LT. q ) )
-* $ V(p,q) = TEMP1 * ( V(p,q) / ABS(V(p,q)) )
- $ V(p,q) = CTEMP
- IF ( p .LT. q ) V(p,q) = - V(p,q)
- 5968 CONTINUE
- 5969 CONTINUE
- ELSE
- CALL CLASET( 'U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV )
- END IF
-
- CALL CGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
- $ LWORK-2*N, IERR )
- CALL CLACPY( 'L', N, NR, V, LDV, CWORK(2*N+1), N )
-*
- DO 7969 p = 1, NR
- CALL CCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )
- CALL CLACGV( NR-p+1, U(p,p), 1 )
- 7969 CONTINUE
-
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL/EPSLN)
- DO 9970 q = 2, NR
- DO 9971 p = 1, q - 1
- CTEMP = CMPLX(XSC * MIN(ABS(U(p,p)),ABS(U(q,q))),
- $ ZERO)
-* U(p,q) = - TEMP1 * ( U(q,p) / ABS(U(q,p)) )
- U(p,q) = - CTEMP
- 9971 CONTINUE
- 9970 CONTINUE
- ELSE
- CALL CLASET('U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
- END IF
-
- CALL CGESVJ( 'L', 'U', 'V', NR, NR, U, LDU, SVA,
- $ N, V, LDV, CWORK(2*N+N*NR+1), LWORK-2*N-N*NR,
- $ RWORK, LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
-
- IF ( NR .LT. N ) THEN
- CALL CLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
- CALL CLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
- CALL CLASET( 'A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV )
- END IF
-
- CALL CUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
- $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
-*
-* Permute the rows of V using the (column) permutation from the
-* first QRF. Also, scale the columns to make them unit in
-* Euclidean norm. This applies to all cases.
-*
- TEMP1 = SQRT(REAL(N)) * EPSLN
- DO 7972 q = 1, N
- DO 8972 p = 1, N
- CWORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
- 8972 CONTINUE
- DO 8973 p = 1, N
- V(p,q) = CWORK(2*N+N*NR+NR+p)
- 8973 CONTINUE
- XSC = ONE / SCNRM2( N, V(1,q), 1 )
- IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
- $ CALL CSSCAL( N, XSC, V(1,q), 1 )
- 7972 CONTINUE
-*
-* At this moment, V contains the right singular vectors of A.
-* Next, assemble the left singular vector matrix U (M x N).
-*
- IF ( NR .LT. M ) THEN
- CALL CLASET( 'A', M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU )
- IF ( NR .LT. N1 ) THEN
- CALL CLASET('A',NR, N1-NR, CZERO, CZERO, U(1,NR+1),LDU)
- CALL CLASET('A',M-NR,N1-NR, CZERO, CONE,U(NR+1,NR+1),LDU)
- END IF
- END IF
-*
- CALL CUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
- $ LDU, CWORK(N+1), LWORK-N, IERR )
-*
- IF ( ROWPIV )
- $ CALL CLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
-*
-*
- END IF
- IF ( TRANSP ) THEN
-* .. swap U and V because the procedure worked on A^*
- DO 6974 p = 1, N
- CALL CSWAP( N, U(1,p), 1, V(1,p), 1 )
- 6974 CONTINUE
- END IF
-*
- END IF
-* end of the full SVD
-*
-* Undo scaling, if necessary (and possible)
-*
- IF ( USCAL2 .LE. (BIG/SVA(1))*USCAL1 ) THEN
- CALL SLASCL( 'G', 0, 0, USCAL1, USCAL2, NR, 1, SVA, N, IERR )
- USCAL1 = ONE
- USCAL2 = ONE
- END IF
-*
- IF ( NR .LT. N ) THEN
- DO 3004 p = NR+1, N
- SVA(p) = ZERO
- 3004 CONTINUE
- END IF
-*
- RWORK(1) = USCAL2 * SCALEM
- RWORK(2) = USCAL1
- IF ( ERREST ) RWORK(3) = SCONDA
- IF ( LSVEC .AND. RSVEC ) THEN
- RWORK(4) = CONDR1
- RWORK(5) = CONDR2
- END IF
- IF ( L2TRAN ) THEN
- RWORK(6) = ENTRA
- RWORK(7) = ENTRAT
- END IF
-*
- IWORK(1) = NR
- IWORK(2) = NUMRANK
- IWORK(3) = WARNING
- IF ( TRANSP ) THEN
- IWORK(4) = 1
- ELSE
- IWORK(4) = -1
- END IF
-
-*
- RETURN
-* ..
-* .. END OF CGEJSV
-* ..
- END
-*
+*> \brief \b CGEJSV
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download CGEJSV + dependencies
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgejsv.f">
+*> [TGZ]</a>
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgejsv.f">
+*> [ZIP]</a>
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgejsv.f">
+*> [TXT]</a>
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE CGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
+* M, N, A, LDA, SVA, U, LDU, V, LDV,
+* CWORK, LWORK, RWORK, LRWORK, IWORK, INFO )
+*
+* .. Scalar Arguments ..
+* IMPLICIT NONE
+* INTEGER INFO, LDA, LDU, LDV, LWORK, M, N
+* ..
+* .. Array Arguments ..
+* COMPLEX A( LDA, * ), U( LDU, * ), V( LDV, * ), CWORK( LWORK )
+* REAL SVA( N ), RWORK( LRWORK )
+* INTEGER IWORK( * )
+* CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> CGEJSV computes the singular value decomposition (SVD) of a complex M-by-N
+*> matrix [A], where M >= N. The SVD of [A] is written as
+*>
+*> [A] = [U] * [SIGMA] * [V]^*,
+*>
+*> where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
+*> diagonal elements, [U] is an M-by-N (or M-by-M) unitary matrix, and
+*> [V] is an N-by-N unitary matrix. The diagonal elements of [SIGMA] are
+*> the singular values of [A]. The columns of [U] and [V] are the left and
+*> the right singular vectors of [A], respectively. The matrices [U] and [V]
+*> are computed and stored in the arrays U and V, respectively. The diagonal
+*> of [SIGMA] is computed and stored in the array SVA.
+*> \endverbatim
+*>
+*> Arguments:
+*> ==========
+*>
+*> \param[in] JOBA
+*> \verbatim
+*> JOBA is CHARACTER*1
+*> Specifies the level of accuracy:
+*> = 'C': This option works well (high relative accuracy) if A = B * D,
+*> with well-conditioned B and arbitrary diagonal matrix D.
+*> The accuracy cannot be spoiled by COLUMN scaling. The
+*> accuracy of the computed output depends on the condition of
+*> B, and the procedure aims at the best theoretical accuracy.
+*> The relative error max_{i=1:N}|d sigma_i| / sigma_i is
+*> bounded by f(M,N)*epsilon* cond(B), independent of D.
+*> The input matrix is preprocessed with the QRF with column
+*> pivoting. This initial preprocessing and preconditioning by
+*> a rank revealing QR factorization is common for all values of
+*> JOBA. Additional actions are specified as follows:
+*> = 'E': Computation as with 'C' with an additional estimate of the
+*> condition number of B. It provides a realistic error bound.
+*> = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
+*> D1, D2, and well-conditioned matrix C, this option gives
+*> higher accuracy than the 'C' option. If the structure of the
+*> input matrix is not known, and relative accuracy is
+*> desirable, then this option is advisable. The input matrix A
+*> is preprocessed with QR factorization with FULL (row and
+*> column) pivoting.
+*> = 'G' Computation as with 'F' with an additional estimate of the
+*> condition number of B, where A=B*D. If A has heavily weighted
+*> rows, then using this condition number gives too pessimistic
+*> error bound.
+*> = 'A': Small singular values are not well determined by the data
+*> and are considered as noisy; the matrix is treated as
+*> numerically rank defficient. The error in the computed
+*> singular values is bounded by f(m,n)*epsilon*||A||.
+*> The computed SVD A = U * S * V^* restores A up to
+*> f(m,n)*epsilon*||A||.
+*> This gives the procedure the licence to discard (set to zero)
+*> all singular values below N*epsilon*||A||.
+*> = 'R': Similar as in 'A'. Rank revealing property of the initial
+*> QR factorization is used do reveal (using triangular factor)
+*> a gap sigma_{r+1} < epsilon * sigma_r in which case the
+*> numerical RANK is declared to be r. The SVD is computed with
+*> absolute error bounds, but more accurately than with 'A'.
+*> \endverbatim
+*>
+*> \param[in] JOBU
+*> \verbatim
+*> JOBU is CHARACTER*1
+*> Specifies whether to compute the columns of U:
+*> = 'U': N columns of U are returned in the array U.
+*> = 'F': full set of M left sing. vectors is returned in the array U.
+*> = 'W': U may be used as workspace of length M*N. See the description
+*> of U.
+*> = 'N': U is not computed.
+*> \endverbatim
+*>
+*> \param[in] JOBV
+*> \verbatim
+*> JOBV is CHARACTER*1
+*> Specifies whether to compute the matrix V:
+*> = 'V': N columns of V are returned in the array V; Jacobi rotations
+*> are not explicitly accumulated.
+*> = 'J': N columns of V are returned in the array V, but they are
+*> computed as the product of Jacobi rotations, if JOBT .EQ. 'N'.
+*> = 'W': V may be used as workspace of length N*N. See the description
+*> of V.
+*> = 'N': V is not computed.
+*> \endverbatim
+*>
+*> \param[in] JOBR
+*> \verbatim
+*> JOBR is CHARACTER*1
+*> Specifies the RANGE for the singular values. Issues the licence to
+*> set to zero small positive singular values if they are outside
+*> specified range. If A .NE. 0 is scaled so that the largest singular
+*> value of c*A is around SQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
+*> the licence to kill columns of A whose norm in c*A is less than
+*> SQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,
+*> where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
+*> = 'N': Do not kill small columns of c*A. This option assumes that
+*> BLAS and QR factorizations and triangular solvers are
+*> implemented to work in that range. If the condition of A
+*> is greater than BIG, use CGESVJ.
+*> = 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)]
+*> (roughly, as described above). This option is recommended.
+*> ===========================
+*> For computing the singular values in the FULL range [SFMIN,BIG]
+*> use CGESVJ.
+*> \endverbatim
+*>
+*> \param[in] JOBT
+*> \verbatim
+*> JOBT is CHARACTER*1
+*> If the matrix is square then the procedure may determine to use
+*> transposed A if A^* seems to be better with respect to convergence.
+*> If the matrix is not square, JOBT is ignored.
+*> The decision is based on two values of entropy over the adjoint
+*> orbit of A^* * A. See the descriptions of WORK(6) and WORK(7).
+*> = 'T': transpose if entropy test indicates possibly faster
+*> convergence of Jacobi process if A^* is taken as input. If A is
+*> replaced with A^*, then the row pivoting is included automatically.
+*> = 'N': do not speculate.
+*> The option 'T' can be used to compute only the singular values, or
+*> the full SVD (U, SIGMA and V). For only one set of singular vectors
+*> (U or V), the caller should provide both U and V, as one of the
+*> matrices is used as workspace if the matrix A is transposed.
+*> The implementer can easily remove this constraint and make the
+*> code more complicated. See the descriptions of U and V.
+*> In general, this option is considered experimental, and 'N'; should
+*> be preferred. This is subject to changes in the future.
+*> \endverbatim
+*>
+*> \param[in] JOBP
+*> \verbatim
+*> JOBP is CHARACTER*1
+*> Issues the licence to introduce structured perturbations to drown
+*> denormalized numbers. This licence should be active if the
+*> denormals are poorly implemented, causing slow computation,
+*> especially in cases of fast convergence (!). For details see [1,2].
+*> For the sake of simplicity, this perturbations are included only
+*> when the full SVD or only the singular values are requested. The
+*> implementer/user can easily add the perturbation for the cases of
+*> computing one set of singular vectors.
+*> = 'P': introduce perturbation
+*> = 'N': do not perturb
+*> \endverbatim
+*>
+*> \param[in] M
+*> \verbatim
+*> M is INTEGER
+*> The number of rows of the input matrix A. M >= 0.
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The number of columns of the input matrix A. M >= N >= 0.
+*> \endverbatim
+*>
+*> \param[in,out] A
+*> \verbatim
+*> A is COMPLEX array, dimension (LDA,N)
+*> On entry, the M-by-N matrix A.
+*> \endverbatim
+*>
+*> \param[in] LDA
+*> \verbatim
+*> LDA is INTEGER
+*> The leading dimension of the array A. LDA >= max(1,M).
+*> \endverbatim
+*>
+*> \param[out] SVA
+*> \verbatim
+*> SVA is REAL array, dimension (N)
+*> On exit,
+*> - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
+*> computation SVA contains Euclidean column norms of the
+*> iterated matrices in the array A.
+*> - For WORK(1) .NE. WORK(2): The singular values of A are
+*> (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
+*> sigma_max(A) overflows or if small singular values have been
+*> saved from underflow by scaling the input matrix A.
+*> - If JOBR='R' then some of the singular values may be returned
+*> as exact zeros obtained by "set to zero" because they are
+*> below the numerical rank threshold or are denormalized numbers.
+*> \endverbatim
+*>
+*> \param[out] U
+*> \verbatim
+*> U is COMPLEX array, dimension ( LDU, N ) or ( LDU, M )
+*> If JOBU = 'U', then U contains on exit the M-by-N matrix of
+*> the left singular vectors.
+*> If JOBU = 'F', then U contains on exit the M-by-M matrix of
+*> the left singular vectors, including an ONB
+*> of the orthogonal complement of the Range(A).
+*> If JOBU = 'W' .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),
+*> then U is used as workspace if the procedure
+*> replaces A with A^*. In that case, [V] is computed
+*> in U as left singular vectors of A^* and then
+*> copied back to the V array. This 'W' option is just
+*> a reminder to the caller that in this case U is
+*> reserved as workspace of length N*N.
+*> If JOBU = 'N' U is not referenced, unless JOBT='T'.
+*> \endverbatim
+*>
+*> \param[in] LDU
+*> \verbatim
+*> LDU is INTEGER
+*> The leading dimension of the array U, LDU >= 1.
+*> IF JOBU = 'U' or 'F' or 'W', then LDU >= M.
+*> \endverbatim
+*>
+*> \param[out] V
+*> \verbatim
+*> V is COMPLEX array, dimension ( LDV, N )
+*> If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
+*> the right singular vectors;
+*> If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),
+*> then V is used as workspace if the pprocedure
+*> replaces A with A^*. In that case, [U] is computed
+*> in V as right singular vectors of A^* and then
+*> copied back to the U array. This 'W' option is just
+*> a reminder to the caller that in this case V is
+*> reserved as workspace of length N*N.
+*> If JOBV = 'N' V is not referenced, unless JOBT='T'.
+*> \endverbatim
+*>
+*> \param[in] LDV
+*> \verbatim
+*> LDV is INTEGER
+*> The leading dimension of the array V, LDV >= 1.
+*> If JOBV = 'V' or 'J' or 'W', then LDV >= N.
+*> \endverbatim
+*>
+*> \param[out] CWORK
+*> \verbatim
+*> CWORK is COMPLEX array, dimension (MAX(2,LWORK))
+*> If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
+*> LRWORK=-1), then on exit CWORK(1) contains the required length of
+*> CWORK for the job parameters used in the call.
+*> \endverbatim
+*>
+*> \param[in] LWORK
+*> \verbatim
+*> LWORK is INTEGER
+*> Length of CWORK to confirm proper allocation of workspace.
+*> LWORK depends on the job:
+*>
+*> 1. If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
+*> 1.1 .. no scaled condition estimate required (JOBA.NE.'E'.AND.JOBA.NE.'G'):
+*> LWORK >= 2*N+1. This is the minimal requirement.
+*> ->> For optimal performance (blocked code) the optimal value
+*> is LWORK >= N + (N+1)*NB. Here NB is the optimal
+*> block size for CGEQP3 and CGEQRF.
+*> In general, optimal LWORK is computed as
+*> LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF), LWORK(CGESVJ)).
+*> 1.2. .. an estimate of the scaled condition number of A is
+*> required (JOBA='E', or 'G'). In this case, LWORK the minimal
+*> requirement is LWORK >= N*N + 2*N.
+*> ->> For optimal performance (blocked code) the optimal value
+*> is LWORK >= max(N+(N+1)*NB, N*N+2*N)=N**2+2*N.
+*> In general, the optimal length LWORK is computed as
+*> LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF), LWORK(CGESVJ),
+*> N*N+LWORK(CPOCON)).
+*> 2. If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
+*> (JOBU.EQ.'N')
+*> 2.1 .. no scaled condition estimate requested (JOBE.EQ.'N'):
+*> -> the minimal requirement is LWORK >= 3*N.
+*> -> For optimal performance,
+*> LWORK >= max(N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
+*> where NB is the optimal block size for CGEQP3, CGEQRF, CGELQ,
+*> CUNMLQ. In general, the optimal length LWORK is computed as
+*> LWORK >= max(N+LWORK(CGEQP3), N+LWORK(CGESVJ),
+*> N+LWORK(CGELQF), 2*N+LWORK(CGEQRF), N+LWORK(CUNMLQ)).
+*> 2.2 .. an estimate of the scaled condition number of A is
+*> required (JOBA='E', or 'G').
+*> -> the minimal requirement is LWORK >= 3*N.
+*> -> For optimal performance,
+*> LWORK >= max(N+(N+1)*NB, 2*N,2*N+N*NB)=2*N+N*NB,
+*> where NB is the optimal block size for CGEQP3, CGEQRF, CGELQ,
+*> CUNMLQ. In general, the optimal length LWORK is computed as
+*> LWORK >= max(N+LWORK(CGEQP3), LWORK(CPOCON), N+LWORK(CGESVJ),
+*> N+LWORK(CGELQF), 2*N+LWORK(CGEQRF), N+LWORK(CUNMLQ)).
+*> 3. If SIGMA and the left singular vectors are needed
+*> 3.1 .. no scaled condition estimate requested (JOBE.EQ.'N'):
+*> -> the minimal requirement is LWORK >= 3*N.
+*> -> For optimal performance:
+*> if JOBU.EQ.'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
+*> where NB is the optimal block size for CGEQP3, CGEQRF, CUNMQR.
+*> In general, the optimal length LWORK is computed as
+*> LWORK >= max(N+LWORK(CGEQP3), 2*N+LWORK(CGEQRF), N+LWORK(CUNMQR)).
+*> 3.2 .. an estimate of the scaled condition number of A is
+*> required (JOBA='E', or 'G').
+*> -> the minimal requirement is LWORK >= 3*N.
+*> -> For optimal performance:
+*> if JOBU.EQ.'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
+*> where NB is the optimal block size for CGEQP3, CGEQRF, CUNMQR.
+*> In general, the optimal length LWORK is computed as
+*> LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CPOCON),
+*> 2*N+LWORK(CGEQRF), N+LWORK(CUNMQR)).
+*>
+*> 4. If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and
+*> 4.1. if JOBV.EQ.'V'
+*> the minimal requirement is LWORK >= 5*N+2*N*N.
+*> 4.2. if JOBV.EQ.'J' the minimal requirement is
+*> LWORK >= 4*N+N*N.
+*> In both cases, the allocated CWORK can accommodate blocked runs
+*> of CGEQP3, CGEQRF, CGELQF, CUNMQR, CUNMLQ.
+*>
+*> If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
+*> LRWORK=-1), then on exit CWORK(1) contains the optimal and CWORK(2) contains the
+*> minimal length of CWORK for the job parameters used in the call.
+*> \endverbatim
+*>
+*> \param[out] RWORK
+*> \verbatim
+*> RWORK is REAL array, dimension (MAX(7,LWORK))
+*> On exit,
+*> RWORK(1) = Determines the scaling factor SCALE = RWORK(2) / RWORK(1)
+*> such that SCALE*SVA(1:N) are the computed singular values
+*> of A. (See the description of SVA().)
+*> RWORK(2) = See the description of RWORK(1).
+*> RWORK(3) = SCONDA is an estimate for the condition number of
+*> column equilibrated A. (If JOBA .EQ. 'E' or 'G')
+*> SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
+*> It is computed using SPOCON. It holds
+*> N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
+*> where R is the triangular factor from the QRF of A.
+*> However, if R is truncated and the numerical rank is
+*> determined to be strictly smaller than N, SCONDA is
+*> returned as -1, thus indicating that the smallest
+*> singular values might be lost.
+*>
+*> If full SVD is needed, the following two condition numbers are
+*> useful for the analysis of the algorithm. They are provied for
+*> a developer/implementer who is familiar with the details of
+*> the method.
+*>
+*> RWORK(4) = an estimate of the scaled condition number of the
+*> triangular factor in the first QR factorization.
+*> RWORK(5) = an estimate of the scaled condition number of the
+*> triangular factor in the second QR factorization.
+*> The following two parameters are computed if JOBT .EQ. 'T'.
+*> They are provided for a developer/implementer who is familiar
+*> with the details of the method.
+*> RWORK(6) = the entropy of A^* * A :: this is the Shannon entropy
+*> of diag(A^* * A) / Trace(A^* * A) taken as point in the
+*> probability simplex.
+*> RWORK(7) = the entropy of A * A^*. (See the description of RWORK(6).)
+*> If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
+*> LRWORK=-1), then on exit RWORK(1) contains the required length of
+*> RWORK for the job parameters used in the call.
+*> \endverbatim
+*>
+*> \param[in] LRWORK
+*> \verbatim
+*> LRWORK is INTEGER
+*> Length of RWORK to confirm proper allocation of workspace.
+*> LRWORK depends on the job:
+*>
+*> 1. If only the singular values are requested i.e. if
+*> LSAME(JOBU,'N') .AND. LSAME(JOBV,'N')
+*> then:
+*> 1.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
+*> then: LRWORK = max( 7, 2 * M ).
+*> 1.2. Otherwise, LRWORK = max( 7, N ).
+*> 2. If singular values with the right singular vectors are requested
+*> i.e. if
+*> (LSAME(JOBV,'V').OR.LSAME(JOBV,'J')) .AND.
+*> .NOT.(LSAME(JOBU,'U').OR.LSAME(JOBU,'F'))
+*> then:
+*> 2.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
+*> then LRWORK = max( 7, 2 * M ).
+*> 2.2. Otherwise, LRWORK = max( 7, N ).
+*> 3. If singular values with the left singular vectors are requested, i.e. if
+*> (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
+*> .NOT.(LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
+*> then:
+*> 3.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
+*> then LRWORK = max( 7, 2 * M ).
+*> 3.2. Otherwise, LRWORK = max( 7, N ).
+*> 4. If singular values with both the left and the right singular vectors
+*> are requested, i.e. if
+*> (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
+*> (LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
+*> then:
+*> 4.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
+*> then LRWORK = max( 7, 2 * M ).
+*> 4.2. Otherwise, LRWORK = max( 7, N ).
+*>
+*> If, on entry, LRWORK = -1 or LWORK=-1, a workspace query is assumed and
+*> the length of RWORK is returned in RWORK(1).
+*> \endverbatim
+*>
+*> \param[out] IWORK
+*> \verbatim
+*> IWORK is INTEGER array, of dimension at least 4, that further depends
+*> on the job:
+*>
+*> 1. If only the singular values are requested then:
+*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
+*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
+*> 2. If the singular values and the right singular vectors are requested then:
+*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
+*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
+*> 3. If the singular values and the left singular vectors are requested then:
+*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
+*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
+*> 4. If the singular values with both the left and the right singular vectors
+*> are requested, then:
+*> 4.1. If LSAME(JOBV,'J') the length of IWORK is determined as follows:
+*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
+*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
+*> 4.2. If LSAME(JOBV,'V') the length of IWORK is determined as follows:
+*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
+*> then the length of IWORK is 2*N+M; otherwise the length of IWORK is 2*N.
+*>
+*> On exit,
+*> IWORK(1) = the numerical rank determined after the initial
+*> QR factorization with pivoting. See the descriptions
+*> of JOBA and JOBR.
+*> IWORK(2) = the number of the computed nonzero singular values
+*> IWORK(3) = if nonzero, a warning message:
+*> If IWORK(3).EQ.1 then some of the column norms of A
+*> were denormalized floats. The requested high accuracy
+*> is not warranted by the data.
+*> IWORK(4) = 1 or -1. If IWORK(4) .EQ. 1, then the procedure used A^* to
+*> do the job as specified by the JOB parameters.
+*> If the call to CGEJSV is a workspace query (indicated by LWORK .EQ. -1 and
+*> LRWORK .EQ. -1), then on exit IWORK(1) contains the required length of
+*> IWORK for the job parameters used in the call.
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> INFO is INTEGER
+*> < 0 : if INFO = -i, then the i-th argument had an illegal value.
+*> = 0 : successful exit;
+*> > 0 : CGEJSV did not converge in the maximal allowed number
+*> of sweeps. The computed values may be inaccurate.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date June 2016
+*
+*> \ingroup complexGEsing
+*
+*> \par Further Details:
+* =====================
+*>
+*> \verbatim
+*> CGEJSV implements a preconditioned Jacobi SVD algorithm. It uses CGEQP3,
+*> CGEQRF, and CGELQF as preprocessors and preconditioners. Optionally, an
+*> additional row pivoting can be used as a preprocessor, which in some
+*> cases results in much higher accuracy. An example is matrix A with the
+*> structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
+*> diagonal matrices and C is well-conditioned matrix. In that case, complete
+*> pivoting in the first QR factorizations provides accuracy dependent on the
+*> condition number of C, and independent of D1, D2. Such higher accuracy is
+*> not completely understood theoretically, but it works well in practice.
+*> Further, if A can be written as A = B*D, with well-conditioned B and some
+*> diagonal D, then the high accuracy is guaranteed, both theoretically and
+*> in software, independent of D. For more details see [1], [2].
+*> The computational range for the singular values can be the full range
+*> ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
+*> & LAPACK routines called by CGEJSV are implemented to work in that range.
+*> If that is not the case, then the restriction for safe computation with
+*> the singular values in the range of normalized IEEE numbers is that the
+*> spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
+*> overflow. This code (CGEJSV) is best used in this restricted range,
+*> meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are
+*> returned as zeros. See JOBR for details on this.
+*> Further, this implementation is somewhat slower than the one described
+*> in [1,2] due to replacement of some non-LAPACK components, and because
+*> the choice of some tuning parameters in the iterative part (CGESVJ) is
+*> left to the implementer on a particular machine.
+*> The rank revealing QR factorization (in this code: CGEQP3) should be
+*> implemented as in [3]. We have a new version of CGEQP3 under development
+*> that is more robust than the current one in LAPACK, with a cleaner cut in
+*> rank deficient cases. It will be available in the SIGMA library [4].
+*> If M is much larger than N, it is obvious that the initial QRF with
+*> column pivoting can be preprocessed by the QRF without pivoting. That
+*> well known trick is not used in CGEJSV because in some cases heavy row
+*> weighting can be treated with complete pivoting. The overhead in cases
+*> M much larger than N is then only due to pivoting, but the benefits in
+*> terms of accuracy have prevailed. The implementer/user can incorporate
+*> this extra QRF step easily. The implementer can also improve data movement
+*> (matrix transpose, matrix copy, matrix transposed copy) - this
+*> implementation of CGEJSV uses only the simplest, naive data movement.
+*> \endverbatim
+*
+*> \par Contributor:
+* ==================
+*>
+*> Zlatko Drmac (Zagreb, Croatia)
+*
+*> \par References:
+* ================
+*>
+*> \verbatim
+*>
+*> [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
+*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
+*> LAPACK Working note 169.
+*> [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
+*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
+*> LAPACK Working note 170.
+*> [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
+*> factorization software - a case study.
+*> ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
+*> LAPACK Working note 176.
+*> [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
+*> QSVD, (H,K)-SVD computations.
+*> Department of Mathematics, University of Zagreb, 2008, 2016.
+*> \endverbatim
+*
+*> \par Bugs, examples and comments:
+* =================================
+*>
+*> Please report all bugs and send interesting examples and/or comments to
+*> drmac@math.hr. Thank you.
+*>
+* =====================================================================
+ SUBROUTINE CGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
+ $ M, N, A, LDA, SVA, U, LDU, V, LDV,
+ $ CWORK, LWORK, RWORK, LRWORK, IWORK, INFO )
+*
+* -- LAPACK computational routine (version 3.7.0) --
+* -- LAPACK is a software package provided by Univ. of Tennessee, --
+* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
+* December 2016
+*
+* .. Scalar Arguments ..
+ IMPLICIT NONE
+ INTEGER INFO, LDA, LDU, LDV, LWORK, LRWORK, M, N
+* ..
+* .. Array Arguments ..
+ COMPLEX A( LDA, * ), U( LDU, * ), V( LDV, * ), CWORK( LWORK )
+ REAL SVA( N ), RWORK( LRWORK )
+ INTEGER IWORK( * )
+ CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
+* ..
+*
+* ===========================================================================
+*
+* .. Local Parameters ..
+ REAL ZERO, ONE
+ PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
+ COMPLEX CZERO, CONE
+ PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ), CONE = ( 1.0E0, 0.0E0 ) )
+* ..
+* .. Local Scalars ..
+ COMPLEX CTEMP
+ REAL AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1, COND_OK,
+ $ CONDR1, CONDR2, ENTRA, ENTRAT, EPSLN, MAXPRJ, SCALEM,
+ $ SCONDA, SFMIN, SMALL, TEMP1, USCAL1, USCAL2, XSC
+ INTEGER IERR, N1, NR, NUMRANK, p, q, WARNING
+ LOGICAL ALMORT, DEFR, ERREST, GOSCAL, JRACC, KILL, LQUERY,
+ $ LSVEC, L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN, NOSCAL,
+ $ ROWPIV, RSVEC, TRANSP
+*
+ INTEGER OPTWRK, MINWRK, MINRWRK, MINIWRK
+ INTEGER LWCON, LWLQF, LWQP3, LWQRF, LWUNMLQ, LWUNMQR, LWUNMQRM,
+ $ LWSVDJ, LWSVDJV, LRWQP3, LRWCON, LRWSVDJ, IWOFF
+ INTEGER LWRK_CGELQF, LWRK_CGEQP3, LWRK_CGEQP3N, LWRK_CGEQRF,
+ $ LWRK_CGESVJ, LWRK_CGESVJV, LWRK_CGESVJU, LWRK_CUNMLQ,
+ $ LWRK_CUNMQR, LWRK_CUNMQRM
+* ..
+* .. Local Arrays
+ COMPLEX CDUMMY(1)
+ REAL RDUMMY(1)
+*
+* .. Intrinsic Functions ..
+ INTRINSIC ABS, CMPLX, CONJG, ALOG, MAX, MIN, REAL, NINT, SQRT
+* ..
+* .. External Functions ..
+ REAL SLAMCH, SCNRM2
+ INTEGER ISAMAX, ICAMAX
+ LOGICAL LSAME
+ EXTERNAL ISAMAX, ICAMAX, LSAME, SLAMCH, SCNRM2
+* ..
+* .. External Subroutines ..
+ EXTERNAL SLASSQ, CCOPY, CGELQF, CGEQP3, CGEQRF, CLACPY, CLAPMR,
+ $ CLASCL, SLASCL, CLASET, CLASSQ, CLASWP, CUNGQR, CUNMLQ,
+ $ CUNMQR, CPOCON, SSCAL, CSSCAL, CSWAP, CTRSM, CLACGV,
+ $ XERBLA
+*
+ EXTERNAL CGESVJ
+* ..
+*
+* Test the input arguments
+*
+ LSVEC = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' )
+ JRACC = LSAME( JOBV, 'J' )
+ RSVEC = LSAME( JOBV, 'V' ) .OR. JRACC
+ ROWPIV = LSAME( JOBA, 'F' ) .OR. LSAME( JOBA, 'G' )
+ L2RANK = LSAME( JOBA, 'R' )
+ L2ABER = LSAME( JOBA, 'A' )
+ ERREST = LSAME( JOBA, 'E' ) .OR. LSAME( JOBA, 'G' )
+ L2TRAN = LSAME( JOBT, 'T' ) .AND. ( M .EQ. N )
+ L2KILL = LSAME( JOBR, 'R' )
+ DEFR = LSAME( JOBR, 'N' )
+ L2PERT = LSAME( JOBP, 'P' )
+*
+ LQUERY = ( LWORK .EQ. -1 ) .OR. ( LRWORK .EQ. -1 )
+*
+ IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR.
+ $ ERREST .OR. LSAME( JOBA, 'C' ) )) THEN
+ INFO = - 1
+ ELSE IF ( .NOT.( LSVEC .OR. LSAME( JOBU, 'N' ) .OR.
+ $ ( LSAME( JOBU, 'W' ) .AND. RSVEC .AND. L2TRAN ) ) ) THEN
+ INFO = - 2
+ ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.
+ $ ( LSAME( JOBV, 'W' ) .AND. LSVEC .AND. L2TRAN ) ) ) THEN
+ INFO = - 3
+ ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) ) THEN
+ INFO = - 4
+ ELSE IF ( .NOT. ( LSAME(JOBT,'T') .OR. LSAME(JOBT,'N') ) ) THEN
+ INFO = - 5
+ ELSE IF ( .NOT. ( L2PERT .OR. LSAME( JOBP, 'N' ) ) ) THEN
+ INFO = - 6
+ ELSE IF ( M .LT. 0 ) THEN
+ INFO = - 7
+ ELSE IF ( ( N .LT. 0 ) .OR. ( N .GT. M ) ) THEN
+ INFO = - 8
+ ELSE IF ( LDA .LT. M ) THEN
+ INFO = - 10
+ ELSE IF ( LSVEC .AND. ( LDU .LT. M ) ) THEN
+ INFO = - 13
+ ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN
+ INFO = - 15
+ ELSE
+* #:)
+ INFO = 0
+ END IF
+*
+ IF ( INFO .EQ. 0 ) THEN
+* .. compute the minimal and the optimal workspace lengths
+* [[The expressions for computing the minimal and the optimal
+* values of LCWORK, LRWORK are written with a lot of redundancy and
+* can be simplified. However, this verbose form is useful for
+* maintenance and modifications of the code.]]
+*
+* .. minimal workspace length for CGEQP3 of an M x N matrix,
+* CGEQRF of an N x N matrix, CGELQF of an N x N matrix,
+* CUNMLQ for computing N x N matrix, CUNMQR for computing N x N
+* matrix, CUNMQR for computing M x N matrix, respectively.
+ LWQP3 = N+1
+ LWQRF = MAX( 1, N )
+ LWLQF = MAX( 1, N )
+ LWUNMLQ = MAX( 1, N )
+ LWUNMQR = MAX( 1, N )
+ LWUNMQRM = MAX( 1, M )
+* .. minimal workspace length for CPOCON of an N x N matrix
+ LWCON = 2 * N
+* .. minimal workspace length for CGESVJ of an N x N matrix,
+* without and with explicit accumulation of Jacobi rotations
+ LWSVDJ = MAX( 2 * N, 1 )
+ LWSVDJV = MAX( 2 * N, 1 )
+* .. minimal REAL workspace length for CGEQP3, CPOCON, CGESVJ
+ LRWQP3 = N
+ LRWCON = N
+ LRWSVDJ = N
+ IF ( LQUERY ) THEN
+ CALL CGEQP3( M, N, A, LDA, IWORK, CDUMMY, CDUMMY, -1,
+ $ RDUMMY, IERR )
+ LWRK_CGEQP3 = CDUMMY(1)
+ CALL CGEQRF( N, N, A, LDA, CDUMMY, CDUMMY,-1, IERR )
+ LWRK_CGEQRF = CDUMMY(1)
+ CALL CGELQF( N, N, A, LDA, CDUMMY, CDUMMY,-1, IERR )
+ LWRK_CGELQF = CDUMMY(1)
+ END IF
+ MINWRK = 2
+ OPTWRK = 2
+ MINIWRK = N
+ IF ( .NOT. (LSVEC .OR. RSVEC ) ) THEN
+* .. minimal and optimal sizes of the complex workspace if
+* only the singular values are requested
+ IF ( ERREST ) THEN
+ MINWRK = MAX( N+LWQP3, N**2+LWCON, N+LWQRF, LWSVDJ )
+ ELSE
+ MINWRK = MAX( N+LWQP3, N+LWQRF, LWSVDJ )
+ END IF
+ IF ( LQUERY ) THEN
+ CALL CGESVJ( 'L', 'N', 'N', N, N, A, LDA, SVA, N, V,
+ $ LDV, CDUMMY, -1, RDUMMY, -1, IERR )
+ LWRK_CGESVJ = CDUMMY(1)
+ IF ( ERREST ) THEN
+ OPTWRK = MAX( N+LWRK_CGEQP3, N**2+LWCON,
+ $ N+LWRK_CGEQRF, LWRK_CGESVJ )
+ ELSE
+ OPTWRK = MAX( N+LWRK_CGEQP3, N+LWRK_CGEQRF,
+ $ LWRK_CGESVJ )
+ END IF
+ END IF
+ IF ( L2TRAN .OR. ROWPIV ) THEN
+ IF ( ERREST ) THEN
+ MINRWRK = MAX( 7, 2*M, LRWQP3, LRWCON, LRWSVDJ )
+ ELSE
+ MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
+ END IF
+ ELSE
+ IF ( ERREST ) THEN
+ MINRWRK = MAX( 7, LRWQP3, LRWCON, LRWSVDJ )
+ ELSE
+ MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
+ END IF
+ END IF
+ IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
+ ELSE IF ( RSVEC .AND. (.NOT.LSVEC) ) THEN
+* .. minimal and optimal sizes of the complex workspace if the
+* singular values and the right singular vectors are requested
+ IF ( ERREST ) THEN
+ MINWRK = MAX( N+LWQP3, LWCON, LWSVDJ, N+LWLQF,
+ $ 2*N+LWQRF, N+LWSVDJ, N+LWUNMLQ )
+ ELSE
+ MINWRK = MAX( N+LWQP3, LWSVDJ, N+LWLQF, 2*N+LWQRF,
+ $ N+LWSVDJ, N+LWUNMLQ )
+ END IF
+ IF ( LQUERY ) THEN
+ CALL CGESVJ( 'L', 'U', 'N', N,N, U, LDU, SVA, N, A,
+ $ LDA, CDUMMY, -1, RDUMMY, -1, IERR )
+ LWRK_CGESVJ = CDUMMY(1)
+ CALL CUNMLQ( 'L', 'C', N, N, N, A, LDA, CDUMMY,
+ $ V, LDV, CDUMMY, -1, IERR )
+ LWRK_CUNMLQ = CDUMMY(1)
+ IF ( ERREST ) THEN
+ OPTWRK = MAX( N+LWRK_CGEQP3, LWCON, LWRK_CGESVJ,
+ $ N+LWRK_CGELQF, 2*N+LWRK_CGEQRF,
+ $ N+LWRK_CGESVJ, N+LWRK_CUNMLQ )
+ ELSE
+ OPTWRK = MAX( N+LWRK_CGEQP3, LWRK_CGESVJ,N+LWRK_CGELQF,
+ $ 2*N+LWRK_CGEQRF, N+LWRK_CGESVJ,
+ $ N+LWRK_CUNMLQ )
+ END IF
+ END IF
+ IF ( L2TRAN .OR. ROWPIV ) THEN
+ IF ( ERREST ) THEN
+ MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
+ ELSE
+ MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
+ END IF
+ ELSE
+ IF ( ERREST ) THEN
+ MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
+ ELSE
+ MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
+ END IF
+ END IF
+ IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
+ ELSE IF ( LSVEC .AND. (.NOT.RSVEC) ) THEN
+* .. minimal and optimal sizes of the complex workspace if the
+* singular values and the left singular vectors are requested
+ IF ( ERREST ) THEN
+ MINWRK = N + MAX( LWQP3,LWCON,N+LWQRF,LWSVDJ,LWUNMQRM )
+ ELSE
+ MINWRK = N + MAX( LWQP3, N+LWQRF, LWSVDJ, LWUNMQRM )
+ END IF
+ IF ( LQUERY ) THEN
+ CALL CGESVJ( 'L', 'U', 'N', N,N, U, LDU, SVA, N, A,
+ $ LDA, CDUMMY, -1, RDUMMY, -1, IERR )
+ LWRK_CGESVJ = CDUMMY(1)
+ CALL CUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
+ $ LDU, CDUMMY, -1, IERR )
+ LWRK_CUNMQRM = CDUMMY(1)
+ IF ( ERREST ) THEN
+ OPTWRK = N + MAX( LWRK_CGEQP3, LWCON, N+LWRK_CGEQRF,
+ $ LWRK_CGESVJ, LWRK_CUNMQRM )
+ ELSE
+ OPTWRK = N + MAX( LWRK_CGEQP3, N+LWRK_CGEQRF,
+ $ LWRK_CGESVJ, LWRK_CUNMQRM )
+ END IF
+ END IF
+ IF ( L2TRAN .OR. ROWPIV ) THEN
+ IF ( ERREST ) THEN
+ MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
+ ELSE
+ MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
+ END IF
+ ELSE
+ IF ( ERREST ) THEN
+ MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
+ ELSE
+ MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
+ END IF
+ END IF
+ IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
+ ELSE
+* .. minimal and optimal sizes of the complex workspace if the
+* full SVD is requested
+ IF ( .NOT. JRACC ) THEN
+ IF ( ERREST ) THEN
+ MINWRK = MAX( N+LWQP3, N+LWCON, 2*N+N**2+LWCON,
+ $ 2*N+LWQRF, 2*N+LWQP3,
+ $ 2*N+N**2+N+LWLQF, 2*N+N**2+N+N**2+LWCON,
+ $ 2*N+N**2+N+LWSVDJ, 2*N+N**2+N+LWSVDJV,
+ $ 2*N+N**2+N+LWUNMQR,2*N+N**2+N+LWUNMLQ,
+ $ N+N**2+LWSVDJ, N+LWUNMQRM )
+ ELSE
+ MINWRK = MAX( N+LWQP3, 2*N+N**2+LWCON,
+ $ 2*N+LWQRF, 2*N+LWQP3,
+ $ 2*N+N**2+N+LWLQF, 2*N+N**2+N+N**2+LWCON,
+ $ 2*N+N**2+N+LWSVDJ, 2*N+N**2+N+LWSVDJV,
+ $ 2*N+N**2+N+LWUNMQR,2*N+N**2+N+LWUNMLQ,
+ $ N+N**2+LWSVDJ, N+LWUNMQRM )
+ END IF
+ MINIWRK = MINIWRK + N
+ IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
+ ELSE
+ IF ( ERREST ) THEN
+ MINWRK = MAX( N+LWQP3, N+LWCON, 2*N+LWQRF,
+ $ 2*N+N**2+LWSVDJV, 2*N+N**2+N+LWUNMQR,
+ $ N+LWUNMQRM )
+ ELSE
+ MINWRK = MAX( N+LWQP3, 2*N+LWQRF,
+ $ 2*N+N**2+LWSVDJV, 2*N+N**2+N+LWUNMQR,
+ $ N+LWUNMQRM )
+ END IF
+ IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
+ END IF
+ IF ( LQUERY ) THEN
+ CALL CUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
+ $ LDU, CDUMMY, -1, IERR )
+ LWRK_CUNMQRM = CDUMMY(1)
+ CALL CUNMQR( 'L', 'N', N, N, N, A, LDA, CDUMMY, U,
+ $ LDU, CDUMMY, -1, IERR )
+ LWRK_CUNMQR = CDUMMY(1)
+ IF ( .NOT. JRACC ) THEN
+ CALL CGEQP3( N,N, A, LDA, IWORK, CDUMMY,CDUMMY, -1,
+ $ RDUMMY, IERR )
+ LWRK_CGEQP3N = CDUMMY(1)
+ CALL CGESVJ( 'L', 'U', 'N', N, N, U, LDU, SVA,
+ $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
+ LWRK_CGESVJ = CDUMMY(1)
+ CALL CGESVJ( 'U', 'U', 'N', N, N, U, LDU, SVA,
+ $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
+ LWRK_CGESVJU = CDUMMY(1)
+ CALL CGESVJ( 'L', 'U', 'V', N, N, U, LDU, SVA,
+ $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
+ LWRK_CGESVJV = CDUMMY(1)
+ CALL CUNMLQ( 'L', 'C', N, N, N, A, LDA, CDUMMY,
+ $ V, LDV, CDUMMY, -1, IERR )
+ LWRK_CUNMLQ = CDUMMY(1)
+ IF ( ERREST ) THEN
+ OPTWRK = MAX( N+LWRK_CGEQP3, N+LWCON,
+ $ 2*N+N**2+LWCON, 2*N+LWRK_CGEQRF,
+ $ 2*N+LWRK_CGEQP3N,
+ $ 2*N+N**2+N+LWRK_CGELQF,
+ $ 2*N+N**2+N+N**2+LWCON,
+ $ 2*N+N**2+N+LWRK_CGESVJ,
+ $ 2*N+N**2+N+LWRK_CGESVJV,
+ $ 2*N+N**2+N+LWRK_CUNMQR,
+ $ 2*N+N**2+N+LWRK_CUNMLQ,
+ $ N+N**2+LWRK_CGESVJU,
+ $ N+LWRK_CUNMQRM )
+ ELSE
+ OPTWRK = MAX( N+LWRK_CGEQP3,
+ $ 2*N+N**2+LWCON, 2*N+LWRK_CGEQRF,
+ $ 2*N+LWRK_CGEQP3N,
+ $ 2*N+N**2+N+LWRK_CGELQF,
+ $ 2*N+N**2+N+N**2+LWCON,
+ $ 2*N+N**2+N+LWRK_CGESVJ,
+ $ 2*N+N**2+N+LWRK_CGESVJV,
+ $ 2*N+N**2+N+LWRK_CUNMQR,
+ $ 2*N+N**2+N+LWRK_CUNMLQ,
+ $ N+N**2+LWRK_CGESVJU,
+ $ N+LWRK_CUNMQRM )
+ END IF
+ ELSE
+ CALL CGESVJ( 'L', 'U', 'V', N, N, U, LDU, SVA,
+ $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
+ LWRK_CGESVJV = CDUMMY(1)
+ CALL CUNMQR( 'L', 'N', N, N, N, CDUMMY, N, CDUMMY,
+ $ V, LDV, CDUMMY, -1, IERR )
+ LWRK_CUNMQR = CDUMMY(1)
+ CALL CUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
+ $ LDU, CDUMMY, -1, IERR )
+ LWRK_CUNMQRM = CDUMMY(1)
+ IF ( ERREST ) THEN
+ OPTWRK = MAX( N+LWRK_CGEQP3, N+LWCON,
+ $ 2*N+LWRK_CGEQRF, 2*N+N**2,
+ $ 2*N+N**2+LWRK_CGESVJV,
+ $ 2*N+N**2+N+LWRK_CUNMQR,N+LWRK_CUNMQRM )
+ ELSE
+ OPTWRK = MAX( N+LWRK_CGEQP3, 2*N+LWRK_CGEQRF,
+ $ 2*N+N**2, 2*N+N**2+LWRK_CGESVJV,
+ $ 2*N+N**2+N+LWRK_CUNMQR,
+ $ N+LWRK_CUNMQRM )
+ END IF
+ END IF
+ END IF
+ IF ( L2TRAN .OR. ROWPIV ) THEN
+ MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
+ ELSE
+ MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
+ END IF
+ END IF
+ MINWRK = MAX( 2, MINWRK )
+ OPTWRK = MAX( 2, OPTWRK )
+ IF ( LWORK .LT. MINWRK .AND. (.NOT.LQUERY) ) INFO = - 17
+ IF ( LRWORK .LT. MINRWRK .AND. (.NOT.LQUERY) ) INFO = - 19
+ END IF
+*
+ IF ( INFO .NE. 0 ) THEN
+* #:(
+ CALL XERBLA( 'CGEJSV', - INFO )
+ RETURN
+ ELSE IF ( LQUERY ) THEN
+ CWORK(1) = OPTWRK
+ CWORK(2) = MINWRK
+ RWORK(1) = MINRWRK
+ IWORK(1) = MAX( 4, MINIWRK )
+ RETURN
+ END IF
+*
+* Quick return for void matrix (Y3K safe)
+* #:)
+ IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) THEN
+ IWORK(1:4) = 0
+ RWORK(1:7) = 0
+ RETURN
+ ENDIF
+*
+* Determine whether the matrix U should be M x N or M x M
+*
+ IF ( LSVEC ) THEN
+ N1 = N
+ IF ( LSAME( JOBU, 'F' ) ) N1 = M
+ END IF
+*
+* Set numerical parameters
+*
+*! NOTE: Make sure SLAMCH() does not fail on the target architecture.
+*
+ EPSLN = SLAMCH('Epsilon')
+ SFMIN = SLAMCH('SafeMinimum')
+ SMALL = SFMIN / EPSLN
+ BIG = SLAMCH('O')
+* BIG = ONE / SFMIN
+*
+* Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N
+*
+*(!) If necessary, scale SVA() to protect the largest norm from
+* overflow. It is possible that this scaling pushes the smallest
+* column norm left from the underflow threshold (extreme case).
+*
+ SCALEM = ONE / SQRT(REAL(M)*REAL(N))
+ NOSCAL = .TRUE.
+ GOSCAL = .TRUE.
+ DO 1874 p = 1, N
+ AAPP = ZERO
+ AAQQ = ONE
+ CALL CLASSQ( M, A(1,p), 1, AAPP, AAQQ )
+ IF ( AAPP .GT. BIG ) THEN
+ INFO = - 9
+ CALL XERBLA( 'CGEJSV', -INFO )
+ RETURN
+ END IF
+ AAQQ = SQRT(AAQQ)
+ IF ( ( AAPP .LT. (BIG / AAQQ) ) .AND. NOSCAL ) THEN
+ SVA(p) = AAPP * AAQQ
+ ELSE
+ NOSCAL = .FALSE.
+ SVA(p) = AAPP * ( AAQQ * SCALEM )
+ IF ( GOSCAL ) THEN
+ GOSCAL = .FALSE.
+ CALL SSCAL( p-1, SCALEM, SVA, 1 )
+ END IF
+ END IF
+ 1874 CONTINUE
+*
+ IF ( NOSCAL ) SCALEM = ONE
+*
+ AAPP = ZERO
+ AAQQ = BIG
+ DO 4781 p = 1, N
+ AAPP = MAX( AAPP, SVA(p) )
+ IF ( SVA(p) .NE. ZERO ) AAQQ = MIN( AAQQ, SVA(p) )
+ 4781 CONTINUE
+*
+* Quick return for zero M x N matrix
+* #:)
+ IF ( AAPP .EQ. ZERO ) THEN
+ IF ( LSVEC ) CALL CLASET( 'G', M, N1, CZERO, CONE, U, LDU )
+ IF ( RSVEC ) CALL CLASET( 'G', N, N, CZERO, CONE, V, LDV )
+ RWORK(1) = ONE
+ RWORK(2) = ONE
+ IF ( ERREST ) RWORK(3) = ONE
+ IF ( LSVEC .AND. RSVEC ) THEN
+ RWORK(4) = ONE
+ RWORK(5) = ONE
+ END IF
+ IF ( L2TRAN ) THEN
+ RWORK(6) = ZERO
+ RWORK(7) = ZERO
+ END IF
+ IWORK(1) = 0
+ IWORK(2) = 0
+ IWORK(3) = 0
+ IWORK(4) = -1
+ RETURN
+ END IF
+*
+* Issue warning if denormalized column norms detected. Override the
+* high relative accuracy request. Issue licence to kill nonzero columns
+* (set them to zero) whose norm is less than sigma_max / BIG (roughly).
+* #:(
+ WARNING = 0
+ IF ( AAQQ .LE. SFMIN ) THEN
+ L2RANK = .TRUE.
+ L2KILL = .TRUE.
+ WARNING = 1
+ END IF
+*
+* Quick return for one-column matrix
+* #:)
+ IF ( N .EQ. 1 ) THEN
+*
+ IF ( LSVEC ) THEN
+ CALL CLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR )
+ CALL CLACPY( 'A', M, 1, A, LDA, U, LDU )
+* computing all M left singular vectors of the M x 1 matrix
+ IF ( N1 .NE. N ) THEN
+ CALL CGEQRF( M, N, U,LDU, CWORK, CWORK(N+1),LWORK-N,IERR )
+ CALL CUNGQR( M,N1,1, U,LDU,CWORK,CWORK(N+1),LWORK-N,IERR )
+ CALL CCOPY( M, A(1,1), 1, U(1,1), 1 )
+ END IF
+ END IF
+ IF ( RSVEC ) THEN
+ V(1,1) = CONE
+ END IF
+ IF ( SVA(1) .LT. (BIG*SCALEM) ) THEN
+ SVA(1) = SVA(1) / SCALEM
+ SCALEM = ONE
+ END IF
+ RWORK(1) = ONE / SCALEM
+ RWORK(2) = ONE
+ IF ( SVA(1) .NE. ZERO ) THEN
+ IWORK(1) = 1
+ IF ( ( SVA(1) / SCALEM) .GE. SFMIN ) THEN
+ IWORK(2) = 1
+ ELSE
+ IWORK(2) = 0
+ END IF
+ ELSE
+ IWORK(1) = 0
+ IWORK(2) = 0
+ END IF
+ IWORK(3) = 0
+ IWORK(4) = -1
+ IF ( ERREST ) RWORK(3) = ONE
+ IF ( LSVEC .AND. RSVEC ) THEN
+ RWORK(4) = ONE
+ RWORK(5) = ONE
+ END IF
+ IF ( L2TRAN ) THEN
+ RWORK(6) = ZERO
+ RWORK(7) = ZERO
+ END IF
+ RETURN
+*
+ END IF
+*
+ TRANSP = .FALSE.
+*
+ AATMAX = -ONE
+ AATMIN = BIG
+ IF ( ROWPIV .OR. L2TRAN ) THEN
+*
+* Compute the row norms, needed to determine row pivoting sequence
+* (in the case of heavily row weighted A, row pivoting is strongly
+* advised) and to collect information needed to compare the
+* structures of A * A^* and A^* * A (in the case L2TRAN.EQ..TRUE.).
+*
+ IF ( L2TRAN ) THEN
+ DO 1950 p = 1, M
+ XSC = ZERO
+ TEMP1 = ONE
+ CALL CLASSQ( N, A(p,1), LDA, XSC, TEMP1 )
+* CLASSQ gets both the ell_2 and the ell_infinity norm
+* in one pass through the vector
+ RWORK(M+p) = XSC * SCALEM
+ RWORK(p) = XSC * (SCALEM*SQRT(TEMP1))
+ AATMAX = MAX( AATMAX, RWORK(p) )
+ IF (RWORK(p) .NE. ZERO)
+ $ AATMIN = MIN(AATMIN,RWORK(p))
+ 1950 CONTINUE
+ ELSE
+ DO 1904 p = 1, M
+ RWORK(M+p) = SCALEM*ABS( A(p,ICAMAX(N,A(p,1),LDA)) )
+ AATMAX = MAX( AATMAX, RWORK(M+p) )
+ AATMIN = MIN( AATMIN, RWORK(M+p) )
+ 1904 CONTINUE
+ END IF
+*
+ END IF
+*
+* For square matrix A try to determine whether A^* would be better
+* input for the preconditioned Jacobi SVD, with faster convergence.
+* The decision is based on an O(N) function of the vector of column
+* and row norms of A, based on the Shannon entropy. This should give
+* the right choice in most cases when the difference actually matters.
+* It may fail and pick the slower converging side.
+*
+ ENTRA = ZERO
+ ENTRAT = ZERO
+ IF ( L2TRAN ) THEN
+*
+ XSC = ZERO
+ TEMP1 = ONE
+ CALL SLASSQ( N, SVA, 1, XSC, TEMP1 )
+ TEMP1 = ONE / TEMP1
+*
+ ENTRA = ZERO
+ DO 1113 p = 1, N
+ BIG1 = ( ( SVA(p) / XSC )**2 ) * TEMP1
+ IF ( BIG1 .NE. ZERO ) ENTRA = ENTRA + BIG1 * ALOG(BIG1)
+ 1113 CONTINUE
+ ENTRA = - ENTRA / ALOG(REAL(N))
+*
+* Now, SVA().^2/Trace(A^* * A) is a point in the probability simplex.
+* It is derived from the diagonal of A^* * A. Do the same with the
+* diagonal of A * A^*, compute the entropy of the corresponding
+* probability distribution. Note that A * A^* and A^* * A have the
+* same trace.
+*
+ ENTRAT = ZERO
+ DO 1114 p = 1, M
+ BIG1 = ( ( RWORK(p) / XSC )**2 ) * TEMP1
+ IF ( BIG1 .NE. ZERO ) ENTRAT = ENTRAT + BIG1 * ALOG(BIG1)
+ 1114 CONTINUE
+ ENTRAT = - ENTRAT / ALOG(REAL(M))
+*
+* Analyze the entropies and decide A or A^*. Smaller entropy
+* usually means better input for the algorithm.
+*
+ TRANSP = ( ENTRAT .LT. ENTRA )
+*
+* If A^* is better than A, take the adjoint of A. This is allowed
+* only for square matrices, M=N.
+ IF ( TRANSP ) THEN
+* In an optimal implementation, this trivial transpose
+* should be replaced with faster transpose.
+ DO 1115 p = 1, N - 1
+ A(p,p) = CONJG(A(p,p))
+ DO 1116 q = p + 1, N
+ CTEMP = CONJG(A(q,p))
+ A(q,p) = CONJG(A(p,q))
+ A(p,q) = CTEMP
+ 1116 CONTINUE
+ 1115 CONTINUE
+ A(N,N) = CONJG(A(N,N))
+ DO 1117 p = 1, N
+ RWORK(M+p) = SVA(p)
+ SVA(p) = RWORK(p)
+* previously computed row 2-norms are now column 2-norms
+* of the transposed matrix
+ 1117 CONTINUE
+ TEMP1 = AAPP
+ AAPP = AATMAX
+ AATMAX = TEMP1
+ TEMP1 = AAQQ
+ AAQQ = AATMIN
+ AATMIN = TEMP1
+ KILL = LSVEC
+ LSVEC = RSVEC
+ RSVEC = KILL
+ IF ( LSVEC ) N1 = N
+*
+ ROWPIV = .TRUE.
+ END IF
+*
+ END IF
+* END IF L2TRAN
+*
+* Scale the matrix so that its maximal singular value remains less
+* than SQRT(BIG) -- the matrix is scaled so that its maximal column
+* has Euclidean norm equal to SQRT(BIG/N). The only reason to keep
+* SQRT(BIG) instead of BIG is the fact that CGEJSV uses LAPACK and
+* BLAS routines that, in some implementations, are not capable of
+* working in the full interval [SFMIN,BIG] and that they may provoke
+* overflows in the intermediate results. If the singular values spread
+* from SFMIN to BIG, then CGESVJ will compute them. So, in that case,
+* one should use CGESVJ instead of CGEJSV.
+ BIG1 = SQRT( BIG )
+ TEMP1 = SQRT( BIG / REAL(N) )
+* >> for future updates: allow bigger range, i.e. the largest column
+* will be allowed up to BIG/N and CGESVJ will do the rest. However, for
+* this all other (LAPACK) components must allow such a range.
+* TEMP1 = BIG/REAL(N)
+* TEMP1 = BIG * EPSLN this should 'almost' work with current LAPACK components
+ CALL SLASCL( 'G', 0, 0, AAPP, TEMP1, N, 1, SVA, N, IERR )
+ IF ( AAQQ .GT. (AAPP * SFMIN) ) THEN
+ AAQQ = ( AAQQ / AAPP ) * TEMP1
+ ELSE
+ AAQQ = ( AAQQ * TEMP1 ) / AAPP
+ END IF
+ TEMP1 = TEMP1 * SCALEM
+ CALL CLASCL( 'G', 0, 0, AAPP, TEMP1, M, N, A, LDA, IERR )
+*
+* To undo scaling at the end of this procedure, multiply the
+* computed singular values with USCAL2 / USCAL1.
+*
+ USCAL1 = TEMP1
+ USCAL2 = AAPP
+*
+ IF ( L2KILL ) THEN
+* L2KILL enforces computation of nonzero singular values in
+* the restricted range of condition number of the initial A,
+* sigma_max(A) / sigma_min(A) approx. SQRT(BIG)/SQRT(SFMIN).
+ XSC = SQRT( SFMIN )
+ ELSE
+ XSC = SMALL
+*
+* Now, if the condition number of A is too big,
+* sigma_max(A) / sigma_min(A) .GT. SQRT(BIG/N) * EPSLN / SFMIN,
+* as a precaution measure, the full SVD is computed using CGESVJ
+* with accumulated Jacobi rotations. This provides numerically
+* more robust computation, at the cost of slightly increased run
+* time. Depending on the concrete implementation of BLAS and LAPACK
+* (i.e. how they behave in presence of extreme ill-conditioning) the
+* implementor may decide to remove this switch.
+ IF ( ( AAQQ.LT.SQRT(SFMIN) ) .AND. LSVEC .AND. RSVEC ) THEN
+ JRACC = .TRUE.
+ END IF
+*
+ END IF
+ IF ( AAQQ .LT. XSC ) THEN
+ DO 700 p = 1, N
+ IF ( SVA(p) .LT. XSC ) THEN
+ CALL CLASET( 'A', M, 1, CZERO, CZERO, A(1,p), LDA )
+ SVA(p) = ZERO
+ END IF
+ 700 CONTINUE
+ END IF
+*
+* Preconditioning using QR factorization with pivoting
+*
+ IF ( ROWPIV ) THEN
+* Optional row permutation (Bjoerck row pivoting):
+* A result by Cox and Higham shows that the Bjoerck's
+* row pivoting combined with standard column pivoting
+* has similar effect as Powell-Reid complete pivoting.
+* The ell-infinity norms of A are made nonincreasing.
+ IF ( ( LSVEC .AND. RSVEC ) .AND. .NOT.( JRACC ) ) THEN
+ IWOFF = 2*N
+ ELSE
+ IWOFF = N
+ END IF
+ DO 1952 p = 1, M - 1
+ q = ISAMAX( M-p+1, RWORK(M+p), 1 ) + p - 1
+ IWORK(IWOFF+p) = q
+ IF ( p .NE. q ) THEN
+ TEMP1 = RWORK(M+p)
+ RWORK(M+p) = RWORK(M+q)
+ RWORK(M+q) = TEMP1
+ END IF
+ 1952 CONTINUE
+ CALL CLASWP( N, A, LDA, 1, M-1, IWORK(IWOFF+1), 1 )
+ END IF
+*
+* End of the preparation phase (scaling, optional sorting and
+* transposing, optional flushing of small columns).
+*
+* Preconditioning
+*
+* If the full SVD is needed, the right singular vectors are computed
+* from a matrix equation, and for that we need theoretical analysis
+* of the Businger-Golub pivoting. So we use CGEQP3 as the first RR QRF.
+* In all other cases the first RR QRF can be chosen by other criteria
+* (eg speed by replacing global with restricted window pivoting, such
+* as in xGEQPX from TOMS # 782). Good results will be obtained using
+* xGEQPX with properly (!) chosen numerical parameters.
+* Any improvement of CGEQP3 improves overal performance of CGEJSV.
+*
+* A * P1 = Q1 * [ R1^* 0]^*:
+ DO 1963 p = 1, N
+* .. all columns are free columns
+ IWORK(p) = 0
+ 1963 CONTINUE
+ CALL CGEQP3( M, N, A, LDA, IWORK, CWORK, CWORK(N+1), LWORK-N,
+ $ RWORK, IERR )
+*
+* The upper triangular matrix R1 from the first QRF is inspected for
+* rank deficiency and possibilities for deflation, or possible
+* ill-conditioning. Depending on the user specified flag L2RANK,
+* the procedure explores possibilities to reduce the numerical
+* rank by inspecting the computed upper triangular factor. If
+* L2RANK or L2ABER are up, then CGEJSV will compute the SVD of
+* A + dA, where ||dA|| <= f(M,N)*EPSLN.
+*
+ NR = 1
+ IF ( L2ABER ) THEN
+* Standard absolute error bound suffices. All sigma_i with
+* sigma_i < N*EPSLN*||A|| are flushed to zero. This is an
+* agressive enforcement of lower numerical rank by introducing a
+* backward error of the order of N*EPSLN*||A||.
+ TEMP1 = SQRT(REAL(N))*EPSLN
+ DO 3001 p = 2, N
+ IF ( ABS(A(p,p)) .GE. (TEMP1*ABS(A(1,1))) ) THEN
+ NR = NR + 1
+ ELSE
+ GO TO 3002
+ END IF
+ 3001 CONTINUE
+ 3002 CONTINUE
+ ELSE IF ( L2RANK ) THEN
+* .. similarly as above, only slightly more gentle (less agressive).
+* Sudden drop on the diagonal of R1 is used as the criterion for
+* close-to-rank-defficient.
+ TEMP1 = SQRT(SFMIN)
+ DO 3401 p = 2, N
+ IF ( ( ABS(A(p,p)) .LT. (EPSLN*ABS(A(p-1,p-1))) ) .OR.
+ $ ( ABS(A(p,p)) .LT. SMALL ) .OR.
+ $ ( L2KILL .AND. (ABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3402
+ NR = NR + 1
+ 3401 CONTINUE
+ 3402 CONTINUE
+*
+ ELSE
+* The goal is high relative accuracy. However, if the matrix
+* has high scaled condition number the relative accuracy is in
+* general not feasible. Later on, a condition number estimator
+* will be deployed to estimate the scaled condition number.
+* Here we just remove the underflowed part of the triangular
+* factor. This prevents the situation in which the code is
+* working hard to get the accuracy not warranted by the data.
+ TEMP1 = SQRT(SFMIN)
+ DO 3301 p = 2, N
+ IF ( ( ABS(A(p,p)) .LT. SMALL ) .OR.
+ $ ( L2KILL .AND. (ABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3302
+ NR = NR + 1
+ 3301 CONTINUE
+ 3302 CONTINUE
+*
+ END IF
+*
+ ALMORT = .FALSE.
+ IF ( NR .EQ. N ) THEN
+ MAXPRJ = ONE
+ DO 3051 p = 2, N
+ TEMP1 = ABS(A(p,p)) / SVA(IWORK(p))
+ MAXPRJ = MIN( MAXPRJ, TEMP1 )
+ 3051 CONTINUE
+ IF ( MAXPRJ**2 .GE. ONE - REAL(N)*EPSLN ) ALMORT = .TRUE.
+ END IF
+*
+*
+ SCONDA = - ONE
+ CONDR1 = - ONE
+ CONDR2 = - ONE
+*
+ IF ( ERREST ) THEN
+ IF ( N .EQ. NR ) THEN
+ IF ( RSVEC ) THEN
+* .. V is available as workspace
+ CALL CLACPY( 'U', N, N, A, LDA, V, LDV )
+ DO 3053 p = 1, N
+ TEMP1 = SVA(IWORK(p))
+ CALL CSSCAL( p, ONE/TEMP1, V(1,p), 1 )
+ 3053 CONTINUE
+ IF ( LSVEC )THEN
+ CALL CPOCON( 'U', N, V, LDV, ONE, TEMP1,
+ $ CWORK(N+1), RWORK, IERR )
+ ELSE
+ CALL CPOCON( 'U', N, V, LDV, ONE, TEMP1,
+ $ CWORK, RWORK, IERR )
+ END IF
+*
+ ELSE IF ( LSVEC ) THEN
+* .. U is available as workspace
+ CALL CLACPY( 'U', N, N, A, LDA, U, LDU )
+ DO 3054 p = 1, N
+ TEMP1 = SVA(IWORK(p))
+ CALL CSSCAL( p, ONE/TEMP1, U(1,p), 1 )
+ 3054 CONTINUE
+ CALL CPOCON( 'U', N, U, LDU, ONE, TEMP1,
+ $ CWORK(N+1), RWORK, IERR )
+ ELSE
+ CALL CLACPY( 'U', N, N, A, LDA, CWORK, N )
+*[] CALL CLACPY( 'U', N, N, A, LDA, CWORK(N+1), N )
+* Change: here index shifted by N to the left, CWORK(1:N)
+* not needed for SIGMA only computation
+ DO 3052 p = 1, N
+ TEMP1 = SVA(IWORK(p))
+*[] CALL CSSCAL( p, ONE/TEMP1, CWORK(N+(p-1)*N+1), 1 )
+ CALL CSSCAL( p, ONE/TEMP1, CWORK((p-1)*N+1), 1 )
+ 3052 CONTINUE
+* .. the columns of R are scaled to have unit Euclidean lengths.
+*[] CALL CPOCON( 'U', N, CWORK(N+1), N, ONE, TEMP1,
+*[] $ CWORK(N+N*N+1), RWORK, IERR )
+ CALL CPOCON( 'U', N, CWORK, N, ONE, TEMP1,
+ $ CWORK(N*N+1), RWORK, IERR )
+*
+ END IF
+ IF ( TEMP1 .NE. ZERO ) THEN
+ SCONDA = ONE / SQRT(TEMP1)
+ ELSE
+ SCONDA = - ONE
+ END IF
+* SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
+* N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
+ ELSE
+ SCONDA = - ONE
+ END IF
+ END IF
+*
+ L2PERT = L2PERT .AND. ( ABS( A(1,1)/A(NR,NR) ) .GT. SQRT(BIG1) )
+* If there is no violent scaling, artificial perturbation is not needed.
+*
+* Phase 3:
+*
+ IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN
+*
+* Singular Values only
+*
+* .. transpose A(1:NR,1:N)
+ DO 1946 p = 1, MIN( N-1, NR )
+ CALL CCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )
+ CALL CLACGV( N-p+1, A(p,p), 1 )
+ 1946 CONTINUE
+ IF ( NR .EQ. N ) A(N,N) = CONJG(A(N,N))
+*
+* The following two DO-loops introduce small relative perturbation
+* into the strict upper triangle of the lower triangular matrix.
+* Small entries below the main diagonal are also changed.
+* This modification is useful if the computing environment does not
+* provide/allow FLUSH TO ZERO underflow, for it prevents many
+* annoying denormalized numbers in case of strongly scaled matrices.
+* The perturbation is structured so that it does not introduce any
+* new perturbation of the singular values, and it does not destroy
+* the job done by the preconditioner.
+* The licence for this perturbation is in the variable L2PERT, which
+* should be .FALSE. if FLUSH TO ZERO underflow is active.
+*
+ IF ( .NOT. ALMORT ) THEN
+*
+ IF ( L2PERT ) THEN
+* XSC = SQRT(SMALL)
+ XSC = EPSLN / REAL(N)
+ DO 4947 q = 1, NR
+ CTEMP = CMPLX(XSC*ABS(A(q,q)),ZERO)
+ DO 4949 p = 1, N
+ IF ( ( (p.GT.q) .AND. (ABS(A(p,q)).LE.TEMP1) )
+ $ .OR. ( p .LT. q ) )
+* $ A(p,q) = TEMP1 * ( A(p,q) / ABS(A(p,q)) )
+ $ A(p,q) = CTEMP
+ 4949 CONTINUE
+ 4947 CONTINUE
+ ELSE
+ CALL CLASET( 'U', NR-1,NR-1, CZERO,CZERO, A(1,2),LDA )
+ END IF
+*
+* .. second preconditioning using the QR factorization
+*
+ CALL CGEQRF( N,NR, A,LDA, CWORK, CWORK(N+1),LWORK-N, IERR )
+*
+* .. and transpose upper to lower triangular
+ DO 1948 p = 1, NR - 1
+ CALL CCOPY( NR-p, A(p,p+1), LDA, A(p+1,p), 1 )
+ CALL CLACGV( NR-p+1, A(p,p), 1 )
+ 1948 CONTINUE
+*
+ END IF
+*
+* Row-cyclic Jacobi SVD algorithm with column pivoting
+*
+* .. again some perturbation (a "background noise") is added
+* to drown denormals
+ IF ( L2PERT ) THEN
+* XSC = SQRT(SMALL)
+ XSC = EPSLN / REAL(N)
+ DO 1947 q = 1, NR
+ CTEMP = CMPLX(XSC*ABS(A(q,q)),ZERO)
+ DO 1949 p = 1, NR
+ IF ( ( (p.GT.q) .AND. (ABS(A(p,q)).LE.TEMP1) )
+ $ .OR. ( p .LT. q ) )
+* $ A(p,q) = TEMP1 * ( A(p,q) / ABS(A(p,q)) )
+ $ A(p,q) = CTEMP
+ 1949 CONTINUE
+ 1947 CONTINUE
+ ELSE
+ CALL CLASET( 'U', NR-1, NR-1, CZERO, CZERO, A(1,2), LDA )
+ END IF
+*
+* .. and one-sided Jacobi rotations are started on a lower
+* triangular matrix (plus perturbation which is ignored in
+* the part which destroys triangular form (confusing?!))
+*
+ CALL CGESVJ( 'L', 'N', 'N', NR, NR, A, LDA, SVA,
+ $ N, V, LDV, CWORK, LWORK, RWORK, LRWORK, INFO )
+*
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+*
+*
+ ELSE IF ( ( RSVEC .AND. ( .NOT. LSVEC ) .AND. ( .NOT. JRACC ) )
+ $ .OR.
+ $ ( JRACC .AND. ( .NOT. LSVEC ) .AND. ( NR .NE. N ) ) ) THEN
+*
+* -> Singular Values and Right Singular Vectors <-
+*
+ IF ( ALMORT ) THEN
+*
+* .. in this case NR equals N
+ DO 1998 p = 1, NR
+ CALL CCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
+ CALL CLACGV( N-p+1, V(p,p), 1 )
+ 1998 CONTINUE
+ CALL CLASET( 'U', NR-1,NR-1, CZERO, CZERO, V(1,2), LDV )
+*
+ CALL CGESVJ( 'L','U','N', N, NR, V, LDV, SVA, NR, A, LDA,
+ $ CWORK, LWORK, RWORK, LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+
+ ELSE
+*
+* .. two more QR factorizations ( one QRF is not enough, two require
+* accumulated product of Jacobi rotations, three are perfect )
+*
+ CALL CLASET( 'L', NR-1,NR-1, CZERO, CZERO, A(2,1), LDA )
+ CALL CGELQF( NR,N, A, LDA, CWORK, CWORK(N+1), LWORK-N, IERR)
+ CALL CLACPY( 'L', NR, NR, A, LDA, V, LDV )
+ CALL CLASET( 'U', NR-1,NR-1, CZERO, CZERO, V(1,2), LDV )
+ CALL CGEQRF( NR, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
+ $ LWORK-2*N, IERR )
+ DO 8998 p = 1, NR
+ CALL CCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )
+ CALL CLACGV( NR-p+1, V(p,p), 1 )
+ 8998 CONTINUE
+ CALL CLASET('U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV)
+*
+ CALL CGESVJ( 'L', 'U','N', NR, NR, V,LDV, SVA, NR, U,
+ $ LDU, CWORK(N+1), LWORK-N, RWORK, LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+ IF ( NR .LT. N ) THEN
+ CALL CLASET( 'A',N-NR, NR, CZERO,CZERO, V(NR+1,1), LDV )
+ CALL CLASET( 'A',NR, N-NR, CZERO,CZERO, V(1,NR+1), LDV )
+ CALL CLASET( 'A',N-NR,N-NR,CZERO,CONE, V(NR+1,NR+1),LDV )
+ END IF
+*
+ CALL CUNMLQ( 'L', 'C', N, N, NR, A, LDA, CWORK,
+ $ V, LDV, CWORK(N+1), LWORK-N, IERR )
+*
+ END IF
+* .. permute the rows of V
+* DO 8991 p = 1, N
+* CALL CCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA )
+* 8991 CONTINUE
+* CALL CLACPY( 'All', N, N, A, LDA, V, LDV )
+ CALL CLAPMR( .FALSE., N, N, V, LDV, IWORK )
+*
+ IF ( TRANSP ) THEN
+ CALL CLACPY( 'A', N, N, V, LDV, U, LDU )
+ END IF
+*
+ ELSE IF ( JRACC .AND. (.NOT. LSVEC) .AND. ( NR.EQ. N ) ) THEN
+*
+ CALL CLASET( 'L', N-1,N-1, CZERO, CZERO, A(2,1), LDA )
+*
+ CALL CGESVJ( 'U','N','V', N, N, A, LDA, SVA, N, V, LDV,
+ $ CWORK, LWORK, RWORK, LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+ CALL CLAPMR( .FALSE., N, N, V, LDV, IWORK )
+*
+ ELSE IF ( LSVEC .AND. ( .NOT. RSVEC ) ) THEN
+*
+* .. Singular Values and Left Singular Vectors ..
+*
+* .. second preconditioning step to avoid need to accumulate
+* Jacobi rotations in the Jacobi iterations.
+ DO 1965 p = 1, NR
+ CALL CCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 )
+ CALL CLACGV( N-p+1, U(p,p), 1 )
+ 1965 CONTINUE
+ CALL CLASET( 'U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
+*
+ CALL CGEQRF( N, NR, U, LDU, CWORK(N+1), CWORK(2*N+1),
+ $ LWORK-2*N, IERR )
+*
+ DO 1967 p = 1, NR - 1
+ CALL CCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )
+ CALL CLACGV( N-p+1, U(p,p), 1 )
+ 1967 CONTINUE
+ CALL CLASET( 'U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
+*
+ CALL CGESVJ( 'L', 'U', 'N', NR,NR, U, LDU, SVA, NR, A,
+ $ LDA, CWORK(N+1), LWORK-N, RWORK, LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+*
+ IF ( NR .LT. M ) THEN
+ CALL CLASET( 'A', M-NR, NR,CZERO, CZERO, U(NR+1,1), LDU )
+ IF ( NR .LT. N1 ) THEN
+ CALL CLASET( 'A',NR, N1-NR, CZERO, CZERO, U(1,NR+1),LDU )
+ CALL CLASET( 'A',M-NR,N1-NR,CZERO,CONE,U(NR+1,NR+1),LDU )
+ END IF
+ END IF
+*
+ CALL CUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
+ $ LDU, CWORK(N+1), LWORK-N, IERR )
+*
+ IF ( ROWPIV )
+ $ CALL CLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
+*
+ DO 1974 p = 1, N1
+ XSC = ONE / SCNRM2( M, U(1,p), 1 )
+ CALL CSSCAL( M, XSC, U(1,p), 1 )
+ 1974 CONTINUE
+*
+ IF ( TRANSP ) THEN
+ CALL CLACPY( 'A', N, N, U, LDU, V, LDV )
+ END IF
+*
+ ELSE
+*
+* .. Full SVD ..
+*
+ IF ( .NOT. JRACC ) THEN
+*
+ IF ( .NOT. ALMORT ) THEN
+*
+* Second Preconditioning Step (QRF [with pivoting])
+* Note that the composition of TRANSPOSE, QRF and TRANSPOSE is
+* equivalent to an LQF CALL. Since in many libraries the QRF
+* seems to be better optimized than the LQF, we do explicit
+* transpose and use the QRF. This is subject to changes in an
+* optimized implementation of CGEJSV.
+*
+ DO 1968 p = 1, NR
+ CALL CCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
+ CALL CLACGV( N-p+1, V(p,p), 1 )
+ 1968 CONTINUE
+*
+* .. the following two loops perturb small entries to avoid
+* denormals in the second QR factorization, where they are
+* as good as zeros. This is done to avoid painfully slow
+* computation with denormals. The relative size of the perturbation
+* is a parameter that can be changed by the implementer.
+* This perturbation device will be obsolete on machines with
+* properly implemented arithmetic.
+* To switch it off, set L2PERT=.FALSE. To remove it from the
+* code, remove the action under L2PERT=.TRUE., leave the ELSE part.
+* The following two loops should be blocked and fused with the
+* transposed copy above.
+*
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL)
+ DO 2969 q = 1, NR
+ CTEMP = CMPLX(XSC*ABS( V(q,q) ),ZERO)
+ DO 2968 p = 1, N
+ IF ( ( p .GT. q ) .AND. ( ABS(V(p,q)) .LE. TEMP1 )
+ $ .OR. ( p .LT. q ) )
+* $ V(p,q) = TEMP1 * ( V(p,q) / ABS(V(p,q)) )
+ $ V(p,q) = CTEMP
+ IF ( p .LT. q ) V(p,q) = - V(p,q)
+ 2968 CONTINUE
+ 2969 CONTINUE
+ ELSE
+ CALL CLASET( 'U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV )
+ END IF
+*
+* Estimate the row scaled condition number of R1
+* (If R1 is rectangular, N > NR, then the condition number
+* of the leading NR x NR submatrix is estimated.)
+*
+ CALL CLACPY( 'L', NR, NR, V, LDV, CWORK(2*N+1), NR )
+ DO 3950 p = 1, NR
+ TEMP1 = SCNRM2(NR-p+1,CWORK(2*N+(p-1)*NR+p),1)
+ CALL CSSCAL(NR-p+1,ONE/TEMP1,CWORK(2*N+(p-1)*NR+p),1)
+ 3950 CONTINUE
+ CALL CPOCON('L',NR,CWORK(2*N+1),NR,ONE,TEMP1,
+ $ CWORK(2*N+NR*NR+1),RWORK,IERR)
+ CONDR1 = ONE / SQRT(TEMP1)
+* .. here need a second oppinion on the condition number
+* .. then assume worst case scenario
+* R1 is OK for inverse <=> CONDR1 .LT. REAL(N)
+* more conservative <=> CONDR1 .LT. SQRT(REAL(N))
+*
+ COND_OK = SQRT(SQRT(REAL(NR)))
+*[TP] COND_OK is a tuning parameter.
+*
+ IF ( CONDR1 .LT. COND_OK ) THEN
+* .. the second QRF without pivoting. Note: in an optimized
+* implementation, this QRF should be implemented as the QRF
+* of a lower triangular matrix.
+* R1^* = Q2 * R2
+ CALL CGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
+ $ LWORK-2*N, IERR )
+*
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL)/EPSLN
+ DO 3959 p = 2, NR
+ DO 3958 q = 1, p - 1
+ CTEMP=CMPLX(XSC*MIN(ABS(V(p,p)),ABS(V(q,q))),
+ $ ZERO)
+ IF ( ABS(V(q,p)) .LE. TEMP1 )
+* $ V(q,p) = TEMP1 * ( V(q,p) / ABS(V(q,p)) )
+ $ V(q,p) = CTEMP
+ 3958 CONTINUE
+ 3959 CONTINUE
+ END IF
+*
+ IF ( NR .NE. N )
+ $ CALL CLACPY( 'A', N, NR, V, LDV, CWORK(2*N+1), N )
+* .. save ...
+*
+* .. this transposed copy should be better than naive
+ DO 1969 p = 1, NR - 1
+ CALL CCOPY( NR-p, V(p,p+1), LDV, V(p+1,p), 1 )
+ CALL CLACGV(NR-p+1, V(p,p), 1 )
+ 1969 CONTINUE
+ V(NR,NR)=CONJG(V(NR,NR))
+*
+ CONDR2 = CONDR1
+*
+ ELSE
+*
+* .. ill-conditioned case: second QRF with pivoting
+* Note that windowed pivoting would be equaly good
+* numerically, and more run-time efficient. So, in
+* an optimal implementation, the next call to CGEQP3
+* should be replaced with eg. CALL CGEQPX (ACM TOMS #782)
+* with properly (carefully) chosen parameters.
+*
+* R1^* * P2 = Q2 * R2
+ DO 3003 p = 1, NR
+ IWORK(N+p) = 0
+ 3003 CONTINUE
+ CALL CGEQP3( N, NR, V, LDV, IWORK(N+1), CWORK(N+1),
+ $ CWORK(2*N+1), LWORK-2*N, RWORK, IERR )
+** CALL CGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
+** $ LWORK-2*N, IERR )
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL)
+ DO 3969 p = 2, NR
+ DO 3968 q = 1, p - 1
+ CTEMP=CMPLX(XSC*MIN(ABS(V(p,p)),ABS(V(q,q))),
+ $ ZERO)
+ IF ( ABS(V(q,p)) .LE. TEMP1 )
+* $ V(q,p) = TEMP1 * ( V(q,p) / ABS(V(q,p)) )
+ $ V(q,p) = CTEMP
+ 3968 CONTINUE
+ 3969 CONTINUE
+ END IF
+*
+ CALL CLACPY( 'A', N, NR, V, LDV, CWORK(2*N+1), N )
+*
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL)
+ DO 8970 p = 2, NR
+ DO 8971 q = 1, p - 1
+ CTEMP=CMPLX(XSC*MIN(ABS(V(p,p)),ABS(V(q,q))),
+ $ ZERO)
+* V(p,q) = - TEMP1*( V(q,p) / ABS(V(q,p)) )
+ V(p,q) = - CTEMP
+ 8971 CONTINUE
+ 8970 CONTINUE
+ ELSE
+ CALL CLASET( 'L',NR-1,NR-1,CZERO,CZERO,V(2,1),LDV )
+ END IF
+* Now, compute R2 = L3 * Q3, the LQ factorization.
+ CALL CGELQF( NR, NR, V, LDV, CWORK(2*N+N*NR+1),
+ $ CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, IERR )
+* .. and estimate the condition number
+ CALL CLACPY( 'L',NR,NR,V,LDV,CWORK(2*N+N*NR+NR+1),NR )
+ DO 4950 p = 1, NR
+ TEMP1 = SCNRM2( p, CWORK(2*N+N*NR+NR+p), NR )
+ CALL CSSCAL( p, ONE/TEMP1, CWORK(2*N+N*NR+NR+p), NR )
+ 4950 CONTINUE
+ CALL CPOCON( 'L',NR,CWORK(2*N+N*NR+NR+1),NR,ONE,TEMP1,
+ $ CWORK(2*N+N*NR+NR+NR*NR+1),RWORK,IERR )
+ CONDR2 = ONE / SQRT(TEMP1)
+*
+*
+ IF ( CONDR2 .GE. COND_OK ) THEN
+* .. save the Householder vectors used for Q3
+* (this overwrittes the copy of R2, as it will not be
+* needed in this branch, but it does not overwritte the
+* Huseholder vectors of Q2.).
+ CALL CLACPY( 'U', NR, NR, V, LDV, CWORK(2*N+1), N )
+* .. and the rest of the information on Q3 is in
+* WORK(2*N+N*NR+1:2*N+N*NR+N)
+ END IF
+*
+ END IF
+*
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL)
+ DO 4968 q = 2, NR
+ CTEMP = XSC * V(q,q)
+ DO 4969 p = 1, q - 1
+* V(p,q) = - TEMP1*( V(p,q) / ABS(V(p,q)) )
+ V(p,q) = - CTEMP
+ 4969 CONTINUE
+ 4968 CONTINUE
+ ELSE
+ CALL CLASET( 'U', NR-1,NR-1, CZERO,CZERO, V(1,2), LDV )
+ END IF
+*
+* Second preconditioning finished; continue with Jacobi SVD
+* The input matrix is lower trinagular.
+*
+* Recover the right singular vectors as solution of a well
+* conditioned triangular matrix equation.
+*
+ IF ( CONDR1 .LT. COND_OK ) THEN
+*
+ CALL CGESVJ( 'L','U','N',NR,NR,V,LDV,SVA,NR,U, LDU,
+ $ CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,RWORK,
+ $ LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+ DO 3970 p = 1, NR
+ CALL CCOPY( NR, V(1,p), 1, U(1,p), 1 )
+ CALL CSSCAL( NR, SVA(p), V(1,p), 1 )
+ 3970 CONTINUE
+
+* .. pick the right matrix equation and solve it
+*
+ IF ( NR .EQ. N ) THEN
+* :)) .. best case, R1 is inverted. The solution of this matrix
+* equation is Q2*V2 = the product of the Jacobi rotations
+* used in CGESVJ, premultiplied with the orthogonal matrix
+* from the second QR factorization.
+ CALL CTRSM('L','U','N','N', NR,NR,CONE, A,LDA, V,LDV)
+ ELSE
+* .. R1 is well conditioned, but non-square. Adjoint of R2
+* is inverted to get the product of the Jacobi rotations
+* used in CGESVJ. The Q-factor from the second QR
+* factorization is then built in explicitly.
+ CALL CTRSM('L','U','C','N',NR,NR,CONE,CWORK(2*N+1),
+ $ N,V,LDV)
+ IF ( NR .LT. N ) THEN
+ CALL CLASET('A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV)
+ CALL CLASET('A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV)
+ CALL CLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
+ END IF
+ CALL CUNMQR('L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
+ $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR)
+ END IF
+*
+ ELSE IF ( CONDR2 .LT. COND_OK ) THEN
+*
+* The matrix R2 is inverted. The solution of the matrix equation
+* is Q3^* * V3 = the product of the Jacobi rotations (appplied to
+* the lower triangular L3 from the LQ factorization of
+* R2=L3*Q3), pre-multiplied with the transposed Q3.
+ CALL CGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR, U,
+ $ LDU, CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR,
+ $ RWORK, LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+ DO 3870 p = 1, NR
+ CALL CCOPY( NR, V(1,p), 1, U(1,p), 1 )
+ CALL CSSCAL( NR, SVA(p), U(1,p), 1 )
+ 3870 CONTINUE
+ CALL CTRSM('L','U','N','N',NR,NR,CONE,CWORK(2*N+1),N,
+ $ U,LDU)
+* .. apply the permutation from the second QR factorization
+ DO 873 q = 1, NR
+ DO 872 p = 1, NR
+ CWORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
+ 872 CONTINUE
+ DO 874 p = 1, NR
+ U(p,q) = CWORK(2*N+N*NR+NR+p)
+ 874 CONTINUE
+ 873 CONTINUE
+ IF ( NR .LT. N ) THEN
+ CALL CLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
+ CALL CLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
+ CALL CLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
+ END IF
+ CALL CUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
+ $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
+ ELSE
+* Last line of defense.
+* #:( This is a rather pathological case: no scaled condition
+* improvement after two pivoted QR factorizations. Other
+* possibility is that the rank revealing QR factorization
+* or the condition estimator has failed, or the COND_OK
+* is set very close to ONE (which is unnecessary). Normally,
+* this branch should never be executed, but in rare cases of
+* failure of the RRQR or condition estimator, the last line of
+* defense ensures that CGEJSV completes the task.
+* Compute the full SVD of L3 using CGESVJ with explicit
+* accumulation of Jacobi rotations.
+ CALL CGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR, U,
+ $ LDU, CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR,
+ $ RWORK, LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+ IF ( NR .LT. N ) THEN
+ CALL CLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
+ CALL CLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
+ CALL CLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
+ END IF
+ CALL CUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
+ $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
+*
+ CALL CUNMLQ( 'L', 'C', NR, NR, NR, CWORK(2*N+1), N,
+ $ CWORK(2*N+N*NR+1), U, LDU, CWORK(2*N+N*NR+NR+1),
+ $ LWORK-2*N-N*NR-NR, IERR )
+ DO 773 q = 1, NR
+ DO 772 p = 1, NR
+ CWORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
+ 772 CONTINUE
+ DO 774 p = 1, NR
+ U(p,q) = CWORK(2*N+N*NR+NR+p)
+ 774 CONTINUE
+ 773 CONTINUE
+*
+ END IF
+*
+* Permute the rows of V using the (column) permutation from the
+* first QRF. Also, scale the columns to make them unit in
+* Euclidean norm. This applies to all cases.
+*
+ TEMP1 = SQRT(REAL(N)) * EPSLN
+ DO 1972 q = 1, N
+ DO 972 p = 1, N
+ CWORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
+ 972 CONTINUE
+ DO 973 p = 1, N
+ V(p,q) = CWORK(2*N+N*NR+NR+p)
+ 973 CONTINUE
+ XSC = ONE / SCNRM2( N, V(1,q), 1 )
+ IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
+ $ CALL CSSCAL( N, XSC, V(1,q), 1 )
+ 1972 CONTINUE
+* At this moment, V contains the right singular vectors of A.
+* Next, assemble the left singular vector matrix U (M x N).
+ IF ( NR .LT. M ) THEN
+ CALL CLASET('A', M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU)
+ IF ( NR .LT. N1 ) THEN
+ CALL CLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)
+ CALL CLASET('A',M-NR,N1-NR,CZERO,CONE,
+ $ U(NR+1,NR+1),LDU)
+ END IF
+ END IF
+*
+* The Q matrix from the first QRF is built into the left singular
+* matrix U. This applies to all cases.
+*
+ CALL CUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
+ $ LDU, CWORK(N+1), LWORK-N, IERR )
+
+* The columns of U are normalized. The cost is O(M*N) flops.
+ TEMP1 = SQRT(REAL(M)) * EPSLN
+ DO 1973 p = 1, NR
+ XSC = ONE / SCNRM2( M, U(1,p), 1 )
+ IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
+ $ CALL CSSCAL( M, XSC, U(1,p), 1 )
+ 1973 CONTINUE
+*
+* If the initial QRF is computed with row pivoting, the left
+* singular vectors must be adjusted.
+*
+ IF ( ROWPIV )
+ $ CALL CLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
+*
+ ELSE
+*
+* .. the initial matrix A has almost orthogonal columns and
+* the second QRF is not needed
+*
+ CALL CLACPY( 'U', N, N, A, LDA, CWORK(N+1), N )
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL)
+ DO 5970 p = 2, N
+ CTEMP = XSC * CWORK( N + (p-1)*N + p )
+ DO 5971 q = 1, p - 1
+* CWORK(N+(q-1)*N+p)=-TEMP1 * ( CWORK(N+(p-1)*N+q) /
+* $ ABS(CWORK(N+(p-1)*N+q)) )
+ CWORK(N+(q-1)*N+p)=-CTEMP
+ 5971 CONTINUE
+ 5970 CONTINUE
+ ELSE
+ CALL CLASET( 'L',N-1,N-1,CZERO,CZERO,CWORK(N+2),N )
+ END IF
+*
+ CALL CGESVJ( 'U', 'U', 'N', N, N, CWORK(N+1), N, SVA,
+ $ N, U, LDU, CWORK(N+N*N+1), LWORK-N-N*N, RWORK, LRWORK,
+ $ INFO )
+*
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+ DO 6970 p = 1, N
+ CALL CCOPY( N, CWORK(N+(p-1)*N+1), 1, U(1,p), 1 )
+ CALL CSSCAL( N, SVA(p), CWORK(N+(p-1)*N+1), 1 )
+ 6970 CONTINUE
+*
+ CALL CTRSM( 'L', 'U', 'N', 'N', N, N,
+ $ CONE, A, LDA, CWORK(N+1), N )
+ DO 6972 p = 1, N
+ CALL CCOPY( N, CWORK(N+p), N, V(IWORK(p),1), LDV )
+ 6972 CONTINUE
+ TEMP1 = SQRT(REAL(N))*EPSLN
+ DO 6971 p = 1, N
+ XSC = ONE / SCNRM2( N, V(1,p), 1 )
+ IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
+ $ CALL CSSCAL( N, XSC, V(1,p), 1 )
+ 6971 CONTINUE
+*
+* Assemble the left singular vector matrix U (M x N).
+*
+ IF ( N .LT. M ) THEN
+ CALL CLASET( 'A', M-N, N, CZERO, CZERO, U(N+1,1), LDU )
+ IF ( N .LT. N1 ) THEN
+ CALL CLASET('A',N, N1-N, CZERO, CZERO, U(1,N+1),LDU)
+ CALL CLASET( 'A',M-N,N1-N, CZERO, CONE,U(N+1,N+1),LDU)
+ END IF
+ END IF
+ CALL CUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
+ $ LDU, CWORK(N+1), LWORK-N, IERR )
+ TEMP1 = SQRT(REAL(M))*EPSLN
+ DO 6973 p = 1, N1
+ XSC = ONE / SCNRM2( M, U(1,p), 1 )
+ IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
+ $ CALL CSSCAL( M, XSC, U(1,p), 1 )
+ 6973 CONTINUE
+*
+ IF ( ROWPIV )
+ $ CALL CLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
+*
+ END IF
+*
+* end of the >> almost orthogonal case << in the full SVD
+*
+ ELSE
+*
+* This branch deploys a preconditioned Jacobi SVD with explicitly
+* accumulated rotations. It is included as optional, mainly for
+* experimental purposes. It does perfom well, and can also be used.
+* In this implementation, this branch will be automatically activated
+* if the condition number sigma_max(A) / sigma_min(A) is predicted
+* to be greater than the overflow threshold. This is because the
+* a posteriori computation of the singular vectors assumes robust
+* implementation of BLAS and some LAPACK procedures, capable of working
+* in presence of extreme values, e.g. when the singular values spread from
+* the underflow to the overflow threshold.
+*
+ DO 7968 p = 1, NR
+ CALL CCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
+ CALL CLACGV( N-p+1, V(p,p), 1 )
+ 7968 CONTINUE
+*
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL/EPSLN)
+ DO 5969 q = 1, NR
+ CTEMP = CMPLX(XSC*ABS( V(q,q) ),ZERO)
+ DO 5968 p = 1, N
+ IF ( ( p .GT. q ) .AND. ( ABS(V(p,q)) .LE. TEMP1 )
+ $ .OR. ( p .LT. q ) )
+* $ V(p,q) = TEMP1 * ( V(p,q) / ABS(V(p,q)) )
+ $ V(p,q) = CTEMP
+ IF ( p .LT. q ) V(p,q) = - V(p,q)
+ 5968 CONTINUE
+ 5969 CONTINUE
+ ELSE
+ CALL CLASET( 'U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV )
+ END IF
+
+ CALL CGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
+ $ LWORK-2*N, IERR )
+ CALL CLACPY( 'L', N, NR, V, LDV, CWORK(2*N+1), N )
+*
+ DO 7969 p = 1, NR
+ CALL CCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )
+ CALL CLACGV( NR-p+1, U(p,p), 1 )
+ 7969 CONTINUE
+
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL/EPSLN)
+ DO 9970 q = 2, NR
+ DO 9971 p = 1, q - 1
+ CTEMP = CMPLX(XSC * MIN(ABS(U(p,p)),ABS(U(q,q))),
+ $ ZERO)
+* U(p,q) = - TEMP1 * ( U(q,p) / ABS(U(q,p)) )
+ U(p,q) = - CTEMP
+ 9971 CONTINUE
+ 9970 CONTINUE
+ ELSE
+ CALL CLASET('U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
+ END IF
+
+ CALL CGESVJ( 'L', 'U', 'V', NR, NR, U, LDU, SVA,
+ $ N, V, LDV, CWORK(2*N+N*NR+1), LWORK-2*N-N*NR,
+ $ RWORK, LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+
+ IF ( NR .LT. N ) THEN
+ CALL CLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
+ CALL CLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
+ CALL CLASET( 'A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV )
+ END IF
+
+ CALL CUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
+ $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
+*
+* Permute the rows of V using the (column) permutation from the
+* first QRF. Also, scale the columns to make them unit in
+* Euclidean norm. This applies to all cases.
+*
+ TEMP1 = SQRT(REAL(N)) * EPSLN
+ DO 7972 q = 1, N
+ DO 8972 p = 1, N
+ CWORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
+ 8972 CONTINUE
+ DO 8973 p = 1, N
+ V(p,q) = CWORK(2*N+N*NR+NR+p)
+ 8973 CONTINUE
+ XSC = ONE / SCNRM2( N, V(1,q), 1 )
+ IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
+ $ CALL CSSCAL( N, XSC, V(1,q), 1 )
+ 7972 CONTINUE
+*
+* At this moment, V contains the right singular vectors of A.
+* Next, assemble the left singular vector matrix U (M x N).
+*
+ IF ( NR .LT. M ) THEN
+ CALL CLASET( 'A', M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU )
+ IF ( NR .LT. N1 ) THEN
+ CALL CLASET('A',NR, N1-NR, CZERO, CZERO, U(1,NR+1),LDU)
+ CALL CLASET('A',M-NR,N1-NR, CZERO, CONE,U(NR+1,NR+1),LDU)
+ END IF
+ END IF
+*
+ CALL CUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
+ $ LDU, CWORK(N+1), LWORK-N, IERR )
+*
+ IF ( ROWPIV )
+ $ CALL CLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
+*
+*
+ END IF
+ IF ( TRANSP ) THEN
+* .. swap U and V because the procedure worked on A^*
+ DO 6974 p = 1, N
+ CALL CSWAP( N, U(1,p), 1, V(1,p), 1 )
+ 6974 CONTINUE
+ END IF
+*
+ END IF
+* end of the full SVD
+*
+* Undo scaling, if necessary (and possible)
+*
+ IF ( USCAL2 .LE. (BIG/SVA(1))*USCAL1 ) THEN
+ CALL SLASCL( 'G', 0, 0, USCAL1, USCAL2, NR, 1, SVA, N, IERR )
+ USCAL1 = ONE
+ USCAL2 = ONE
+ END IF
+*
+ IF ( NR .LT. N ) THEN
+ DO 3004 p = NR+1, N
+ SVA(p) = ZERO
+ 3004 CONTINUE
+ END IF
+*
+ RWORK(1) = USCAL2 * SCALEM
+ RWORK(2) = USCAL1
+ IF ( ERREST ) RWORK(3) = SCONDA
+ IF ( LSVEC .AND. RSVEC ) THEN
+ RWORK(4) = CONDR1
+ RWORK(5) = CONDR2
+ END IF
+ IF ( L2TRAN ) THEN
+ RWORK(6) = ENTRA
+ RWORK(7) = ENTRAT
+ END IF
+*
+ IWORK(1) = NR
+ IWORK(2) = NUMRANK
+ IWORK(3) = WARNING
+ IF ( TRANSP ) THEN
+ IWORK(4) = 1
+ ELSE
+ IWORK(4) = -1
+ END IF
+
+*
+ RETURN
+* ..
+* .. END OF CGEJSV
+* ..
+ END
+*
diff --git a/SRC/clarrv.f b/SRC/clarrv.f
index 1e1a309..d650301 100644
--- a/SRC/clarrv.f
+++ b/SRC/clarrv.f
@@ -348,6 +348,13 @@
* ..
INFO = 0
+*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
* The first N entries of WORK are reserved for the eigenvalues
INDLD = N+1
INDLLD= 2*N+1
diff --git a/SRC/dlarra.f b/SRC/dlarra.f
index 31a0bfb..933386e 100644
--- a/SRC/dlarra.f
+++ b/SRC/dlarra.f
@@ -167,7 +167,13 @@
* .. Executable Statements ..
*
INFO = 0
-
+*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
* Compute splitting points
NSPLIT = 1
IF(SPLTOL.LT.ZERO) THEN
diff --git a/SRC/dlarrb.f b/SRC/dlarrb.f
index 2733922..eff244a 100644
--- a/SRC/dlarrb.f
+++ b/SRC/dlarrb.f
@@ -237,6 +237,12 @@
*
INFO = 0
*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
$ LOG( TWO ) ) + 2
MNWDTH = TWO * PIVMIN
diff --git a/SRC/dlarrc.f b/SRC/dlarrc.f
index 9635e41..c99cf89 100644
--- a/SRC/dlarrc.f
+++ b/SRC/dlarrc.f
@@ -170,6 +170,13 @@
* .. Executable Statements ..
*
INFO = 0
+*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
LCNT = 0
RCNT = 0
EIGCNT = 0
diff --git a/SRC/dlarrd.f b/SRC/dlarrd.f
index 57abf74..bbc9e29 100644
--- a/SRC/dlarrd.f
+++ b/SRC/dlarrd.f
@@ -385,6 +385,12 @@
*
INFO = 0
*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
* Decode RANGE
*
IF( LSAME( RANGE, 'A' ) ) THEN
diff --git a/SRC/dlarre.f b/SRC/dlarre.f
index f01b25f..ed87692 100644
--- a/SRC/dlarre.f
+++ b/SRC/dlarre.f
@@ -370,7 +370,12 @@
*
INFO = 0
-
+*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
*
* Decode RANGE
*
diff --git a/SRC/dlarrf.f b/SRC/dlarrf.f
index 5ad4337..335d0c9 100644
--- a/SRC/dlarrf.f
+++ b/SRC/dlarrf.f
@@ -239,6 +239,13 @@
* .. Executable Statements ..
*
INFO = 0
+*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
FACT = DBLE(2**KTRYMAX)
EPS = DLAMCH( 'Precision' )
SHIFT = 0
diff --git a/SRC/dlarrj.f b/SRC/dlarrj.f
index ecd136f..c6efe01 100644
--- a/SRC/dlarrj.f
+++ b/SRC/dlarrj.f
@@ -204,6 +204,12 @@
*
INFO = 0
*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
$ LOG( TWO ) ) + 2
*
diff --git a/SRC/dlarrk.f b/SRC/dlarrk.f
index 8b307a4..246cb34 100644
--- a/SRC/dlarrk.f
+++ b/SRC/dlarrk.f
@@ -179,6 +179,13 @@
* ..
* .. Executable Statements ..
*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ INFO = 0
+ RETURN
+ END IF
+*
* Get machine constants
EPS = DLAMCH( 'P' )
diff --git a/SRC/dlarrr.f b/SRC/dlarrr.f
index c12b605..89bbe81 100644
--- a/SRC/dlarrr.f
+++ b/SRC/dlarrr.f
@@ -130,6 +130,13 @@
* ..
* .. Executable Statements ..
*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ INFO = 0
+ RETURN
+ END IF
+*
* As a default, do NOT go for relative-accuracy preserving computations.
INFO = 1
diff --git a/SRC/dlarrv.f b/SRC/dlarrv.f
index edda67d..8006a75 100644
--- a/SRC/dlarrv.f
+++ b/SRC/dlarrv.f
@@ -344,6 +344,13 @@
* ..
INFO = 0
+*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
* The first N entries of WORK are reserved for the eigenvalues
INDLD = N+1
INDLLD= 2*N+1
diff --git a/SRC/slarra.f b/SRC/slarra.f
index fd248c9..e06cdf0 100644
--- a/SRC/slarra.f
+++ b/SRC/slarra.f
@@ -167,7 +167,13 @@
* .. Executable Statements ..
*
INFO = 0
-
+*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
* Compute splitting points
NSPLIT = 1
IF(SPLTOL.LT.ZERO) THEN
diff --git a/SRC/slarrb.f b/SRC/slarrb.f
index c2d130b..ea6e12b 100644
--- a/SRC/slarrb.f
+++ b/SRC/slarrb.f
@@ -237,6 +237,12 @@
*
INFO = 0
*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
$ LOG( TWO ) ) + 2
MNWDTH = TWO * PIVMIN
diff --git a/SRC/slarrc.f b/SRC/slarrc.f
index 8469660..75e568e 100644
--- a/SRC/slarrc.f
+++ b/SRC/slarrc.f
@@ -170,6 +170,13 @@
* .. Executable Statements ..
*
INFO = 0
+*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
LCNT = 0
RCNT = 0
EIGCNT = 0
diff --git a/SRC/slarrd.f b/SRC/slarrd.f
index 8da31a9..ab389b4 100644
--- a/SRC/slarrd.f
+++ b/SRC/slarrd.f
@@ -385,6 +385,12 @@
*
INFO = 0
*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
* Decode RANGE
*
IF( LSAME( RANGE, 'A' ) ) THEN
diff --git a/SRC/slarre.f b/SRC/slarre.f
index 3c1b511..6dd8939 100644
--- a/SRC/slarre.f
+++ b/SRC/slarre.f
@@ -370,7 +370,12 @@
*
INFO = 0
-
+*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
*
* Decode RANGE
*
diff --git a/SRC/slarrf.f b/SRC/slarrf.f
index ee8af8c..8a26574 100644
--- a/SRC/slarrf.f
+++ b/SRC/slarrf.f
@@ -239,6 +239,13 @@
* .. Executable Statements ..
*
INFO = 0
+*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
FACT = REAL(2**KTRYMAX)
EPS = SLAMCH( 'Precision' )
SHIFT = 0
diff --git a/SRC/slarrj.f b/SRC/slarrj.f
index 6ce1516..665942d 100644
--- a/SRC/slarrj.f
+++ b/SRC/slarrj.f
@@ -204,6 +204,12 @@
*
INFO = 0
*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
$ LOG( TWO ) ) + 2
*
diff --git a/SRC/slarrk.f b/SRC/slarrk.f
index 4d625c5..3fdb776 100644
--- a/SRC/slarrk.f
+++ b/SRC/slarrk.f
@@ -179,6 +179,13 @@
* ..
* .. Executable Statements ..
*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ INFO = 0
+ RETURN
+ END IF
+*
* Get machine constants
EPS = SLAMCH( 'P' )
diff --git a/SRC/slarrr.f b/SRC/slarrr.f
index e4181ea..dd4b835 100644
--- a/SRC/slarrr.f
+++ b/SRC/slarrr.f
@@ -130,6 +130,13 @@
* ..
* .. Executable Statements ..
*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ INFO = 0
+ RETURN
+ END IF
+*
* As a default, do NOT go for relative-accuracy preserving computations.
INFO = 1
diff --git a/SRC/slarrv.f b/SRC/slarrv.f
index e574da5..73bebb3 100644
--- a/SRC/slarrv.f
+++ b/SRC/slarrv.f
@@ -344,6 +344,13 @@
* ..
INFO = 0
+*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
* The first N entries of WORK are reserved for the eigenvalues
INDLD = N+1
INDLLD= 2*N+1
diff --git a/SRC/zgejsv.f b/SRC/zgejsv.f
index fcf0735..0d05c99 100644
--- a/SRC/zgejsv.f
+++ b/SRC/zgejsv.f
@@ -1,2237 +1,2237 @@
-*> \brief \b ZGEJSV
-*
-* =========== DOCUMENTATION ===========
-*
-* Online html documentation available at
-* http://www.netlib.org/lapack/explore-html/
-*
-*> \htmlonly
-*> Download ZGEJSV + dependencies
-*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgejsv.f">
-*> [TGZ]</a>
-*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgejsv.f">
-*> [ZIP]</a>
-*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgejsv.f">
-*> [TXT]</a>
-*> \endhtmlonly
-*
-* Definition:
-* ===========
-*
-* SUBROUTINE ZGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
-* M, N, A, LDA, SVA, U, LDU, V, LDV,
-* CWORK, LWORK, RWORK, LRWORK, IWORK, INFO )
-*
-* .. Scalar Arguments ..
-* IMPLICIT NONE
-* INTEGER INFO, LDA, LDU, LDV, LWORK, M, N
-* ..
-* .. Array Arguments ..
-* COMPLEX*16 A( LDA, * ), U( LDU, * ), V( LDV, * ), CWORK( LWORK )
-* DOUBLE PRECISION SVA( N ), RWORK( LRWORK )
-* INTEGER IWORK( * )
-* CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
-* ..
-*
-*
-*> \par Purpose:
-* =============
-*>
-*> \verbatim
-*>
-*> ZGEJSV computes the singular value decomposition (SVD) of a complex M-by-N
-*> matrix [A], where M >= N. The SVD of [A] is written as
-*>
-*> [A] = [U] * [SIGMA] * [V]^*,
-*>
-*> where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
-*> diagonal elements, [U] is an M-by-N (or M-by-M) unitary matrix, and
-*> [V] is an N-by-N unitary matrix. The diagonal elements of [SIGMA] are
-*> the singular values of [A]. The columns of [U] and [V] are the left and
-*> the right singular vectors of [A], respectively. The matrices [U] and [V]
-*> are computed and stored in the arrays U and V, respectively. The diagonal
-*> of [SIGMA] is computed and stored in the array SVA.
-*> \endverbatim
-*>
-*> Arguments:
-*> ==========
-*>
-*> \param[in] JOBA
-*> \verbatim
-*> JOBA is CHARACTER*1
-*> Specifies the level of accuracy:
-*> = 'C': This option works well (high relative accuracy) if A = B * D,
-*> with well-conditioned B and arbitrary diagonal matrix D.
-*> The accuracy cannot be spoiled by COLUMN scaling. The
-*> accuracy of the computed output depends on the condition of
-*> B, and the procedure aims at the best theoretical accuracy.
-*> The relative error max_{i=1:N}|d sigma_i| / sigma_i is
-*> bounded by f(M,N)*epsilon* cond(B), independent of D.
-*> The input matrix is preprocessed with the QRF with column
-*> pivoting. This initial preprocessing and preconditioning by
-*> a rank revealing QR factorization is common for all values of
-*> JOBA. Additional actions are specified as follows:
-*> = 'E': Computation as with 'C' with an additional estimate of the
-*> condition number of B. It provides a realistic error bound.
-*> = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
-*> D1, D2, and well-conditioned matrix C, this option gives
-*> higher accuracy than the 'C' option. If the structure of the
-*> input matrix is not known, and relative accuracy is
-*> desirable, then this option is advisable. The input matrix A
-*> is preprocessed with QR factorization with FULL (row and
-*> column) pivoting.
-*> = 'G' Computation as with 'F' with an additional estimate of the
-*> condition number of B, where A=B*D. If A has heavily weighted
-*> rows, then using this condition number gives too pessimistic
-*> error bound.
-*> = 'A': Small singular values are not well determined by the data
-*> and are considered as noisy; the matrix is treated as
-*> numerically rank defficient. The error in the computed
-*> singular values is bounded by f(m,n)*epsilon*||A||.
-*> The computed SVD A = U * S * V^* restores A up to
-*> f(m,n)*epsilon*||A||.
-*> This gives the procedure the licence to discard (set to zero)
-*> all singular values below N*epsilon*||A||.
-*> = 'R': Similar as in 'A'. Rank revealing property of the initial
-*> QR factorization is used do reveal (using triangular factor)
-*> a gap sigma_{r+1} < epsilon * sigma_r in which case the
-*> numerical RANK is declared to be r. The SVD is computed with
-*> absolute error bounds, but more accurately than with 'A'.
-*> \endverbatim
-*>
-*> \param[in] JOBU
-*> \verbatim
-*> JOBU is CHARACTER*1
-*> Specifies whether to compute the columns of U:
-*> = 'U': N columns of U are returned in the array U.
-*> = 'F': full set of M left sing. vectors is returned in the array U.
-*> = 'W': U may be used as workspace of length M*N. See the description
-*> of U.
-*> = 'N': U is not computed.
-*> \endverbatim
-*>
-*> \param[in] JOBV
-*> \verbatim
-*> JOBV is CHARACTER*1
-*> Specifies whether to compute the matrix V:
-*> = 'V': N columns of V are returned in the array V; Jacobi rotations
-*> are not explicitly accumulated.
-*> = 'J': N columns of V are returned in the array V, but they are
-*> computed as the product of Jacobi rotations, if JOBT .EQ. 'N'.
-*> = 'W': V may be used as workspace of length N*N. See the description
-*> of V.
-*> = 'N': V is not computed.
-*> \endverbatim
-*>
-*> \param[in] JOBR
-*> \verbatim
-*> JOBR is CHARACTER*1
-*> Specifies the RANGE for the singular values. Issues the licence to
-*> set to zero small positive singular values if they are outside
-*> specified range. If A .NE. 0 is scaled so that the largest singular
-*> value of c*A is around SQRT(BIG), BIG=DLAMCH('O'), then JOBR issues
-*> the licence to kill columns of A whose norm in c*A is less than
-*> SQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,
-*> where SFMIN=DLAMCH('S'), EPSLN=DLAMCH('E').
-*> = 'N': Do not kill small columns of c*A. This option assumes that
-*> BLAS and QR factorizations and triangular solvers are
-*> implemented to work in that range. If the condition of A
-*> is greater than BIG, use ZGESVJ.
-*> = 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)]
-*> (roughly, as described above). This option is recommended.
-*> ===========================
-*> For computing the singular values in the FULL range [SFMIN,BIG]
-*> use ZGESVJ.
-*> \endverbatim
-*>
-*> \param[in] JOBT
-*> \verbatim
-*> JOBT is CHARACTER*1
-*> If the matrix is square then the procedure may determine to use
-*> transposed A if A^* seems to be better with respect to convergence.
-*> If the matrix is not square, JOBT is ignored.
-*> The decision is based on two values of entropy over the adjoint
-*> orbit of A^* * A. See the descriptions of WORK(6) and WORK(7).
-*> = 'T': transpose if entropy test indicates possibly faster
-*> convergence of Jacobi process if A^* is taken as input. If A is
-*> replaced with A^*, then the row pivoting is included automatically.
-*> = 'N': do not speculate.
-*> The option 'T' can be used to compute only the singular values, or
-*> the full SVD (U, SIGMA and V). For only one set of singular vectors
-*> (U or V), the caller should provide both U and V, as one of the
-*> matrices is used as workspace if the matrix A is transposed.
-*> The implementer can easily remove this constraint and make the
-*> code more complicated. See the descriptions of U and V.
-*> In general, this option is considered experimental, and 'N'; should
-*> be preferred. This is subject to changes in the future.
-*> \endverbatim
-*>
-*> \param[in] JOBP
-*> \verbatim
-*> JOBP is CHARACTER*1
-*> Issues the licence to introduce structured perturbations to drown
-*> denormalized numbers. This licence should be active if the
-*> denormals are poorly implemented, causing slow computation,
-*> especially in cases of fast convergence (!). For details see [1,2].
-*> For the sake of simplicity, this perturbations are included only
-*> when the full SVD or only the singular values are requested. The
-*> implementer/user can easily add the perturbation for the cases of
-*> computing one set of singular vectors.
-*> = 'P': introduce perturbation
-*> = 'N': do not perturb
-*> \endverbatim
-*>
-*> \param[in] M
-*> \verbatim
-*> M is INTEGER
-*> The number of rows of the input matrix A. M >= 0.
-*> \endverbatim
-*>
-*> \param[in] N
-*> \verbatim
-*> N is INTEGER
-*> The number of columns of the input matrix A. M >= N >= 0.
-*> \endverbatim
-*>
-*> \param[in,out] A
-*> \verbatim
-*> A is COMPLEX*16 array, dimension (LDA,N)
-*> On entry, the M-by-N matrix A.
-*> \endverbatim
-*>
-*> \param[in] LDA
-*> \verbatim
-*> LDA is INTEGER
-*> The leading dimension of the array A. LDA >= max(1,M).
-*> \endverbatim
-*>
-*> \param[out] SVA
-*> \verbatim
-*> SVA is DOUBLE PRECISION array, dimension (N)
-*> On exit,
-*> - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
-*> computation SVA contains Euclidean column norms of the
-*> iterated matrices in the array A.
-*> - For WORK(1) .NE. WORK(2): The singular values of A are
-*> (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
-*> sigma_max(A) overflows or if small singular values have been
-*> saved from underflow by scaling the input matrix A.
-*> - If JOBR='R' then some of the singular values may be returned
-*> as exact zeros obtained by "set to zero" because they are
-*> below the numerical rank threshold or are denormalized numbers.
-*> \endverbatim
-*>
-*> \param[out] U
-*> \verbatim
-*> U is COMPLEX*16 array, dimension ( LDU, N )
-*> If JOBU = 'U', then U contains on exit the M-by-N matrix of
-*> the left singular vectors.
-*> If JOBU = 'F', then U contains on exit the M-by-M matrix of
-*> the left singular vectors, including an ONB
-*> of the orthogonal complement of the Range(A).
-*> If JOBU = 'W' .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),
-*> then U is used as workspace if the procedure
-*> replaces A with A^*. In that case, [V] is computed
-*> in U as left singular vectors of A^* and then
-*> copied back to the V array. This 'W' option is just
-*> a reminder to the caller that in this case U is
-*> reserved as workspace of length N*N.
-*> If JOBU = 'N' U is not referenced, unless JOBT='T'.
-*> \endverbatim
-*>
-*> \param[in] LDU
-*> \verbatim
-*> LDU is INTEGER
-*> The leading dimension of the array U, LDU >= 1.
-*> IF JOBU = 'U' or 'F' or 'W', then LDU >= M.
-*> \endverbatim
-*>
-*> \param[out] V
-*> \verbatim
-*> V is COMPLEX*16 array, dimension ( LDV, N )
-*> If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
-*> the right singular vectors;
-*> If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),
-*> then V is used as workspace if the pprocedure
-*> replaces A with A^*. In that case, [U] is computed
-*> in V as right singular vectors of A^* and then
-*> copied back to the U array. This 'W' option is just
-*> a reminder to the caller that in this case V is
-*> reserved as workspace of length N*N.
-*> If JOBV = 'N' V is not referenced, unless JOBT='T'.
-*> \endverbatim
-*>
-*> \param[in] LDV
-*> \verbatim
-*> LDV is INTEGER
-*> The leading dimension of the array V, LDV >= 1.
-*> If JOBV = 'V' or 'J' or 'W', then LDV >= N.
-*> \endverbatim
-*>
-*> \param[out] CWORK
-*> \verbatim
-*> CWORK is COMPLEX*16 array, dimension (MAX(2,LWORK))
-*> If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or
-*> LRWORK=-1), then on exit CWORK(1) contains the required length of
-*> CWORK for the job parameters used in the call.
-*> \endverbatim
-*>
-*> \param[in] LWORK
-*> \verbatim
-*> LWORK is INTEGER
-*> Length of CWORK to confirm proper allocation of workspace.
-*> LWORK depends on the job:
-*>
-*> 1. If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
-*> 1.1 .. no scaled condition estimate required (JOBA.NE.'E'.AND.JOBA.NE.'G'):
-*> LWORK >= 2*N+1. This is the minimal requirement.
-*> ->> For optimal performance (blocked code) the optimal value
-*> is LWORK >= N + (N+1)*NB. Here NB is the optimal
-*> block size for ZGEQP3 and ZGEQRF.
-*> In general, optimal LWORK is computed as
-*> LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZGEQRF), LWORK(ZGESVJ)).
-*> 1.2. .. an estimate of the scaled condition number of A is
-*> required (JOBA='E', or 'G'). In this case, LWORK the minimal
-*> requirement is LWORK >= N*N + 2*N.
-*> ->> For optimal performance (blocked code) the optimal value
-*> is LWORK >= max(N+(N+1)*NB, N*N+2*N)=N**2+2*N.
-*> In general, the optimal length LWORK is computed as
-*> LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZGEQRF), LWORK(ZGESVJ),
-*> N*N+LWORK(ZPOCON)).
-*> 2. If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
-*> (JOBU.EQ.'N')
-*> 2.1 .. no scaled condition estimate requested (JOBE.EQ.'N'):
-*> -> the minimal requirement is LWORK >= 3*N.
-*> -> For optimal performance,
-*> LWORK >= max(N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
-*> where NB is the optimal block size for ZGEQP3, ZGEQRF, ZGELQ,
-*> ZUNMLQ. In general, the optimal length LWORK is computed as
-*> LWORK >= max(N+LWORK(ZGEQP3), N+LWORK(ZGESVJ),
-*> N+LWORK(ZGELQF), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMLQ)).
-*> 2.2 .. an estimate of the scaled condition number of A is
-*> required (JOBA='E', or 'G').
-*> -> the minimal requirement is LWORK >= 3*N.
-*> -> For optimal performance,
-*> LWORK >= max(N+(N+1)*NB, 2*N,2*N+N*NB)=2*N+N*NB,
-*> where NB is the optimal block size for ZGEQP3, ZGEQRF, ZGELQ,
-*> ZUNMLQ. In general, the optimal length LWORK is computed as
-*> LWORK >= max(N+LWORK(ZGEQP3), LWORK(ZPOCON), N+LWORK(ZGESVJ),
-*> N+LWORK(ZGELQF), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMLQ)).
-*> 3. If SIGMA and the left singular vectors are needed
-*> 3.1 .. no scaled condition estimate requested (JOBE.EQ.'N'):
-*> -> the minimal requirement is LWORK >= 3*N.
-*> -> For optimal performance:
-*> if JOBU.EQ.'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
-*> where NB is the optimal block size for ZGEQP3, ZGEQRF, ZUNMQR.
-*> In general, the optimal length LWORK is computed as
-*> LWORK >= max(N+LWORK(ZGEQP3), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMQR)).
-*> 3.2 .. an estimate of the scaled condition number of A is
-*> required (JOBA='E', or 'G').
-*> -> the minimal requirement is LWORK >= 3*N.
-*> -> For optimal performance:
-*> if JOBU.EQ.'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
-*> where NB is the optimal block size for ZGEQP3, ZGEQRF, ZUNMQR.
-*> In general, the optimal length LWORK is computed as
-*> LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZPOCON),
-*> 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMQR)).
-*> 4. If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and
-*> 4.1. if JOBV.EQ.'V'
-*> the minimal requirement is LWORK >= 5*N+2*N*N.
-*> 4.2. if JOBV.EQ.'J' the minimal requirement is
-*> LWORK >= 4*N+N*N.
-*> In both cases, the allocated CWORK can accommodate blocked runs
-*> of ZGEQP3, ZGEQRF, ZGELQF, SUNMQR, ZUNMLQ.
-*>
-*> If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or
-*> LRWORK=-1), then on exit CWORK(1) contains the optimal and CWORK(2) contains the
-*> minimal length of CWORK for the job parameters used in the call.
-*> \endverbatim
-*>
-*> \param[out] RWORK
-*> \verbatim
-*> RWORK is DOUBLE PRECISION array, dimension (MAX(7,LWORK))
-*> On exit,
-*> RWORK(1) = Determines the scaling factor SCALE = RWORK(2) / RWORK(1)
-*> such that SCALE*SVA(1:N) are the computed singular values
-*> of A. (See the description of SVA().)
-*> RWORK(2) = See the description of RWORK(1).
-*> RWORK(3) = SCONDA is an estimate for the condition number of
-*> column equilibrated A. (If JOBA .EQ. 'E' or 'G')
-*> SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
-*> It is computed using SPOCON. It holds
-*> N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
-*> where R is the triangular factor from the QRF of A.
-*> However, if R is truncated and the numerical rank is
-*> determined to be strictly smaller than N, SCONDA is
-*> returned as -1, thus indicating that the smallest
-*> singular values might be lost.
-*>
-*> If full SVD is needed, the following two condition numbers are
-*> useful for the analysis of the algorithm. They are provied for
-*> a developer/implementer who is familiar with the details of
-*> the method.
-*>
-*> RWORK(4) = an estimate of the scaled condition number of the
-*> triangular factor in the first QR factorization.
-*> RWORK(5) = an estimate of the scaled condition number of the
-*> triangular factor in the second QR factorization.
-*> The following two parameters are computed if JOBT .EQ. 'T'.
-*> They are provided for a developer/implementer who is familiar
-*> with the details of the method.
-*> RWORK(6) = the entropy of A^* * A :: this is the Shannon entropy
-*> of diag(A^* * A) / Trace(A^* * A) taken as point in the
-*> probability simplex.
-*> RWORK(7) = the entropy of A * A^*. (See the description of RWORK(6).)
-*> If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or
-*> LRWORK=-1), then on exit RWORK(1) contains the required length of
-*> RWORK for the job parameters used in the call.
-*> \endverbatim
-*>
-*> \param[in] LRWORK
-*> \verbatim
-*> LRWORK is INTEGER
-*> Length of RWORK to confirm proper allocation of workspace.
-*> LRWORK depends on the job:
-*>
-*> 1. If only the singular values are requested i.e. if
-*> LSAME(JOBU,'N') .AND. LSAME(JOBV,'N')
-*> then:
-*> 1.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
-*> then: LRWORK = max( 7, 2 * M ).
-*> 1.2. Otherwise, LRWORK = max( 7, N ).
-*> 2. If singular values with the right singular vectors are requested
-*> i.e. if
-*> (LSAME(JOBV,'V').OR.LSAME(JOBV,'J')) .AND.
-*> .NOT.(LSAME(JOBU,'U').OR.LSAME(JOBU,'F'))
-*> then:
-*> 2.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
-*> then LRWORK = max( 7, 2 * M ).
-*> 2.2. Otherwise, LRWORK = max( 7, N ).
-*> 3. If singular values with the left singular vectors are requested, i.e. if
-*> (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
-*> .NOT.(LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
-*> then:
-*> 3.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
-*> then LRWORK = max( 7, 2 * M ).
-*> 3.2. Otherwise, LRWORK = max( 7, N ).
-*> 4. If singular values with both the left and the right singular vectors
-*> are requested, i.e. if
-*> (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
-*> (LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
-*> then:
-*> 4.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
-*> then LRWORK = max( 7, 2 * M ).
-*> 4.2. Otherwise, LRWORK = max( 7, N ).
-*>
-*> If, on entry, LRWORK = -1 or LWORK=-1, a workspace query is assumed and
-*> the length of RWORK is returned in RWORK(1).
-*> \endverbatim
-*>
-*> \param[out] IWORK
-*> \verbatim
-*> IWORK is INTEGER array, of dimension at least 4, that further depends
-*> on the job:
-*>
-*> 1. If only the singular values are requested then:
-*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
-*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
-*> 2. If the singular values and the right singular vectors are requested then:
-*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
-*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
-*> 3. If the singular values and the left singular vectors are requested then:
-*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
-*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
-*> 4. If the singular values with both the left and the right singular vectors
-*> are requested, then:
-*> 4.1. If LSAME(JOBV,'J') the length of IWORK is determined as follows:
-*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
-*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
-*> 4.2. If LSAME(JOBV,'V') the length of IWORK is determined as follows:
-*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
-*> then the length of IWORK is 2*N+M; otherwise the length of IWORK is 2*N.
-*>
-*> On exit,
-*> IWORK(1) = the numerical rank determined after the initial
-*> QR factorization with pivoting. See the descriptions
-*> of JOBA and JOBR.
-*> IWORK(2) = the number of the computed nonzero singular values
-*> IWORK(3) = if nonzero, a warning message:
-*> If IWORK(3).EQ.1 then some of the column norms of A
-*> were denormalized floats. The requested high accuracy
-*> is not warranted by the data.
-*> IWORK(4) = 1 or -1. If IWORK(4) .EQ. 1, then the procedure used A^* to
-*> do the job as specified by the JOB parameters.
-*> If the call to ZGEJSV is a workspace query (indicated by LWORK .EQ. -1 or
-*> LRWORK .EQ. -1), then on exit IWORK(1) contains the required length of
-*> IWORK for the job parameters used in the call.
-*> \endverbatim
-*>
-*> \param[out] INFO
-*> \verbatim
-*> INFO is INTEGER
-*> < 0 : if INFO = -i, then the i-th argument had an illegal value.
-*> = 0 : successful exit;
-*> > 0 : ZGEJSV did not converge in the maximal allowed number
-*> of sweeps. The computed values may be inaccurate.
-*> \endverbatim
-*
-* Authors:
-* ========
-*
-*> \author Univ. of Tennessee
-*> \author Univ. of California Berkeley
-*> \author Univ. of Colorado Denver
-*> \author NAG Ltd.
-*
-*> \date June 2016
-*
-*> \ingroup complex16GEsing
-*
-*> \par Further Details:
-* =====================
-*>
-*> \verbatim
-*>
-*> ZGEJSV implements a preconditioned Jacobi SVD algorithm. It uses ZGEQP3,
-*> ZGEQRF, and ZGELQF as preprocessors and preconditioners. Optionally, an
-*> additional row pivoting can be used as a preprocessor, which in some
-*> cases results in much higher accuracy. An example is matrix A with the
-*> structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
-*> diagonal matrices and C is well-conditioned matrix. In that case, complete
-*> pivoting in the first QR factorizations provides accuracy dependent on the
-*> condition number of C, and independent of D1, D2. Such higher accuracy is
-*> not completely understood theoretically, but it works well in practice.
-*> Further, if A can be written as A = B*D, with well-conditioned B and some
-*> diagonal D, then the high accuracy is guaranteed, both theoretically and
-*> in software, independent of D. For more details see [1], [2].
-*> The computational range for the singular values can be the full range
-*> ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
-*> & LAPACK routines called by ZGEJSV are implemented to work in that range.
-*> If that is not the case, then the restriction for safe computation with
-*> the singular values in the range of normalized IEEE numbers is that the
-*> spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
-*> overflow. This code (ZGEJSV) is best used in this restricted range,
-*> meaning that singular values of magnitude below ||A||_2 / DLAMCH('O') are
-*> returned as zeros. See JOBR for details on this.
-*> Further, this implementation is somewhat slower than the one described
-*> in [1,2] due to replacement of some non-LAPACK components, and because
-*> the choice of some tuning parameters in the iterative part (ZGESVJ) is
-*> left to the implementer on a particular machine.
-*> The rank revealing QR factorization (in this code: ZGEQP3) should be
-*> implemented as in [3]. We have a new version of ZGEQP3 under development
-*> that is more robust than the current one in LAPACK, with a cleaner cut in
-*> rank deficient cases. It will be available in the SIGMA library [4].
-*> If M is much larger than N, it is obvious that the initial QRF with
-*> column pivoting can be preprocessed by the QRF without pivoting. That
-*> well known trick is not used in ZGEJSV because in some cases heavy row
-*> weighting can be treated with complete pivoting. The overhead in cases
-*> M much larger than N is then only due to pivoting, but the benefits in
-*> terms of accuracy have prevailed. The implementer/user can incorporate
-*> this extra QRF step easily. The implementer can also improve data movement
-*> (matrix transpose, matrix copy, matrix transposed copy) - this
-*> implementation of ZGEJSV uses only the simplest, naive data movement.
-*> \endverbatim
-*
-*> \par Contributor:
-* ==================
-*>
-*> Zlatko Drmac, Department of Mathematics, Faculty of Science,
-*> University of Zagreb (Zagreb, Croatia); drmac@math.hr
-*
-*> \par References:
-* ================
-*>
-*> \verbatim
-*>
-*> [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
-*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
-*> LAPACK Working note 169.
-*> [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
-*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
-*> LAPACK Working note 170.
-*> [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
-*> factorization software - a case study.
-*> ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
-*> LAPACK Working note 176.
-*> [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
-*> QSVD, (H,K)-SVD computations.
-*> Department of Mathematics, University of Zagreb, 2008, 2016.
-*> \endverbatim
-*
-*> \par Bugs, examples and comments:
-* =================================
-*>
-*> Please report all bugs and send interesting examples and/or comments to
-*> drmac@math.hr. Thank you.
-*>
-* =====================================================================
- SUBROUTINE ZGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
- $ M, N, A, LDA, SVA, U, LDU, V, LDV,
- $ CWORK, LWORK, RWORK, LRWORK, IWORK, INFO )
-*
-* -- LAPACK computational routine (version 3.7.0) --
-* -- LAPACK is a software package provided by Univ. of Tennessee, --
-* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* December 2016
-*
-* .. Scalar Arguments ..
- IMPLICIT NONE
- INTEGER INFO, LDA, LDU, LDV, LWORK, LRWORK, M, N
-* ..
-* .. Array Arguments ..
- COMPLEX*16 A( LDA, * ), U( LDU, * ), V( LDV, * ),
- $ CWORK( LWORK )
- DOUBLE PRECISION SVA( N ), RWORK( LRWORK )
- INTEGER IWORK( * )
- CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
-* ..
-*
-* ===========================================================================
-*
-* .. Local Parameters ..
- DOUBLE PRECISION ZERO, ONE
- PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
- COMPLEX*16 CZERO, CONE
- PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ), CONE = ( 1.0D0, 0.0D0 ) )
-* ..
-* .. Local Scalars ..
- COMPLEX*16 CTEMP
- DOUBLE PRECISION AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1,
- $ COND_OK, CONDR1, CONDR2, ENTRA, ENTRAT, EPSLN,
- $ MAXPRJ, SCALEM, SCONDA, SFMIN, SMALL, TEMP1,
- $ USCAL1, USCAL2, XSC
- INTEGER IERR, N1, NR, NUMRANK, p, q, WARNING
- LOGICAL ALMORT, DEFR, ERREST, GOSCAL, JRACC, KILL, LQUERY,
- $ LSVEC, L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN, NOSCAL,
- $ ROWPIV, RSVEC, TRANSP
-*
- INTEGER OPTWRK, MINWRK, MINRWRK, MINIWRK
- INTEGER LWCON, LWLQF, LWQP3, LWQRF, LWUNMLQ, LWUNMQR, LWUNMQRM,
- $ LWSVDJ, LWSVDJV, LRWQP3, LRWCON, LRWSVDJ, IWOFF
- INTEGER LWRK_ZGELQF, LWRK_ZGEQP3, LWRK_ZGEQP3N, LWRK_ZGEQRF,
- $ LWRK_ZGESVJ, LWRK_ZGESVJV, LWRK_ZGESVJU, LWRK_ZUNMLQ,
- $ LWRK_ZUNMQR, LWRK_ZUNMQRM
-* ..
-* .. Local Arrays
- COMPLEX*16 CDUMMY(1)
- DOUBLE PRECISION RDUMMY(1)
-*
-* .. Intrinsic Functions ..
- INTRINSIC ABS, DCMPLX, CONJG, DLOG, MAX, MIN, DBLE, NINT, SQRT
-* ..
-* .. External Functions ..
- DOUBLE PRECISION DLAMCH, DZNRM2
- INTEGER IDAMAX, IZAMAX
- LOGICAL LSAME
- EXTERNAL IDAMAX, IZAMAX, LSAME, DLAMCH, DZNRM2
-* ..
-* .. External Subroutines ..
- EXTERNAL DLASSQ, ZCOPY, ZGELQF, ZGEQP3, ZGEQRF, ZLACPY, ZLAPMR,
- $ ZLASCL, DLASCL, ZLASET, ZLASSQ, ZLASWP, ZUNGQR, ZUNMLQ,
- $ ZUNMQR, ZPOCON, DSCAL, ZDSCAL, ZSWAP, ZTRSM, ZLACGV,
- $ XERBLA
-*
- EXTERNAL ZGESVJ
-* ..
-*
-* Test the input arguments
-*
- LSVEC = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' )
- JRACC = LSAME( JOBV, 'J' )
- RSVEC = LSAME( JOBV, 'V' ) .OR. JRACC
- ROWPIV = LSAME( JOBA, 'F' ) .OR. LSAME( JOBA, 'G' )
- L2RANK = LSAME( JOBA, 'R' )
- L2ABER = LSAME( JOBA, 'A' )
- ERREST = LSAME( JOBA, 'E' ) .OR. LSAME( JOBA, 'G' )
- L2TRAN = LSAME( JOBT, 'T' ) .AND. ( M .EQ. N )
- L2KILL = LSAME( JOBR, 'R' )
- DEFR = LSAME( JOBR, 'N' )
- L2PERT = LSAME( JOBP, 'P' )
-*
- LQUERY = ( LWORK .EQ. -1 ) .OR. ( LRWORK .EQ. -1 )
-*
- IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR.
- $ ERREST .OR. LSAME( JOBA, 'C' ) )) THEN
- INFO = - 1
- ELSE IF ( .NOT.( LSVEC .OR. LSAME( JOBU, 'N' ) .OR.
- $ ( LSAME( JOBU, 'W' ) .AND. RSVEC .AND. L2TRAN ) ) ) THEN
- INFO = - 2
- ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.
- $ ( LSAME( JOBV, 'W' ) .AND. LSVEC .AND. L2TRAN ) ) ) THEN
- INFO = - 3
- ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) ) THEN
- INFO = - 4
- ELSE IF ( .NOT. ( LSAME(JOBT,'T') .OR. LSAME(JOBT,'N') ) ) THEN
- INFO = - 5
- ELSE IF ( .NOT. ( L2PERT .OR. LSAME( JOBP, 'N' ) ) ) THEN
- INFO = - 6
- ELSE IF ( M .LT. 0 ) THEN
- INFO = - 7
- ELSE IF ( ( N .LT. 0 ) .OR. ( N .GT. M ) ) THEN
- INFO = - 8
- ELSE IF ( LDA .LT. M ) THEN
- INFO = - 10
- ELSE IF ( LSVEC .AND. ( LDU .LT. M ) ) THEN
- INFO = - 13
- ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN
- INFO = - 15
- ELSE
-* #:)
- INFO = 0
- END IF
-*
- IF ( INFO .EQ. 0 ) THEN
-* .. compute the minimal and the optimal workspace lengths
-* [[The expressions for computing the minimal and the optimal
-* values of LCWORK, LRWORK are written with a lot of redundancy and
-* can be simplified. However, this verbose form is useful for
-* maintenance and modifications of the code.]]
-*
-* .. minimal workspace length for ZGEQP3 of an M x N matrix,
-* ZGEQRF of an N x N matrix, ZGELQF of an N x N matrix,
-* ZUNMLQ for computing N x N matrix, ZUNMQR for computing N x N
-* matrix, ZUNMQR for computing M x N matrix, respectively.
- LWQP3 = N+1
- LWQRF = MAX( 1, N )
- LWLQF = MAX( 1, N )
- LWUNMLQ = MAX( 1, N )
- LWUNMQR = MAX( 1, N )
- LWUNMQRM = MAX( 1, M )
-* .. minimal workspace length for ZPOCON of an N x N matrix
- LWCON = 2 * N
-* .. minimal workspace length for ZGESVJ of an N x N matrix,
-* without and with explicit accumulation of Jacobi rotations
- LWSVDJ = MAX( 2 * N, 1 )
- LWSVDJV = MAX( 2 * N, 1 )
-* .. minimal REAL workspace length for ZGEQP3, ZPOCON, ZGESVJ
- LRWQP3 = N
- LRWCON = N
- LRWSVDJ = N
- IF ( LQUERY ) THEN
- CALL ZGEQP3( M, N, A, LDA, IWORK, CDUMMY, CDUMMY, -1,
- $ RDUMMY, IERR )
- LWRK_ZGEQP3 = CDUMMY(1)
- CALL ZGEQRF( N, N, A, LDA, CDUMMY, CDUMMY,-1, IERR )
- LWRK_ZGEQRF = CDUMMY(1)
- CALL ZGELQF( N, N, A, LDA, CDUMMY, CDUMMY,-1, IERR )
- LWRK_ZGELQF = CDUMMY(1)
- END IF
- MINWRK = 2
- OPTWRK = 2
- MINIWRK = N
- IF ( .NOT. (LSVEC .OR. RSVEC ) ) THEN
-* .. minimal and optimal sizes of the complex workspace if
-* only the singular values are requested
- IF ( ERREST ) THEN
- MINWRK = MAX( N+LWQP3, N**2+LWCON, N+LWQRF, LWSVDJ )
- ELSE
- MINWRK = MAX( N+LWQP3, N+LWQRF, LWSVDJ )
- END IF
- IF ( LQUERY ) THEN
- CALL ZGESVJ( 'L', 'N', 'N', N, N, A, LDA, SVA, N, V,
- $ LDV, CDUMMY, -1, RDUMMY, -1, IERR )
- LWRK_ZGESVJ = CDUMMY(1)
- IF ( ERREST ) THEN
- OPTWRK = MAX( N+LWRK_ZGEQP3, N**2+LWCON,
- $ N+LWRK_ZGEQRF, LWRK_ZGESVJ )
- ELSE
- OPTWRK = MAX( N+LWRK_ZGEQP3, N+LWRK_ZGEQRF,
- $ LWRK_ZGESVJ )
- END IF
- END IF
- IF ( L2TRAN .OR. ROWPIV ) THEN
- IF ( ERREST ) THEN
- MINRWRK = MAX( 7, 2*M, LRWQP3, LRWCON, LRWSVDJ )
- ELSE
- MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
- END IF
- ELSE
- IF ( ERREST ) THEN
- MINRWRK = MAX( 7, LRWQP3, LRWCON, LRWSVDJ )
- ELSE
- MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
- END IF
- END IF
- IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
- ELSE IF ( RSVEC .AND. (.NOT.LSVEC) ) THEN
-* .. minimal and optimal sizes of the complex workspace if the
-* singular values and the right singular vectors are requested
- IF ( ERREST ) THEN
- MINWRK = MAX( N+LWQP3, LWCON, LWSVDJ, N+LWLQF,
- $ 2*N+LWQRF, N+LWSVDJ, N+LWUNMLQ )
- ELSE
- MINWRK = MAX( N+LWQP3, LWSVDJ, N+LWLQF, 2*N+LWQRF,
- $ N+LWSVDJ, N+LWUNMLQ )
- END IF
- IF ( LQUERY ) THEN
- CALL ZGESVJ( 'L', 'U', 'N', N,N, U, LDU, SVA, N, A,
- $ LDA, CDUMMY, -1, RDUMMY, -1, IERR )
- LWRK_ZGESVJ = CDUMMY(1)
- CALL ZUNMLQ( 'L', 'C', N, N, N, A, LDA, CDUMMY,
- $ V, LDV, CDUMMY, -1, IERR )
- LWRK_ZUNMLQ = CDUMMY(1)
- IF ( ERREST ) THEN
- OPTWRK = MAX( N+LWRK_ZGEQP3, LWCON, LWRK_ZGESVJ,
- $ N+LWRK_ZGELQF, 2*N+LWRK_ZGEQRF,
- $ N+LWRK_ZGESVJ, N+LWRK_ZUNMLQ )
- ELSE
- OPTWRK = MAX( N+LWRK_ZGEQP3, LWRK_ZGESVJ,N+LWRK_ZGELQF,
- $ 2*N+LWRK_ZGEQRF, N+LWRK_ZGESVJ,
- $ N+LWRK_ZUNMLQ )
- END IF
- END IF
- IF ( L2TRAN .OR. ROWPIV ) THEN
- IF ( ERREST ) THEN
- MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
- ELSE
- MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
- END IF
- ELSE
- IF ( ERREST ) THEN
- MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
- ELSE
- MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
- END IF
- END IF
- IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
- ELSE IF ( LSVEC .AND. (.NOT.RSVEC) ) THEN
-* .. minimal and optimal sizes of the complex workspace if the
-* singular values and the left singular vectors are requested
- IF ( ERREST ) THEN
- MINWRK = N + MAX( LWQP3,LWCON,N+LWQRF,LWSVDJ,LWUNMQRM )
- ELSE
- MINWRK = N + MAX( LWQP3, N+LWQRF, LWSVDJ, LWUNMQRM )
- END IF
- IF ( LQUERY ) THEN
- CALL ZGESVJ( 'L', 'U', 'N', N,N, U, LDU, SVA, N, A,
- $ LDA, CDUMMY, -1, RDUMMY, -1, IERR )
- LWRK_ZGESVJ = CDUMMY(1)
- CALL ZUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
- $ LDU, CDUMMY, -1, IERR )
- LWRK_ZUNMQRM = CDUMMY(1)
- IF ( ERREST ) THEN
- OPTWRK = N + MAX( LWRK_ZGEQP3, LWCON, N+LWRK_ZGEQRF,
- $ LWRK_ZGESVJ, LWRK_ZUNMQRM )
- ELSE
- OPTWRK = N + MAX( LWRK_ZGEQP3, N+LWRK_ZGEQRF,
- $ LWRK_ZGESVJ, LWRK_ZUNMQRM )
- END IF
- END IF
- IF ( L2TRAN .OR. ROWPIV ) THEN
- IF ( ERREST ) THEN
- MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
- ELSE
- MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
- END IF
- ELSE
- IF ( ERREST ) THEN
- MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
- ELSE
- MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
- END IF
- END IF
- IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
- ELSE
-* .. minimal and optimal sizes of the complex workspace if the
-* full SVD is requested
- IF ( .NOT. JRACC ) THEN
- IF ( ERREST ) THEN
- MINWRK = MAX( N+LWQP3, N+LWCON, 2*N+N**2+LWCON,
- $ 2*N+LWQRF, 2*N+LWQP3,
- $ 2*N+N**2+N+LWLQF, 2*N+N**2+N+N**2+LWCON,
- $ 2*N+N**2+N+LWSVDJ, 2*N+N**2+N+LWSVDJV,
- $ 2*N+N**2+N+LWUNMQR,2*N+N**2+N+LWUNMLQ,
- $ N+N**2+LWSVDJ, N+LWUNMQRM )
- ELSE
- MINWRK = MAX( N+LWQP3, 2*N+N**2+LWCON,
- $ 2*N+LWQRF, 2*N+LWQP3,
- $ 2*N+N**2+N+LWLQF, 2*N+N**2+N+N**2+LWCON,
- $ 2*N+N**2+N+LWSVDJ, 2*N+N**2+N+LWSVDJV,
- $ 2*N+N**2+N+LWUNMQR,2*N+N**2+N+LWUNMLQ,
- $ N+N**2+LWSVDJ, N+LWUNMQRM )
- END IF
- MINIWRK = MINIWRK + N
- IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
- ELSE
- IF ( ERREST ) THEN
- MINWRK = MAX( N+LWQP3, N+LWCON, 2*N+LWQRF,
- $ 2*N+N**2+LWSVDJV, 2*N+N**2+N+LWUNMQR,
- $ N+LWUNMQRM )
- ELSE
- MINWRK = MAX( N+LWQP3, 2*N+LWQRF,
- $ 2*N+N**2+LWSVDJV, 2*N+N**2+N+LWUNMQR,
- $ N+LWUNMQRM )
- END IF
- IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
- END IF
- IF ( LQUERY ) THEN
- CALL ZUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
- $ LDU, CDUMMY, -1, IERR )
- LWRK_ZUNMQRM = CDUMMY(1)
- CALL ZUNMQR( 'L', 'N', N, N, N, A, LDA, CDUMMY, U,
- $ LDU, CDUMMY, -1, IERR )
- LWRK_ZUNMQR = CDUMMY(1)
- IF ( .NOT. JRACC ) THEN
- CALL ZGEQP3( N,N, A, LDA, IWORK, CDUMMY,CDUMMY, -1,
- $ RDUMMY, IERR )
- LWRK_ZGEQP3N = CDUMMY(1)
- CALL ZGESVJ( 'L', 'U', 'N', N, N, U, LDU, SVA,
- $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
- LWRK_ZGESVJ = CDUMMY(1)
- CALL ZGESVJ( 'U', 'U', 'N', N, N, U, LDU, SVA,
- $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
- LWRK_ZGESVJU = CDUMMY(1)
- CALL ZGESVJ( 'L', 'U', 'V', N, N, U, LDU, SVA,
- $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
- LWRK_ZGESVJV = CDUMMY(1)
- CALL ZUNMLQ( 'L', 'C', N, N, N, A, LDA, CDUMMY,
- $ V, LDV, CDUMMY, -1, IERR )
- LWRK_ZUNMLQ = CDUMMY(1)
- IF ( ERREST ) THEN
- OPTWRK = MAX( N+LWRK_ZGEQP3, N+LWCON,
- $ 2*N+N**2+LWCON, 2*N+LWRK_ZGEQRF,
- $ 2*N+LWRK_ZGEQP3N,
- $ 2*N+N**2+N+LWRK_ZGELQF,
- $ 2*N+N**2+N+N**2+LWCON,
- $ 2*N+N**2+N+LWRK_ZGESVJ,
- $ 2*N+N**2+N+LWRK_ZGESVJV,
- $ 2*N+N**2+N+LWRK_ZUNMQR,
- $ 2*N+N**2+N+LWRK_ZUNMLQ,
- $ N+N**2+LWRK_ZGESVJU,
- $ N+LWRK_ZUNMQRM )
- ELSE
- OPTWRK = MAX( N+LWRK_ZGEQP3,
- $ 2*N+N**2+LWCON, 2*N+LWRK_ZGEQRF,
- $ 2*N+LWRK_ZGEQP3N,
- $ 2*N+N**2+N+LWRK_ZGELQF,
- $ 2*N+N**2+N+N**2+LWCON,
- $ 2*N+N**2+N+LWRK_ZGESVJ,
- $ 2*N+N**2+N+LWRK_ZGESVJV,
- $ 2*N+N**2+N+LWRK_ZUNMQR,
- $ 2*N+N**2+N+LWRK_ZUNMLQ,
- $ N+N**2+LWRK_ZGESVJU,
- $ N+LWRK_ZUNMQRM )
- END IF
- ELSE
- CALL ZGESVJ( 'L', 'U', 'V', N, N, U, LDU, SVA,
- $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
- LWRK_ZGESVJV = CDUMMY(1)
- CALL ZUNMQR( 'L', 'N', N, N, N, CDUMMY, N, CDUMMY,
- $ V, LDV, CDUMMY, -1, IERR )
- LWRK_ZUNMQR = CDUMMY(1)
- CALL ZUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
- $ LDU, CDUMMY, -1, IERR )
- LWRK_ZUNMQRM = CDUMMY(1)
- IF ( ERREST ) THEN
- OPTWRK = MAX( N+LWRK_ZGEQP3, N+LWCON,
- $ 2*N+LWRK_ZGEQRF, 2*N+N**2,
- $ 2*N+N**2+LWRK_ZGESVJV,
- $ 2*N+N**2+N+LWRK_ZUNMQR,N+LWRK_ZUNMQRM )
- ELSE
- OPTWRK = MAX( N+LWRK_ZGEQP3, 2*N+LWRK_ZGEQRF,
- $ 2*N+N**2, 2*N+N**2+LWRK_ZGESVJV,
- $ 2*N+N**2+N+LWRK_ZUNMQR,
- $ N+LWRK_ZUNMQRM )
- END IF
- END IF
- END IF
- IF ( L2TRAN .OR. ROWPIV ) THEN
- MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
- ELSE
- MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
- END IF
- END IF
- MINWRK = MAX( 2, MINWRK )
- OPTWRK = MAX( 2, OPTWRK )
- IF ( LWORK .LT. MINWRK .AND. (.NOT.LQUERY) ) INFO = - 17
- IF ( LRWORK .LT. MINRWRK .AND. (.NOT.LQUERY) ) INFO = - 19
- END IF
-*
- IF ( INFO .NE. 0 ) THEN
-* #:(
- CALL XERBLA( 'ZGEJSV', - INFO )
- RETURN
- ELSE IF ( LQUERY ) THEN
- CWORK(1) = OPTWRK
- CWORK(2) = MINWRK
- RWORK(1) = MINRWRK
- IWORK(1) = MAX( 4, MINIWRK )
- RETURN
- END IF
-*
-* Quick return for void matrix (Y3K safe)
-* #:)
- IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) THEN
- IWORK(1:4) = 0
- RWORK(1:7) = 0
- RETURN
- ENDIF
-*
-* Determine whether the matrix U should be M x N or M x M
-*
- IF ( LSVEC ) THEN
- N1 = N
- IF ( LSAME( JOBU, 'F' ) ) N1 = M
- END IF
-*
-* Set numerical parameters
-*
-*! NOTE: Make sure DLAMCH() does not fail on the target architecture.
-*
- EPSLN = DLAMCH('Epsilon')
- SFMIN = DLAMCH('SafeMinimum')
- SMALL = SFMIN / EPSLN
- BIG = DLAMCH('O')
-* BIG = ONE / SFMIN
-*
-* Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N
-*
-*(!) If necessary, scale SVA() to protect the largest norm from
-* overflow. It is possible that this scaling pushes the smallest
-* column norm left from the underflow threshold (extreme case).
-*
- SCALEM = ONE / SQRT(DBLE(M)*DBLE(N))
- NOSCAL = .TRUE.
- GOSCAL = .TRUE.
- DO 1874 p = 1, N
- AAPP = ZERO
- AAQQ = ONE
- CALL ZLASSQ( M, A(1,p), 1, AAPP, AAQQ )
- IF ( AAPP .GT. BIG ) THEN
- INFO = - 9
- CALL XERBLA( 'ZGEJSV', -INFO )
- RETURN
- END IF
- AAQQ = SQRT(AAQQ)
- IF ( ( AAPP .LT. (BIG / AAQQ) ) .AND. NOSCAL ) THEN
- SVA(p) = AAPP * AAQQ
- ELSE
- NOSCAL = .FALSE.
- SVA(p) = AAPP * ( AAQQ * SCALEM )
- IF ( GOSCAL ) THEN
- GOSCAL = .FALSE.
- CALL DSCAL( p-1, SCALEM, SVA, 1 )
- END IF
- END IF
- 1874 CONTINUE
-*
- IF ( NOSCAL ) SCALEM = ONE
-*
- AAPP = ZERO
- AAQQ = BIG
- DO 4781 p = 1, N
- AAPP = MAX( AAPP, SVA(p) )
- IF ( SVA(p) .NE. ZERO ) AAQQ = MIN( AAQQ, SVA(p) )
- 4781 CONTINUE
-*
-* Quick return for zero M x N matrix
-* #:)
- IF ( AAPP .EQ. ZERO ) THEN
- IF ( LSVEC ) CALL ZLASET( 'G', M, N1, CZERO, CONE, U, LDU )
- IF ( RSVEC ) CALL ZLASET( 'G', N, N, CZERO, CONE, V, LDV )
- RWORK(1) = ONE
- RWORK(2) = ONE
- IF ( ERREST ) RWORK(3) = ONE
- IF ( LSVEC .AND. RSVEC ) THEN
- RWORK(4) = ONE
- RWORK(5) = ONE
- END IF
- IF ( L2TRAN ) THEN
- RWORK(6) = ZERO
- RWORK(7) = ZERO
- END IF
- IWORK(1) = 0
- IWORK(2) = 0
- IWORK(3) = 0
- IWORK(4) = -1
- RETURN
- END IF
-*
-* Issue warning if denormalized column norms detected. Override the
-* high relative accuracy request. Issue licence to kill nonzero columns
-* (set them to zero) whose norm is less than sigma_max / BIG (roughly).
-* #:(
- WARNING = 0
- IF ( AAQQ .LE. SFMIN ) THEN
- L2RANK = .TRUE.
- L2KILL = .TRUE.
- WARNING = 1
- END IF
-*
-* Quick return for one-column matrix
-* #:)
- IF ( N .EQ. 1 ) THEN
-*
- IF ( LSVEC ) THEN
- CALL ZLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR )
- CALL ZLACPY( 'A', M, 1, A, LDA, U, LDU )
-* computing all M left singular vectors of the M x 1 matrix
- IF ( N1 .NE. N ) THEN
- CALL ZGEQRF( M, N, U,LDU, CWORK, CWORK(N+1),LWORK-N,IERR )
- CALL ZUNGQR( M,N1,1, U,LDU,CWORK,CWORK(N+1),LWORK-N,IERR )
- CALL ZCOPY( M, A(1,1), 1, U(1,1), 1 )
- END IF
- END IF
- IF ( RSVEC ) THEN
- V(1,1) = CONE
- END IF
- IF ( SVA(1) .LT. (BIG*SCALEM) ) THEN
- SVA(1) = SVA(1) / SCALEM
- SCALEM = ONE
- END IF
- RWORK(1) = ONE / SCALEM
- RWORK(2) = ONE
- IF ( SVA(1) .NE. ZERO ) THEN
- IWORK(1) = 1
- IF ( ( SVA(1) / SCALEM) .GE. SFMIN ) THEN
- IWORK(2) = 1
- ELSE
- IWORK(2) = 0
- END IF
- ELSE
- IWORK(1) = 0
- IWORK(2) = 0
- END IF
- IWORK(3) = 0
- IWORK(4) = -1
- IF ( ERREST ) RWORK(3) = ONE
- IF ( LSVEC .AND. RSVEC ) THEN
- RWORK(4) = ONE
- RWORK(5) = ONE
- END IF
- IF ( L2TRAN ) THEN
- RWORK(6) = ZERO
- RWORK(7) = ZERO
- END IF
- RETURN
-*
- END IF
-*
- TRANSP = .FALSE.
-*
- AATMAX = -ONE
- AATMIN = BIG
- IF ( ROWPIV .OR. L2TRAN ) THEN
-*
-* Compute the row norms, needed to determine row pivoting sequence
-* (in the case of heavily row weighted A, row pivoting is strongly
-* advised) and to collect information needed to compare the
-* structures of A * A^* and A^* * A (in the case L2TRAN.EQ..TRUE.).
-*
- IF ( L2TRAN ) THEN
- DO 1950 p = 1, M
- XSC = ZERO
- TEMP1 = ONE
- CALL ZLASSQ( N, A(p,1), LDA, XSC, TEMP1 )
-* ZLASSQ gets both the ell_2 and the ell_infinity norm
-* in one pass through the vector
- RWORK(M+p) = XSC * SCALEM
- RWORK(p) = XSC * (SCALEM*SQRT(TEMP1))
- AATMAX = MAX( AATMAX, RWORK(p) )
- IF (RWORK(p) .NE. ZERO)
- $ AATMIN = MIN(AATMIN,RWORK(p))
- 1950 CONTINUE
- ELSE
- DO 1904 p = 1, M
- RWORK(M+p) = SCALEM*ABS( A(p,IZAMAX(N,A(p,1),LDA)) )
- AATMAX = MAX( AATMAX, RWORK(M+p) )
- AATMIN = MIN( AATMIN, RWORK(M+p) )
- 1904 CONTINUE
- END IF
-*
- END IF
-*
-* For square matrix A try to determine whether A^* would be better
-* input for the preconditioned Jacobi SVD, with faster convergence.
-* The decision is based on an O(N) function of the vector of column
-* and row norms of A, based on the Shannon entropy. This should give
-* the right choice in most cases when the difference actually matters.
-* It may fail and pick the slower converging side.
-*
- ENTRA = ZERO
- ENTRAT = ZERO
- IF ( L2TRAN ) THEN
-*
- XSC = ZERO
- TEMP1 = ONE
- CALL DLASSQ( N, SVA, 1, XSC, TEMP1 )
- TEMP1 = ONE / TEMP1
-*
- ENTRA = ZERO
- DO 1113 p = 1, N
- BIG1 = ( ( SVA(p) / XSC )**2 ) * TEMP1
- IF ( BIG1 .NE. ZERO ) ENTRA = ENTRA + BIG1 * DLOG(BIG1)
- 1113 CONTINUE
- ENTRA = - ENTRA / DLOG(DBLE(N))
-*
-* Now, SVA().^2/Trace(A^* * A) is a point in the probability simplex.
-* It is derived from the diagonal of A^* * A. Do the same with the
-* diagonal of A * A^*, compute the entropy of the corresponding
-* probability distribution. Note that A * A^* and A^* * A have the
-* same trace.
-*
- ENTRAT = ZERO
- DO 1114 p = 1, M
- BIG1 = ( ( RWORK(p) / XSC )**2 ) * TEMP1
- IF ( BIG1 .NE. ZERO ) ENTRAT = ENTRAT + BIG1 * DLOG(BIG1)
- 1114 CONTINUE
- ENTRAT = - ENTRAT / DLOG(DBLE(M))
-*
-* Analyze the entropies and decide A or A^*. Smaller entropy
-* usually means better input for the algorithm.
-*
- TRANSP = ( ENTRAT .LT. ENTRA )
-*
-* If A^* is better than A, take the adjoint of A. This is allowed
-* only for square matrices, M=N.
- IF ( TRANSP ) THEN
-* In an optimal implementation, this trivial transpose
-* should be replaced with faster transpose.
- DO 1115 p = 1, N - 1
- A(p,p) = CONJG(A(p,p))
- DO 1116 q = p + 1, N
- CTEMP = CONJG(A(q,p))
- A(q,p) = CONJG(A(p,q))
- A(p,q) = CTEMP
- 1116 CONTINUE
- 1115 CONTINUE
- A(N,N) = CONJG(A(N,N))
- DO 1117 p = 1, N
- RWORK(M+p) = SVA(p)
- SVA(p) = RWORK(p)
-* previously computed row 2-norms are now column 2-norms
-* of the transposed matrix
- 1117 CONTINUE
- TEMP1 = AAPP
- AAPP = AATMAX
- AATMAX = TEMP1
- TEMP1 = AAQQ
- AAQQ = AATMIN
- AATMIN = TEMP1
- KILL = LSVEC
- LSVEC = RSVEC
- RSVEC = KILL
- IF ( LSVEC ) N1 = N
-*
- ROWPIV = .TRUE.
- END IF
-*
- END IF
-* END IF L2TRAN
-*
-* Scale the matrix so that its maximal singular value remains less
-* than SQRT(BIG) -- the matrix is scaled so that its maximal column
-* has Euclidean norm equal to SQRT(BIG/N). The only reason to keep
-* SQRT(BIG) instead of BIG is the fact that ZGEJSV uses LAPACK and
-* BLAS routines that, in some implementations, are not capable of
-* working in the full interval [SFMIN,BIG] and that they may provoke
-* overflows in the intermediate results. If the singular values spread
-* from SFMIN to BIG, then ZGESVJ will compute them. So, in that case,
-* one should use ZGESVJ instead of ZGEJSV.
-* >> change in the April 2016 update: allow bigger range, i.e. the
-* largest column is allowed up to BIG/N and ZGESVJ will do the rest.
- BIG1 = SQRT( BIG )
- TEMP1 = SQRT( BIG / DBLE(N) )
-* TEMP1 = BIG/DBLE(N)
-*
- CALL DLASCL( 'G', 0, 0, AAPP, TEMP1, N, 1, SVA, N, IERR )
- IF ( AAQQ .GT. (AAPP * SFMIN) ) THEN
- AAQQ = ( AAQQ / AAPP ) * TEMP1
- ELSE
- AAQQ = ( AAQQ * TEMP1 ) / AAPP
- END IF
- TEMP1 = TEMP1 * SCALEM
- CALL ZLASCL( 'G', 0, 0, AAPP, TEMP1, M, N, A, LDA, IERR )
-*
-* To undo scaling at the end of this procedure, multiply the
-* computed singular values with USCAL2 / USCAL1.
-*
- USCAL1 = TEMP1
- USCAL2 = AAPP
-*
- IF ( L2KILL ) THEN
-* L2KILL enforces computation of nonzero singular values in
-* the restricted range of condition number of the initial A,
-* sigma_max(A) / sigma_min(A) approx. SQRT(BIG)/SQRT(SFMIN).
- XSC = SQRT( SFMIN )
- ELSE
- XSC = SMALL
-*
-* Now, if the condition number of A is too big,
-* sigma_max(A) / sigma_min(A) .GT. SQRT(BIG/N) * EPSLN / SFMIN,
-* as a precaution measure, the full SVD is computed using ZGESVJ
-* with accumulated Jacobi rotations. This provides numerically
-* more robust computation, at the cost of slightly increased run
-* time. Depending on the concrete implementation of BLAS and LAPACK
-* (i.e. how they behave in presence of extreme ill-conditioning) the
-* implementor may decide to remove this switch.
- IF ( ( AAQQ.LT.SQRT(SFMIN) ) .AND. LSVEC .AND. RSVEC ) THEN
- JRACC = .TRUE.
- END IF
-*
- END IF
- IF ( AAQQ .LT. XSC ) THEN
- DO 700 p = 1, N
- IF ( SVA(p) .LT. XSC ) THEN
- CALL ZLASET( 'A', M, 1, CZERO, CZERO, A(1,p), LDA )
- SVA(p) = ZERO
- END IF
- 700 CONTINUE
- END IF
-*
-* Preconditioning using QR factorization with pivoting
-*
- IF ( ROWPIV ) THEN
-* Optional row permutation (Bjoerck row pivoting):
-* A result by Cox and Higham shows that the Bjoerck's
-* row pivoting combined with standard column pivoting
-* has similar effect as Powell-Reid complete pivoting.
-* The ell-infinity norms of A are made nonincreasing.
- IF ( ( LSVEC .AND. RSVEC ) .AND. .NOT.( JRACC ) ) THEN
- IWOFF = 2*N
- ELSE
- IWOFF = N
- END IF
- DO 1952 p = 1, M - 1
- q = IDAMAX( M-p+1, RWORK(M+p), 1 ) + p - 1
- IWORK(IWOFF+p) = q
- IF ( p .NE. q ) THEN
- TEMP1 = RWORK(M+p)
- RWORK(M+p) = RWORK(M+q)
- RWORK(M+q) = TEMP1
- END IF
- 1952 CONTINUE
- CALL ZLASWP( N, A, LDA, 1, M-1, IWORK(IWOFF+1), 1 )
- END IF
-*
-* End of the preparation phase (scaling, optional sorting and
-* transposing, optional flushing of small columns).
-*
-* Preconditioning
-*
-* If the full SVD is needed, the right singular vectors are computed
-* from a matrix equation, and for that we need theoretical analysis
-* of the Businger-Golub pivoting. So we use ZGEQP3 as the first RR QRF.
-* In all other cases the first RR QRF can be chosen by other criteria
-* (eg speed by replacing global with restricted window pivoting, such
-* as in xGEQPX from TOMS # 782). Good results will be obtained using
-* xGEQPX with properly (!) chosen numerical parameters.
-* Any improvement of ZGEQP3 improves overal performance of ZGEJSV.
-*
-* A * P1 = Q1 * [ R1^* 0]^*:
- DO 1963 p = 1, N
-* .. all columns are free columns
- IWORK(p) = 0
- 1963 CONTINUE
- CALL ZGEQP3( M, N, A, LDA, IWORK, CWORK, CWORK(N+1), LWORK-N,
- $ RWORK, IERR )
-*
-* The upper triangular matrix R1 from the first QRF is inspected for
-* rank deficiency and possibilities for deflation, or possible
-* ill-conditioning. Depending on the user specified flag L2RANK,
-* the procedure explores possibilities to reduce the numerical
-* rank by inspecting the computed upper triangular factor. If
-* L2RANK or L2ABER are up, then ZGEJSV will compute the SVD of
-* A + dA, where ||dA|| <= f(M,N)*EPSLN.
-*
- NR = 1
- IF ( L2ABER ) THEN
-* Standard absolute error bound suffices. All sigma_i with
-* sigma_i < N*EPSLN*||A|| are flushed to zero. This is an
-* agressive enforcement of lower numerical rank by introducing a
-* backward error of the order of N*EPSLN*||A||.
- TEMP1 = SQRT(DBLE(N))*EPSLN
- DO 3001 p = 2, N
- IF ( ABS(A(p,p)) .GE. (TEMP1*ABS(A(1,1))) ) THEN
- NR = NR + 1
- ELSE
- GO TO 3002
- END IF
- 3001 CONTINUE
- 3002 CONTINUE
- ELSE IF ( L2RANK ) THEN
-* .. similarly as above, only slightly more gentle (less agressive).
-* Sudden drop on the diagonal of R1 is used as the criterion for
-* close-to-rank-deficient.
- TEMP1 = SQRT(SFMIN)
- DO 3401 p = 2, N
- IF ( ( ABS(A(p,p)) .LT. (EPSLN*ABS(A(p-1,p-1))) ) .OR.
- $ ( ABS(A(p,p)) .LT. SMALL ) .OR.
- $ ( L2KILL .AND. (ABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3402
- NR = NR + 1
- 3401 CONTINUE
- 3402 CONTINUE
-*
- ELSE
-* The goal is high relative accuracy. However, if the matrix
-* has high scaled condition number the relative accuracy is in
-* general not feasible. Later on, a condition number estimator
-* will be deployed to estimate the scaled condition number.
-* Here we just remove the underflowed part of the triangular
-* factor. This prevents the situation in which the code is
-* working hard to get the accuracy not warranted by the data.
- TEMP1 = SQRT(SFMIN)
- DO 3301 p = 2, N
- IF ( ( ABS(A(p,p)) .LT. SMALL ) .OR.
- $ ( L2KILL .AND. (ABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3302
- NR = NR + 1
- 3301 CONTINUE
- 3302 CONTINUE
-*
- END IF
-*
- ALMORT = .FALSE.
- IF ( NR .EQ. N ) THEN
- MAXPRJ = ONE
- DO 3051 p = 2, N
- TEMP1 = ABS(A(p,p)) / SVA(IWORK(p))
- MAXPRJ = MIN( MAXPRJ, TEMP1 )
- 3051 CONTINUE
- IF ( MAXPRJ**2 .GE. ONE - DBLE(N)*EPSLN ) ALMORT = .TRUE.
- END IF
-*
-*
- SCONDA = - ONE
- CONDR1 = - ONE
- CONDR2 = - ONE
-*
- IF ( ERREST ) THEN
- IF ( N .EQ. NR ) THEN
- IF ( RSVEC ) THEN
-* .. V is available as workspace
- CALL ZLACPY( 'U', N, N, A, LDA, V, LDV )
- DO 3053 p = 1, N
- TEMP1 = SVA(IWORK(p))
- CALL ZDSCAL( p, ONE/TEMP1, V(1,p), 1 )
- 3053 CONTINUE
- IF ( LSVEC )THEN
- CALL ZPOCON( 'U', N, V, LDV, ONE, TEMP1,
- $ CWORK(N+1), RWORK, IERR )
- ELSE
- CALL ZPOCON( 'U', N, V, LDV, ONE, TEMP1,
- $ CWORK, RWORK, IERR )
- END IF
-*
- ELSE IF ( LSVEC ) THEN
-* .. U is available as workspace
- CALL ZLACPY( 'U', N, N, A, LDA, U, LDU )
- DO 3054 p = 1, N
- TEMP1 = SVA(IWORK(p))
- CALL ZDSCAL( p, ONE/TEMP1, U(1,p), 1 )
- 3054 CONTINUE
- CALL ZPOCON( 'U', N, U, LDU, ONE, TEMP1,
- $ CWORK(N+1), RWORK, IERR )
- ELSE
- CALL ZLACPY( 'U', N, N, A, LDA, CWORK, N )
-*[] CALL ZLACPY( 'U', N, N, A, LDA, CWORK(N+1), N )
-* Change: here index shifted by N to the left, CWORK(1:N)
-* not needed for SIGMA only computation
- DO 3052 p = 1, N
- TEMP1 = SVA(IWORK(p))
-*[] CALL ZDSCAL( p, ONE/TEMP1, CWORK(N+(p-1)*N+1), 1 )
- CALL ZDSCAL( p, ONE/TEMP1, CWORK((p-1)*N+1), 1 )
- 3052 CONTINUE
-* .. the columns of R are scaled to have unit Euclidean lengths.
-*[] CALL ZPOCON( 'U', N, CWORK(N+1), N, ONE, TEMP1,
-*[] $ CWORK(N+N*N+1), RWORK, IERR )
- CALL ZPOCON( 'U', N, CWORK, N, ONE, TEMP1,
- $ CWORK(N*N+1), RWORK, IERR )
-*
- END IF
- IF ( TEMP1 .NE. ZERO ) THEN
- SCONDA = ONE / SQRT(TEMP1)
- ELSE
- SCONDA = - ONE
- END IF
-* SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
-* N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
- ELSE
- SCONDA = - ONE
- END IF
- END IF
-*
- L2PERT = L2PERT .AND. ( ABS( A(1,1)/A(NR,NR) ) .GT. SQRT(BIG1) )
-* If there is no violent scaling, artificial perturbation is not needed.
-*
-* Phase 3:
-*
- IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN
-*
-* Singular Values only
-*
-* .. transpose A(1:NR,1:N)
- DO 1946 p = 1, MIN( N-1, NR )
- CALL ZCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )
- CALL ZLACGV( N-p+1, A(p,p), 1 )
- 1946 CONTINUE
- IF ( NR .EQ. N ) A(N,N) = CONJG(A(N,N))
-*
-* The following two DO-loops introduce small relative perturbation
-* into the strict upper triangle of the lower triangular matrix.
-* Small entries below the main diagonal are also changed.
-* This modification is useful if the computing environment does not
-* provide/allow FLUSH TO ZERO underflow, for it prevents many
-* annoying denormalized numbers in case of strongly scaled matrices.
-* The perturbation is structured so that it does not introduce any
-* new perturbation of the singular values, and it does not destroy
-* the job done by the preconditioner.
-* The licence for this perturbation is in the variable L2PERT, which
-* should be .FALSE. if FLUSH TO ZERO underflow is active.
-*
- IF ( .NOT. ALMORT ) THEN
-*
- IF ( L2PERT ) THEN
-* XSC = SQRT(SMALL)
- XSC = EPSLN / DBLE(N)
- DO 4947 q = 1, NR
- CTEMP = DCMPLX(XSC*ABS(A(q,q)),ZERO)
- DO 4949 p = 1, N
- IF ( ( (p.GT.q) .AND. (ABS(A(p,q)).LE.TEMP1) )
- $ .OR. ( p .LT. q ) )
-* $ A(p,q) = TEMP1 * ( A(p,q) / ABS(A(p,q)) )
- $ A(p,q) = CTEMP
- 4949 CONTINUE
- 4947 CONTINUE
- ELSE
- CALL ZLASET( 'U', NR-1,NR-1, CZERO,CZERO, A(1,2),LDA )
- END IF
-*
-* .. second preconditioning using the QR factorization
-*
- CALL ZGEQRF( N,NR, A,LDA, CWORK, CWORK(N+1),LWORK-N, IERR )
-*
-* .. and transpose upper to lower triangular
- DO 1948 p = 1, NR - 1
- CALL ZCOPY( NR-p, A(p,p+1), LDA, A(p+1,p), 1 )
- CALL ZLACGV( NR-p+1, A(p,p), 1 )
- 1948 CONTINUE
-*
- END IF
-*
-* Row-cyclic Jacobi SVD algorithm with column pivoting
-*
-* .. again some perturbation (a "background noise") is added
-* to drown denormals
- IF ( L2PERT ) THEN
-* XSC = SQRT(SMALL)
- XSC = EPSLN / DBLE(N)
- DO 1947 q = 1, NR
- CTEMP = DCMPLX(XSC*ABS(A(q,q)),ZERO)
- DO 1949 p = 1, NR
- IF ( ( (p.GT.q) .AND. (ABS(A(p,q)).LE.TEMP1) )
- $ .OR. ( p .LT. q ) )
-* $ A(p,q) = TEMP1 * ( A(p,q) / ABS(A(p,q)) )
- $ A(p,q) = CTEMP
- 1949 CONTINUE
- 1947 CONTINUE
- ELSE
- CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, A(1,2), LDA )
- END IF
-*
-* .. and one-sided Jacobi rotations are started on a lower
-* triangular matrix (plus perturbation which is ignored in
-* the part which destroys triangular form (confusing?!))
-*
- CALL ZGESVJ( 'L', 'N', 'N', NR, NR, A, LDA, SVA,
- $ N, V, LDV, CWORK, LWORK, RWORK, LRWORK, INFO )
-*
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
-*
-*
- ELSE IF ( ( RSVEC .AND. ( .NOT. LSVEC ) .AND. ( .NOT. JRACC ) )
- $ .OR.
- $ ( JRACC .AND. ( .NOT. LSVEC ) .AND. ( NR .NE. N ) ) ) THEN
-*
-* -> Singular Values and Right Singular Vectors <-
-*
- IF ( ALMORT ) THEN
-*
-* .. in this case NR equals N
- DO 1998 p = 1, NR
- CALL ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
- CALL ZLACGV( N-p+1, V(p,p), 1 )
- 1998 CONTINUE
- CALL ZLASET( 'U', NR-1,NR-1, CZERO, CZERO, V(1,2), LDV )
-*
- CALL ZGESVJ( 'L','U','N', N, NR, V, LDV, SVA, NR, A, LDA,
- $ CWORK, LWORK, RWORK, LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
-
- ELSE
-*
-* .. two more QR factorizations ( one QRF is not enough, two require
-* accumulated product of Jacobi rotations, three are perfect )
-*
- CALL ZLASET( 'L', NR-1,NR-1, CZERO, CZERO, A(2,1), LDA )
- CALL ZGELQF( NR,N, A, LDA, CWORK, CWORK(N+1), LWORK-N, IERR)
- CALL ZLACPY( 'L', NR, NR, A, LDA, V, LDV )
- CALL ZLASET( 'U', NR-1,NR-1, CZERO, CZERO, V(1,2), LDV )
- CALL ZGEQRF( NR, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
- $ LWORK-2*N, IERR )
- DO 8998 p = 1, NR
- CALL ZCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )
- CALL ZLACGV( NR-p+1, V(p,p), 1 )
- 8998 CONTINUE
- CALL ZLASET('U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV)
-*
- CALL ZGESVJ( 'L', 'U','N', NR, NR, V,LDV, SVA, NR, U,
- $ LDU, CWORK(N+1), LWORK-N, RWORK, LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
- IF ( NR .LT. N ) THEN
- CALL ZLASET( 'A',N-NR, NR, CZERO,CZERO, V(NR+1,1), LDV )
- CALL ZLASET( 'A',NR, N-NR, CZERO,CZERO, V(1,NR+1), LDV )
- CALL ZLASET( 'A',N-NR,N-NR,CZERO,CONE, V(NR+1,NR+1),LDV )
- END IF
-*
- CALL ZUNMLQ( 'L', 'C', N, N, NR, A, LDA, CWORK,
- $ V, LDV, CWORK(N+1), LWORK-N, IERR )
-*
- END IF
-* .. permute the rows of V
-* DO 8991 p = 1, N
-* CALL ZCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA )
-* 8991 CONTINUE
-* CALL ZLACPY( 'All', N, N, A, LDA, V, LDV )
- CALL ZLAPMR( .FALSE., N, N, V, LDV, IWORK )
-*
- IF ( TRANSP ) THEN
- CALL ZLACPY( 'A', N, N, V, LDV, U, LDU )
- END IF
-*
- ELSE IF ( JRACC .AND. (.NOT. LSVEC) .AND. ( NR.EQ. N ) ) THEN
-*
- CALL ZLASET( 'L', N-1,N-1, CZERO, CZERO, A(2,1), LDA )
-*
- CALL ZGESVJ( 'U','N','V', N, N, A, LDA, SVA, N, V, LDV,
- $ CWORK, LWORK, RWORK, LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
- CALL ZLAPMR( .FALSE., N, N, V, LDV, IWORK )
-*
- ELSE IF ( LSVEC .AND. ( .NOT. RSVEC ) ) THEN
-*
-* .. Singular Values and Left Singular Vectors ..
-*
-* .. second preconditioning step to avoid need to accumulate
-* Jacobi rotations in the Jacobi iterations.
- DO 1965 p = 1, NR
- CALL ZCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 )
- CALL ZLACGV( N-p+1, U(p,p), 1 )
- 1965 CONTINUE
- CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
-*
- CALL ZGEQRF( N, NR, U, LDU, CWORK(N+1), CWORK(2*N+1),
- $ LWORK-2*N, IERR )
-*
- DO 1967 p = 1, NR - 1
- CALL ZCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )
- CALL ZLACGV( N-p+1, U(p,p), 1 )
- 1967 CONTINUE
- CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
-*
- CALL ZGESVJ( 'L', 'U', 'N', NR,NR, U, LDU, SVA, NR, A,
- $ LDA, CWORK(N+1), LWORK-N, RWORK, LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
-*
- IF ( NR .LT. M ) THEN
- CALL ZLASET( 'A', M-NR, NR,CZERO, CZERO, U(NR+1,1), LDU )
- IF ( NR .LT. N1 ) THEN
- CALL ZLASET( 'A',NR, N1-NR, CZERO, CZERO, U(1,NR+1),LDU )
- CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,U(NR+1,NR+1),LDU )
- END IF
- END IF
-*
- CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
- $ LDU, CWORK(N+1), LWORK-N, IERR )
-*
- IF ( ROWPIV )
- $ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
-*
- DO 1974 p = 1, N1
- XSC = ONE / DZNRM2( M, U(1,p), 1 )
- CALL ZDSCAL( M, XSC, U(1,p), 1 )
- 1974 CONTINUE
-*
- IF ( TRANSP ) THEN
- CALL ZLACPY( 'A', N, N, U, LDU, V, LDV )
- END IF
-*
- ELSE
-*
-* .. Full SVD ..
-*
- IF ( .NOT. JRACC ) THEN
-*
- IF ( .NOT. ALMORT ) THEN
-*
-* Second Preconditioning Step (QRF [with pivoting])
-* Note that the composition of TRANSPOSE, QRF and TRANSPOSE is
-* equivalent to an LQF CALL. Since in many libraries the QRF
-* seems to be better optimized than the LQF, we do explicit
-* transpose and use the QRF. This is subject to changes in an
-* optimized implementation of ZGEJSV.
-*
- DO 1968 p = 1, NR
- CALL ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
- CALL ZLACGV( N-p+1, V(p,p), 1 )
- 1968 CONTINUE
-*
-* .. the following two loops perturb small entries to avoid
-* denormals in the second QR factorization, where they are
-* as good as zeros. This is done to avoid painfully slow
-* computation with denormals. The relative size of the perturbation
-* is a parameter that can be changed by the implementer.
-* This perturbation device will be obsolete on machines with
-* properly implemented arithmetic.
-* To switch it off, set L2PERT=.FALSE. To remove it from the
-* code, remove the action under L2PERT=.TRUE., leave the ELSE part.
-* The following two loops should be blocked and fused with the
-* transposed copy above.
-*
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL)
- DO 2969 q = 1, NR
- CTEMP = DCMPLX(XSC*ABS( V(q,q) ),ZERO)
- DO 2968 p = 1, N
- IF ( ( p .GT. q ) .AND. ( ABS(V(p,q)) .LE. TEMP1 )
- $ .OR. ( p .LT. q ) )
-* $ V(p,q) = TEMP1 * ( V(p,q) / ABS(V(p,q)) )
- $ V(p,q) = CTEMP
- IF ( p .LT. q ) V(p,q) = - V(p,q)
- 2968 CONTINUE
- 2969 CONTINUE
- ELSE
- CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV )
- END IF
-*
-* Estimate the row scaled condition number of R1
-* (If R1 is rectangular, N > NR, then the condition number
-* of the leading NR x NR submatrix is estimated.)
-*
- CALL ZLACPY( 'L', NR, NR, V, LDV, CWORK(2*N+1), NR )
- DO 3950 p = 1, NR
- TEMP1 = DZNRM2(NR-p+1,CWORK(2*N+(p-1)*NR+p),1)
- CALL ZDSCAL(NR-p+1,ONE/TEMP1,CWORK(2*N+(p-1)*NR+p),1)
- 3950 CONTINUE
- CALL ZPOCON('L',NR,CWORK(2*N+1),NR,ONE,TEMP1,
- $ CWORK(2*N+NR*NR+1),RWORK,IERR)
- CONDR1 = ONE / SQRT(TEMP1)
-* .. here need a second oppinion on the condition number
-* .. then assume worst case scenario
-* R1 is OK for inverse <=> CONDR1 .LT. DBLE(N)
-* more conservative <=> CONDR1 .LT. SQRT(DBLE(N))
-*
- COND_OK = SQRT(SQRT(DBLE(NR)))
-*[TP] COND_OK is a tuning parameter.
-*
- IF ( CONDR1 .LT. COND_OK ) THEN
-* .. the second QRF without pivoting. Note: in an optimized
-* implementation, this QRF should be implemented as the QRF
-* of a lower triangular matrix.
-* R1^* = Q2 * R2
- CALL ZGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
- $ LWORK-2*N, IERR )
-*
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL)/EPSLN
- DO 3959 p = 2, NR
- DO 3958 q = 1, p - 1
- CTEMP=DCMPLX(XSC*MIN(ABS(V(p,p)),ABS(V(q,q))),
- $ ZERO)
- IF ( ABS(V(q,p)) .LE. TEMP1 )
-* $ V(q,p) = TEMP1 * ( V(q,p) / ABS(V(q,p)) )
- $ V(q,p) = CTEMP
- 3958 CONTINUE
- 3959 CONTINUE
- END IF
-*
- IF ( NR .NE. N )
- $ CALL ZLACPY( 'A', N, NR, V, LDV, CWORK(2*N+1), N )
-* .. save ...
-*
-* .. this transposed copy should be better than naive
- DO 1969 p = 1, NR - 1
- CALL ZCOPY( NR-p, V(p,p+1), LDV, V(p+1,p), 1 )
- CALL ZLACGV(NR-p+1, V(p,p), 1 )
- 1969 CONTINUE
- V(NR,NR)=CONJG(V(NR,NR))
-*
- CONDR2 = CONDR1
-*
- ELSE
-*
-* .. ill-conditioned case: second QRF with pivoting
-* Note that windowed pivoting would be equaly good
-* numerically, and more run-time efficient. So, in
-* an optimal implementation, the next call to ZGEQP3
-* should be replaced with eg. CALL ZGEQPX (ACM TOMS #782)
-* with properly (carefully) chosen parameters.
-*
-* R1^* * P2 = Q2 * R2
- DO 3003 p = 1, NR
- IWORK(N+p) = 0
- 3003 CONTINUE
- CALL ZGEQP3( N, NR, V, LDV, IWORK(N+1), CWORK(N+1),
- $ CWORK(2*N+1), LWORK-2*N, RWORK, IERR )
-** CALL ZGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
-** $ LWORK-2*N, IERR )
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL)
- DO 3969 p = 2, NR
- DO 3968 q = 1, p - 1
- CTEMP=DCMPLX(XSC*MIN(ABS(V(p,p)),ABS(V(q,q))),
- $ ZERO)
- IF ( ABS(V(q,p)) .LE. TEMP1 )
-* $ V(q,p) = TEMP1 * ( V(q,p) / ABS(V(q,p)) )
- $ V(q,p) = CTEMP
- 3968 CONTINUE
- 3969 CONTINUE
- END IF
-*
- CALL ZLACPY( 'A', N, NR, V, LDV, CWORK(2*N+1), N )
-*
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL)
- DO 8970 p = 2, NR
- DO 8971 q = 1, p - 1
- CTEMP=DCMPLX(XSC*MIN(ABS(V(p,p)),ABS(V(q,q))),
- $ ZERO)
-* V(p,q) = - TEMP1*( V(q,p) / ABS(V(q,p)) )
- V(p,q) = - CTEMP
- 8971 CONTINUE
- 8970 CONTINUE
- ELSE
- CALL ZLASET( 'L',NR-1,NR-1,CZERO,CZERO,V(2,1),LDV )
- END IF
-* Now, compute R2 = L3 * Q3, the LQ factorization.
- CALL ZGELQF( NR, NR, V, LDV, CWORK(2*N+N*NR+1),
- $ CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, IERR )
-* .. and estimate the condition number
- CALL ZLACPY( 'L',NR,NR,V,LDV,CWORK(2*N+N*NR+NR+1),NR )
- DO 4950 p = 1, NR
- TEMP1 = DZNRM2( p, CWORK(2*N+N*NR+NR+p), NR )
- CALL ZDSCAL( p, ONE/TEMP1, CWORK(2*N+N*NR+NR+p), NR )
- 4950 CONTINUE
- CALL ZPOCON( 'L',NR,CWORK(2*N+N*NR+NR+1),NR,ONE,TEMP1,
- $ CWORK(2*N+N*NR+NR+NR*NR+1),RWORK,IERR )
- CONDR2 = ONE / SQRT(TEMP1)
-*
-*
- IF ( CONDR2 .GE. COND_OK ) THEN
-* .. save the Householder vectors used for Q3
-* (this overwrittes the copy of R2, as it will not be
-* needed in this branch, but it does not overwritte the
-* Huseholder vectors of Q2.).
- CALL ZLACPY( 'U', NR, NR, V, LDV, CWORK(2*N+1), N )
-* .. and the rest of the information on Q3 is in
-* WORK(2*N+N*NR+1:2*N+N*NR+N)
- END IF
-*
- END IF
-*
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL)
- DO 4968 q = 2, NR
- CTEMP = XSC * V(q,q)
- DO 4969 p = 1, q - 1
-* V(p,q) = - TEMP1*( V(p,q) / ABS(V(p,q)) )
- V(p,q) = - CTEMP
- 4969 CONTINUE
- 4968 CONTINUE
- ELSE
- CALL ZLASET( 'U', NR-1,NR-1, CZERO,CZERO, V(1,2), LDV )
- END IF
-*
-* Second preconditioning finished; continue with Jacobi SVD
-* The input matrix is lower trinagular.
-*
-* Recover the right singular vectors as solution of a well
-* conditioned triangular matrix equation.
-*
- IF ( CONDR1 .LT. COND_OK ) THEN
-*
- CALL ZGESVJ( 'L','U','N',NR,NR,V,LDV,SVA,NR,U, LDU,
- $ CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,RWORK,
- $ LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
- DO 3970 p = 1, NR
- CALL ZCOPY( NR, V(1,p), 1, U(1,p), 1 )
- CALL ZDSCAL( NR, SVA(p), V(1,p), 1 )
- 3970 CONTINUE
-
-* .. pick the right matrix equation and solve it
-*
- IF ( NR .EQ. N ) THEN
-* :)) .. best case, R1 is inverted. The solution of this matrix
-* equation is Q2*V2 = the product of the Jacobi rotations
-* used in ZGESVJ, premultiplied with the orthogonal matrix
-* from the second QR factorization.
- CALL ZTRSM('L','U','N','N', NR,NR,CONE, A,LDA, V,LDV)
- ELSE
-* .. R1 is well conditioned, but non-square. Adjoint of R2
-* is inverted to get the product of the Jacobi rotations
-* used in ZGESVJ. The Q-factor from the second QR
-* factorization is then built in explicitly.
- CALL ZTRSM('L','U','C','N',NR,NR,CONE,CWORK(2*N+1),
- $ N,V,LDV)
- IF ( NR .LT. N ) THEN
- CALL ZLASET('A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV)
- CALL ZLASET('A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV)
- CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
- END IF
- CALL ZUNMQR('L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
- $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR)
- END IF
-*
- ELSE IF ( CONDR2 .LT. COND_OK ) THEN
-*
-* The matrix R2 is inverted. The solution of the matrix equation
-* is Q3^* * V3 = the product of the Jacobi rotations (appplied to
-* the lower triangular L3 from the LQ factorization of
-* R2=L3*Q3), pre-multiplied with the transposed Q3.
- CALL ZGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR, U,
- $ LDU, CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR,
- $ RWORK, LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
- DO 3870 p = 1, NR
- CALL ZCOPY( NR, V(1,p), 1, U(1,p), 1 )
- CALL ZDSCAL( NR, SVA(p), U(1,p), 1 )
- 3870 CONTINUE
- CALL ZTRSM('L','U','N','N',NR,NR,CONE,CWORK(2*N+1),N,
- $ U,LDU)
-* .. apply the permutation from the second QR factorization
- DO 873 q = 1, NR
- DO 872 p = 1, NR
- CWORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
- 872 CONTINUE
- DO 874 p = 1, NR
- U(p,q) = CWORK(2*N+N*NR+NR+p)
- 874 CONTINUE
- 873 CONTINUE
- IF ( NR .LT. N ) THEN
- CALL ZLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
- CALL ZLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
- CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
- END IF
- CALL ZUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
- $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
- ELSE
-* Last line of defense.
-* #:( This is a rather pathological case: no scaled condition
-* improvement after two pivoted QR factorizations. Other
-* possibility is that the rank revealing QR factorization
-* or the condition estimator has failed, or the COND_OK
-* is set very close to ONE (which is unnecessary). Normally,
-* this branch should never be executed, but in rare cases of
-* failure of the RRQR or condition estimator, the last line of
-* defense ensures that ZGEJSV completes the task.
-* Compute the full SVD of L3 using ZGESVJ with explicit
-* accumulation of Jacobi rotations.
- CALL ZGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR, U,
- $ LDU, CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR,
- $ RWORK, LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
- IF ( NR .LT. N ) THEN
- CALL ZLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
- CALL ZLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
- CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
- END IF
- CALL ZUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
- $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
-*
- CALL ZUNMLQ( 'L', 'C', NR, NR, NR, CWORK(2*N+1), N,
- $ CWORK(2*N+N*NR+1), U, LDU, CWORK(2*N+N*NR+NR+1),
- $ LWORK-2*N-N*NR-NR, IERR )
- DO 773 q = 1, NR
- DO 772 p = 1, NR
- CWORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
- 772 CONTINUE
- DO 774 p = 1, NR
- U(p,q) = CWORK(2*N+N*NR+NR+p)
- 774 CONTINUE
- 773 CONTINUE
-*
- END IF
-*
-* Permute the rows of V using the (column) permutation from the
-* first QRF. Also, scale the columns to make them unit in
-* Euclidean norm. This applies to all cases.
-*
- TEMP1 = SQRT(DBLE(N)) * EPSLN
- DO 1972 q = 1, N
- DO 972 p = 1, N
- CWORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
- 972 CONTINUE
- DO 973 p = 1, N
- V(p,q) = CWORK(2*N+N*NR+NR+p)
- 973 CONTINUE
- XSC = ONE / DZNRM2( N, V(1,q), 1 )
- IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
- $ CALL ZDSCAL( N, XSC, V(1,q), 1 )
- 1972 CONTINUE
-* At this moment, V contains the right singular vectors of A.
-* Next, assemble the left singular vector matrix U (M x N).
- IF ( NR .LT. M ) THEN
- CALL ZLASET('A', M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU)
- IF ( NR .LT. N1 ) THEN
- CALL ZLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)
- CALL ZLASET('A',M-NR,N1-NR,CZERO,CONE,
- $ U(NR+1,NR+1),LDU)
- END IF
- END IF
-*
-* The Q matrix from the first QRF is built into the left singular
-* matrix U. This applies to all cases.
-*
- CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
- $ LDU, CWORK(N+1), LWORK-N, IERR )
-
-* The columns of U are normalized. The cost is O(M*N) flops.
- TEMP1 = SQRT(DBLE(M)) * EPSLN
- DO 1973 p = 1, NR
- XSC = ONE / DZNRM2( M, U(1,p), 1 )
- IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
- $ CALL ZDSCAL( M, XSC, U(1,p), 1 )
- 1973 CONTINUE
-*
-* If the initial QRF is computed with row pivoting, the left
-* singular vectors must be adjusted.
-*
- IF ( ROWPIV )
- $ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
-*
- ELSE
-*
-* .. the initial matrix A has almost orthogonal columns and
-* the second QRF is not needed
-*
- CALL ZLACPY( 'U', N, N, A, LDA, CWORK(N+1), N )
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL)
- DO 5970 p = 2, N
- CTEMP = XSC * CWORK( N + (p-1)*N + p )
- DO 5971 q = 1, p - 1
-* CWORK(N+(q-1)*N+p)=-TEMP1 * ( CWORK(N+(p-1)*N+q) /
-* $ ABS(CWORK(N+(p-1)*N+q)) )
- CWORK(N+(q-1)*N+p)=-CTEMP
- 5971 CONTINUE
- 5970 CONTINUE
- ELSE
- CALL ZLASET( 'L',N-1,N-1,CZERO,CZERO,CWORK(N+2),N )
- END IF
-*
- CALL ZGESVJ( 'U', 'U', 'N', N, N, CWORK(N+1), N, SVA,
- $ N, U, LDU, CWORK(N+N*N+1), LWORK-N-N*N, RWORK, LRWORK,
- $ INFO )
-*
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
- DO 6970 p = 1, N
- CALL ZCOPY( N, CWORK(N+(p-1)*N+1), 1, U(1,p), 1 )
- CALL ZDSCAL( N, SVA(p), CWORK(N+(p-1)*N+1), 1 )
- 6970 CONTINUE
-*
- CALL ZTRSM( 'L', 'U', 'N', 'N', N, N,
- $ CONE, A, LDA, CWORK(N+1), N )
- DO 6972 p = 1, N
- CALL ZCOPY( N, CWORK(N+p), N, V(IWORK(p),1), LDV )
- 6972 CONTINUE
- TEMP1 = SQRT(DBLE(N))*EPSLN
- DO 6971 p = 1, N
- XSC = ONE / DZNRM2( N, V(1,p), 1 )
- IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
- $ CALL ZDSCAL( N, XSC, V(1,p), 1 )
- 6971 CONTINUE
-*
-* Assemble the left singular vector matrix U (M x N).
-*
- IF ( N .LT. M ) THEN
- CALL ZLASET( 'A', M-N, N, CZERO, CZERO, U(N+1,1), LDU )
- IF ( N .LT. N1 ) THEN
- CALL ZLASET('A',N, N1-N, CZERO, CZERO, U(1,N+1),LDU)
- CALL ZLASET( 'A',M-N,N1-N, CZERO, CONE,U(N+1,N+1),LDU)
- END IF
- END IF
- CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
- $ LDU, CWORK(N+1), LWORK-N, IERR )
- TEMP1 = SQRT(DBLE(M))*EPSLN
- DO 6973 p = 1, N1
- XSC = ONE / DZNRM2( M, U(1,p), 1 )
- IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
- $ CALL ZDSCAL( M, XSC, U(1,p), 1 )
- 6973 CONTINUE
-*
- IF ( ROWPIV )
- $ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
-*
- END IF
-*
-* end of the >> almost orthogonal case << in the full SVD
-*
- ELSE
-*
-* This branch deploys a preconditioned Jacobi SVD with explicitly
-* accumulated rotations. It is included as optional, mainly for
-* experimental purposes. It does perfom well, and can also be used.
-* In this implementation, this branch will be automatically activated
-* if the condition number sigma_max(A) / sigma_min(A) is predicted
-* to be greater than the overflow threshold. This is because the
-* a posteriori computation of the singular vectors assumes robust
-* implementation of BLAS and some LAPACK procedures, capable of working
-* in presence of extreme values, e.g. when the singular values spread from
-* the underflow to the overflow threshold.
-*
- DO 7968 p = 1, NR
- CALL ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
- CALL ZLACGV( N-p+1, V(p,p), 1 )
- 7968 CONTINUE
-*
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL/EPSLN)
- DO 5969 q = 1, NR
- CTEMP = DCMPLX(XSC*ABS( V(q,q) ),ZERO)
- DO 5968 p = 1, N
- IF ( ( p .GT. q ) .AND. ( ABS(V(p,q)) .LE. TEMP1 )
- $ .OR. ( p .LT. q ) )
-* $ V(p,q) = TEMP1 * ( V(p,q) / ABS(V(p,q)) )
- $ V(p,q) = CTEMP
- IF ( p .LT. q ) V(p,q) = - V(p,q)
- 5968 CONTINUE
- 5969 CONTINUE
- ELSE
- CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV )
- END IF
-
- CALL ZGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
- $ LWORK-2*N, IERR )
- CALL ZLACPY( 'L', N, NR, V, LDV, CWORK(2*N+1), N )
-*
- DO 7969 p = 1, NR
- CALL ZCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )
- CALL ZLACGV( NR-p+1, U(p,p), 1 )
- 7969 CONTINUE
-
- IF ( L2PERT ) THEN
- XSC = SQRT(SMALL/EPSLN)
- DO 9970 q = 2, NR
- DO 9971 p = 1, q - 1
- CTEMP = DCMPLX(XSC * MIN(ABS(U(p,p)),ABS(U(q,q))),
- $ ZERO)
-* U(p,q) = - TEMP1 * ( U(q,p) / ABS(U(q,p)) )
- U(p,q) = - CTEMP
- 9971 CONTINUE
- 9970 CONTINUE
- ELSE
- CALL ZLASET('U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
- END IF
-
- CALL ZGESVJ( 'L', 'U', 'V', NR, NR, U, LDU, SVA,
- $ N, V, LDV, CWORK(2*N+N*NR+1), LWORK-2*N-N*NR,
- $ RWORK, LRWORK, INFO )
- SCALEM = RWORK(1)
- NUMRANK = NINT(RWORK(2))
-
- IF ( NR .LT. N ) THEN
- CALL ZLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
- CALL ZLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
- CALL ZLASET( 'A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV )
- END IF
-
- CALL ZUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
- $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
-*
-* Permute the rows of V using the (column) permutation from the
-* first QRF. Also, scale the columns to make them unit in
-* Euclidean norm. This applies to all cases.
-*
- TEMP1 = SQRT(DBLE(N)) * EPSLN
- DO 7972 q = 1, N
- DO 8972 p = 1, N
- CWORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
- 8972 CONTINUE
- DO 8973 p = 1, N
- V(p,q) = CWORK(2*N+N*NR+NR+p)
- 8973 CONTINUE
- XSC = ONE / DZNRM2( N, V(1,q), 1 )
- IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
- $ CALL ZDSCAL( N, XSC, V(1,q), 1 )
- 7972 CONTINUE
-*
-* At this moment, V contains the right singular vectors of A.
-* Next, assemble the left singular vector matrix U (M x N).
-*
- IF ( NR .LT. M ) THEN
- CALL ZLASET( 'A', M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU )
- IF ( NR .LT. N1 ) THEN
- CALL ZLASET('A',NR, N1-NR, CZERO, CZERO, U(1,NR+1),LDU)
- CALL ZLASET('A',M-NR,N1-NR, CZERO, CONE,U(NR+1,NR+1),LDU)
- END IF
- END IF
-*
- CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
- $ LDU, CWORK(N+1), LWORK-N, IERR )
-*
- IF ( ROWPIV )
- $ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
-*
-*
- END IF
- IF ( TRANSP ) THEN
-* .. swap U and V because the procedure worked on A^*
- DO 6974 p = 1, N
- CALL ZSWAP( N, U(1,p), 1, V(1,p), 1 )
- 6974 CONTINUE
- END IF
-*
- END IF
-* end of the full SVD
-*
-* Undo scaling, if necessary (and possible)
-*
- IF ( USCAL2 .LE. (BIG/SVA(1))*USCAL1 ) THEN
- CALL DLASCL( 'G', 0, 0, USCAL1, USCAL2, NR, 1, SVA, N, IERR )
- USCAL1 = ONE
- USCAL2 = ONE
- END IF
-*
- IF ( NR .LT. N ) THEN
- DO 3004 p = NR+1, N
- SVA(p) = ZERO
- 3004 CONTINUE
- END IF
-*
- RWORK(1) = USCAL2 * SCALEM
- RWORK(2) = USCAL1
- IF ( ERREST ) RWORK(3) = SCONDA
- IF ( LSVEC .AND. RSVEC ) THEN
- RWORK(4) = CONDR1
- RWORK(5) = CONDR2
- END IF
- IF ( L2TRAN ) THEN
- RWORK(6) = ENTRA
- RWORK(7) = ENTRAT
- END IF
-*
- IWORK(1) = NR
- IWORK(2) = NUMRANK
- IWORK(3) = WARNING
- IF ( TRANSP ) THEN
- IWORK(4) = 1
- ELSE
- IWORK(4) = -1
- END IF
-
-*
- RETURN
-* ..
-* .. END OF ZGEJSV
-* ..
- END
-*
+*> \brief \b ZGEJSV
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download ZGEJSV + dependencies
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgejsv.f">
+*> [TGZ]</a>
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgejsv.f">
+*> [ZIP]</a>
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgejsv.f">
+*> [TXT]</a>
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE ZGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
+* M, N, A, LDA, SVA, U, LDU, V, LDV,
+* CWORK, LWORK, RWORK, LRWORK, IWORK, INFO )
+*
+* .. Scalar Arguments ..
+* IMPLICIT NONE
+* INTEGER INFO, LDA, LDU, LDV, LWORK, M, N
+* ..
+* .. Array Arguments ..
+* COMPLEX*16 A( LDA, * ), U( LDU, * ), V( LDV, * ), CWORK( LWORK )
+* DOUBLE PRECISION SVA( N ), RWORK( LRWORK )
+* INTEGER IWORK( * )
+* CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> ZGEJSV computes the singular value decomposition (SVD) of a complex M-by-N
+*> matrix [A], where M >= N. The SVD of [A] is written as
+*>
+*> [A] = [U] * [SIGMA] * [V]^*,
+*>
+*> where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
+*> diagonal elements, [U] is an M-by-N (or M-by-M) unitary matrix, and
+*> [V] is an N-by-N unitary matrix. The diagonal elements of [SIGMA] are
+*> the singular values of [A]. The columns of [U] and [V] are the left and
+*> the right singular vectors of [A], respectively. The matrices [U] and [V]
+*> are computed and stored in the arrays U and V, respectively. The diagonal
+*> of [SIGMA] is computed and stored in the array SVA.
+*> \endverbatim
+*>
+*> Arguments:
+*> ==========
+*>
+*> \param[in] JOBA
+*> \verbatim
+*> JOBA is CHARACTER*1
+*> Specifies the level of accuracy:
+*> = 'C': This option works well (high relative accuracy) if A = B * D,
+*> with well-conditioned B and arbitrary diagonal matrix D.
+*> The accuracy cannot be spoiled by COLUMN scaling. The
+*> accuracy of the computed output depends on the condition of
+*> B, and the procedure aims at the best theoretical accuracy.
+*> The relative error max_{i=1:N}|d sigma_i| / sigma_i is
+*> bounded by f(M,N)*epsilon* cond(B), independent of D.
+*> The input matrix is preprocessed with the QRF with column
+*> pivoting. This initial preprocessing and preconditioning by
+*> a rank revealing QR factorization is common for all values of
+*> JOBA. Additional actions are specified as follows:
+*> = 'E': Computation as with 'C' with an additional estimate of the
+*> condition number of B. It provides a realistic error bound.
+*> = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
+*> D1, D2, and well-conditioned matrix C, this option gives
+*> higher accuracy than the 'C' option. If the structure of the
+*> input matrix is not known, and relative accuracy is
+*> desirable, then this option is advisable. The input matrix A
+*> is preprocessed with QR factorization with FULL (row and
+*> column) pivoting.
+*> = 'G' Computation as with 'F' with an additional estimate of the
+*> condition number of B, where A=B*D. If A has heavily weighted
+*> rows, then using this condition number gives too pessimistic
+*> error bound.
+*> = 'A': Small singular values are not well determined by the data
+*> and are considered as noisy; the matrix is treated as
+*> numerically rank defficient. The error in the computed
+*> singular values is bounded by f(m,n)*epsilon*||A||.
+*> The computed SVD A = U * S * V^* restores A up to
+*> f(m,n)*epsilon*||A||.
+*> This gives the procedure the licence to discard (set to zero)
+*> all singular values below N*epsilon*||A||.
+*> = 'R': Similar as in 'A'. Rank revealing property of the initial
+*> QR factorization is used do reveal (using triangular factor)
+*> a gap sigma_{r+1} < epsilon * sigma_r in which case the
+*> numerical RANK is declared to be r. The SVD is computed with
+*> absolute error bounds, but more accurately than with 'A'.
+*> \endverbatim
+*>
+*> \param[in] JOBU
+*> \verbatim
+*> JOBU is CHARACTER*1
+*> Specifies whether to compute the columns of U:
+*> = 'U': N columns of U are returned in the array U.
+*> = 'F': full set of M left sing. vectors is returned in the array U.
+*> = 'W': U may be used as workspace of length M*N. See the description
+*> of U.
+*> = 'N': U is not computed.
+*> \endverbatim
+*>
+*> \param[in] JOBV
+*> \verbatim
+*> JOBV is CHARACTER*1
+*> Specifies whether to compute the matrix V:
+*> = 'V': N columns of V are returned in the array V; Jacobi rotations
+*> are not explicitly accumulated.
+*> = 'J': N columns of V are returned in the array V, but they are
+*> computed as the product of Jacobi rotations, if JOBT .EQ. 'N'.
+*> = 'W': V may be used as workspace of length N*N. See the description
+*> of V.
+*> = 'N': V is not computed.
+*> \endverbatim
+*>
+*> \param[in] JOBR
+*> \verbatim
+*> JOBR is CHARACTER*1
+*> Specifies the RANGE for the singular values. Issues the licence to
+*> set to zero small positive singular values if they are outside
+*> specified range. If A .NE. 0 is scaled so that the largest singular
+*> value of c*A is around SQRT(BIG), BIG=DLAMCH('O'), then JOBR issues
+*> the licence to kill columns of A whose norm in c*A is less than
+*> SQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,
+*> where SFMIN=DLAMCH('S'), EPSLN=DLAMCH('E').
+*> = 'N': Do not kill small columns of c*A. This option assumes that
+*> BLAS and QR factorizations and triangular solvers are
+*> implemented to work in that range. If the condition of A
+*> is greater than BIG, use ZGESVJ.
+*> = 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)]
+*> (roughly, as described above). This option is recommended.
+*> ===========================
+*> For computing the singular values in the FULL range [SFMIN,BIG]
+*> use ZGESVJ.
+*> \endverbatim
+*>
+*> \param[in] JOBT
+*> \verbatim
+*> JOBT is CHARACTER*1
+*> If the matrix is square then the procedure may determine to use
+*> transposed A if A^* seems to be better with respect to convergence.
+*> If the matrix is not square, JOBT is ignored.
+*> The decision is based on two values of entropy over the adjoint
+*> orbit of A^* * A. See the descriptions of WORK(6) and WORK(7).
+*> = 'T': transpose if entropy test indicates possibly faster
+*> convergence of Jacobi process if A^* is taken as input. If A is
+*> replaced with A^*, then the row pivoting is included automatically.
+*> = 'N': do not speculate.
+*> The option 'T' can be used to compute only the singular values, or
+*> the full SVD (U, SIGMA and V). For only one set of singular vectors
+*> (U or V), the caller should provide both U and V, as one of the
+*> matrices is used as workspace if the matrix A is transposed.
+*> The implementer can easily remove this constraint and make the
+*> code more complicated. See the descriptions of U and V.
+*> In general, this option is considered experimental, and 'N'; should
+*> be preferred. This is subject to changes in the future.
+*> \endverbatim
+*>
+*> \param[in] JOBP
+*> \verbatim
+*> JOBP is CHARACTER*1
+*> Issues the licence to introduce structured perturbations to drown
+*> denormalized numbers. This licence should be active if the
+*> denormals are poorly implemented, causing slow computation,
+*> especially in cases of fast convergence (!). For details see [1,2].
+*> For the sake of simplicity, this perturbations are included only
+*> when the full SVD or only the singular values are requested. The
+*> implementer/user can easily add the perturbation for the cases of
+*> computing one set of singular vectors.
+*> = 'P': introduce perturbation
+*> = 'N': do not perturb
+*> \endverbatim
+*>
+*> \param[in] M
+*> \verbatim
+*> M is INTEGER
+*> The number of rows of the input matrix A. M >= 0.
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The number of columns of the input matrix A. M >= N >= 0.
+*> \endverbatim
+*>
+*> \param[in,out] A
+*> \verbatim
+*> A is COMPLEX*16 array, dimension (LDA,N)
+*> On entry, the M-by-N matrix A.
+*> \endverbatim
+*>
+*> \param[in] LDA
+*> \verbatim
+*> LDA is INTEGER
+*> The leading dimension of the array A. LDA >= max(1,M).
+*> \endverbatim
+*>
+*> \param[out] SVA
+*> \verbatim
+*> SVA is DOUBLE PRECISION array, dimension (N)
+*> On exit,
+*> - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
+*> computation SVA contains Euclidean column norms of the
+*> iterated matrices in the array A.
+*> - For WORK(1) .NE. WORK(2): The singular values of A are
+*> (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
+*> sigma_max(A) overflows or if small singular values have been
+*> saved from underflow by scaling the input matrix A.
+*> - If JOBR='R' then some of the singular values may be returned
+*> as exact zeros obtained by "set to zero" because they are
+*> below the numerical rank threshold or are denormalized numbers.
+*> \endverbatim
+*>
+*> \param[out] U
+*> \verbatim
+*> U is COMPLEX*16 array, dimension ( LDU, N )
+*> If JOBU = 'U', then U contains on exit the M-by-N matrix of
+*> the left singular vectors.
+*> If JOBU = 'F', then U contains on exit the M-by-M matrix of
+*> the left singular vectors, including an ONB
+*> of the orthogonal complement of the Range(A).
+*> If JOBU = 'W' .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),
+*> then U is used as workspace if the procedure
+*> replaces A with A^*. In that case, [V] is computed
+*> in U as left singular vectors of A^* and then
+*> copied back to the V array. This 'W' option is just
+*> a reminder to the caller that in this case U is
+*> reserved as workspace of length N*N.
+*> If JOBU = 'N' U is not referenced, unless JOBT='T'.
+*> \endverbatim
+*>
+*> \param[in] LDU
+*> \verbatim
+*> LDU is INTEGER
+*> The leading dimension of the array U, LDU >= 1.
+*> IF JOBU = 'U' or 'F' or 'W', then LDU >= M.
+*> \endverbatim
+*>
+*> \param[out] V
+*> \verbatim
+*> V is COMPLEX*16 array, dimension ( LDV, N )
+*> If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
+*> the right singular vectors;
+*> If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),
+*> then V is used as workspace if the pprocedure
+*> replaces A with A^*. In that case, [U] is computed
+*> in V as right singular vectors of A^* and then
+*> copied back to the U array. This 'W' option is just
+*> a reminder to the caller that in this case V is
+*> reserved as workspace of length N*N.
+*> If JOBV = 'N' V is not referenced, unless JOBT='T'.
+*> \endverbatim
+*>
+*> \param[in] LDV
+*> \verbatim
+*> LDV is INTEGER
+*> The leading dimension of the array V, LDV >= 1.
+*> If JOBV = 'V' or 'J' or 'W', then LDV >= N.
+*> \endverbatim
+*>
+*> \param[out] CWORK
+*> \verbatim
+*> CWORK is COMPLEX*16 array, dimension (MAX(2,LWORK))
+*> If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or
+*> LRWORK=-1), then on exit CWORK(1) contains the required length of
+*> CWORK for the job parameters used in the call.
+*> \endverbatim
+*>
+*> \param[in] LWORK
+*> \verbatim
+*> LWORK is INTEGER
+*> Length of CWORK to confirm proper allocation of workspace.
+*> LWORK depends on the job:
+*>
+*> 1. If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
+*> 1.1 .. no scaled condition estimate required (JOBA.NE.'E'.AND.JOBA.NE.'G'):
+*> LWORK >= 2*N+1. This is the minimal requirement.
+*> ->> For optimal performance (blocked code) the optimal value
+*> is LWORK >= N + (N+1)*NB. Here NB is the optimal
+*> block size for ZGEQP3 and ZGEQRF.
+*> In general, optimal LWORK is computed as
+*> LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZGEQRF), LWORK(ZGESVJ)).
+*> 1.2. .. an estimate of the scaled condition number of A is
+*> required (JOBA='E', or 'G'). In this case, LWORK the minimal
+*> requirement is LWORK >= N*N + 2*N.
+*> ->> For optimal performance (blocked code) the optimal value
+*> is LWORK >= max(N+(N+1)*NB, N*N+2*N)=N**2+2*N.
+*> In general, the optimal length LWORK is computed as
+*> LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZGEQRF), LWORK(ZGESVJ),
+*> N*N+LWORK(ZPOCON)).
+*> 2. If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
+*> (JOBU.EQ.'N')
+*> 2.1 .. no scaled condition estimate requested (JOBE.EQ.'N'):
+*> -> the minimal requirement is LWORK >= 3*N.
+*> -> For optimal performance,
+*> LWORK >= max(N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
+*> where NB is the optimal block size for ZGEQP3, ZGEQRF, ZGELQ,
+*> ZUNMLQ. In general, the optimal length LWORK is computed as
+*> LWORK >= max(N+LWORK(ZGEQP3), N+LWORK(ZGESVJ),
+*> N+LWORK(ZGELQF), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMLQ)).
+*> 2.2 .. an estimate of the scaled condition number of A is
+*> required (JOBA='E', or 'G').
+*> -> the minimal requirement is LWORK >= 3*N.
+*> -> For optimal performance,
+*> LWORK >= max(N+(N+1)*NB, 2*N,2*N+N*NB)=2*N+N*NB,
+*> where NB is the optimal block size for ZGEQP3, ZGEQRF, ZGELQ,
+*> ZUNMLQ. In general, the optimal length LWORK is computed as
+*> LWORK >= max(N+LWORK(ZGEQP3), LWORK(ZPOCON), N+LWORK(ZGESVJ),
+*> N+LWORK(ZGELQF), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMLQ)).
+*> 3. If SIGMA and the left singular vectors are needed
+*> 3.1 .. no scaled condition estimate requested (JOBE.EQ.'N'):
+*> -> the minimal requirement is LWORK >= 3*N.
+*> -> For optimal performance:
+*> if JOBU.EQ.'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
+*> where NB is the optimal block size for ZGEQP3, ZGEQRF, ZUNMQR.
+*> In general, the optimal length LWORK is computed as
+*> LWORK >= max(N+LWORK(ZGEQP3), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMQR)).
+*> 3.2 .. an estimate of the scaled condition number of A is
+*> required (JOBA='E', or 'G').
+*> -> the minimal requirement is LWORK >= 3*N.
+*> -> For optimal performance:
+*> if JOBU.EQ.'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
+*> where NB is the optimal block size for ZGEQP3, ZGEQRF, ZUNMQR.
+*> In general, the optimal length LWORK is computed as
+*> LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZPOCON),
+*> 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMQR)).
+*> 4. If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and
+*> 4.1. if JOBV.EQ.'V'
+*> the minimal requirement is LWORK >= 5*N+2*N*N.
+*> 4.2. if JOBV.EQ.'J' the minimal requirement is
+*> LWORK >= 4*N+N*N.
+*> In both cases, the allocated CWORK can accommodate blocked runs
+*> of ZGEQP3, ZGEQRF, ZGELQF, SUNMQR, ZUNMLQ.
+*>
+*> If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or
+*> LRWORK=-1), then on exit CWORK(1) contains the optimal and CWORK(2) contains the
+*> minimal length of CWORK for the job parameters used in the call.
+*> \endverbatim
+*>
+*> \param[out] RWORK
+*> \verbatim
+*> RWORK is DOUBLE PRECISION array, dimension (MAX(7,LWORK))
+*> On exit,
+*> RWORK(1) = Determines the scaling factor SCALE = RWORK(2) / RWORK(1)
+*> such that SCALE*SVA(1:N) are the computed singular values
+*> of A. (See the description of SVA().)
+*> RWORK(2) = See the description of RWORK(1).
+*> RWORK(3) = SCONDA is an estimate for the condition number of
+*> column equilibrated A. (If JOBA .EQ. 'E' or 'G')
+*> SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
+*> It is computed using SPOCON. It holds
+*> N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
+*> where R is the triangular factor from the QRF of A.
+*> However, if R is truncated and the numerical rank is
+*> determined to be strictly smaller than N, SCONDA is
+*> returned as -1, thus indicating that the smallest
+*> singular values might be lost.
+*>
+*> If full SVD is needed, the following two condition numbers are
+*> useful for the analysis of the algorithm. They are provied for
+*> a developer/implementer who is familiar with the details of
+*> the method.
+*>
+*> RWORK(4) = an estimate of the scaled condition number of the
+*> triangular factor in the first QR factorization.
+*> RWORK(5) = an estimate of the scaled condition number of the
+*> triangular factor in the second QR factorization.
+*> The following two parameters are computed if JOBT .EQ. 'T'.
+*> They are provided for a developer/implementer who is familiar
+*> with the details of the method.
+*> RWORK(6) = the entropy of A^* * A :: this is the Shannon entropy
+*> of diag(A^* * A) / Trace(A^* * A) taken as point in the
+*> probability simplex.
+*> RWORK(7) = the entropy of A * A^*. (See the description of RWORK(6).)
+*> If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or
+*> LRWORK=-1), then on exit RWORK(1) contains the required length of
+*> RWORK for the job parameters used in the call.
+*> \endverbatim
+*>
+*> \param[in] LRWORK
+*> \verbatim
+*> LRWORK is INTEGER
+*> Length of RWORK to confirm proper allocation of workspace.
+*> LRWORK depends on the job:
+*>
+*> 1. If only the singular values are requested i.e. if
+*> LSAME(JOBU,'N') .AND. LSAME(JOBV,'N')
+*> then:
+*> 1.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
+*> then: LRWORK = max( 7, 2 * M ).
+*> 1.2. Otherwise, LRWORK = max( 7, N ).
+*> 2. If singular values with the right singular vectors are requested
+*> i.e. if
+*> (LSAME(JOBV,'V').OR.LSAME(JOBV,'J')) .AND.
+*> .NOT.(LSAME(JOBU,'U').OR.LSAME(JOBU,'F'))
+*> then:
+*> 2.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
+*> then LRWORK = max( 7, 2 * M ).
+*> 2.2. Otherwise, LRWORK = max( 7, N ).
+*> 3. If singular values with the left singular vectors are requested, i.e. if
+*> (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
+*> .NOT.(LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
+*> then:
+*> 3.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
+*> then LRWORK = max( 7, 2 * M ).
+*> 3.2. Otherwise, LRWORK = max( 7, N ).
+*> 4. If singular values with both the left and the right singular vectors
+*> are requested, i.e. if
+*> (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
+*> (LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
+*> then:
+*> 4.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
+*> then LRWORK = max( 7, 2 * M ).
+*> 4.2. Otherwise, LRWORK = max( 7, N ).
+*>
+*> If, on entry, LRWORK = -1 or LWORK=-1, a workspace query is assumed and
+*> the length of RWORK is returned in RWORK(1).
+*> \endverbatim
+*>
+*> \param[out] IWORK
+*> \verbatim
+*> IWORK is INTEGER array, of dimension at least 4, that further depends
+*> on the job:
+*>
+*> 1. If only the singular values are requested then:
+*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
+*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
+*> 2. If the singular values and the right singular vectors are requested then:
+*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
+*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
+*> 3. If the singular values and the left singular vectors are requested then:
+*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
+*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
+*> 4. If the singular values with both the left and the right singular vectors
+*> are requested, then:
+*> 4.1. If LSAME(JOBV,'J') the length of IWORK is determined as follows:
+*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
+*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
+*> 4.2. If LSAME(JOBV,'V') the length of IWORK is determined as follows:
+*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
+*> then the length of IWORK is 2*N+M; otherwise the length of IWORK is 2*N.
+*>
+*> On exit,
+*> IWORK(1) = the numerical rank determined after the initial
+*> QR factorization with pivoting. See the descriptions
+*> of JOBA and JOBR.
+*> IWORK(2) = the number of the computed nonzero singular values
+*> IWORK(3) = if nonzero, a warning message:
+*> If IWORK(3).EQ.1 then some of the column norms of A
+*> were denormalized floats. The requested high accuracy
+*> is not warranted by the data.
+*> IWORK(4) = 1 or -1. If IWORK(4) .EQ. 1, then the procedure used A^* to
+*> do the job as specified by the JOB parameters.
+*> If the call to ZGEJSV is a workspace query (indicated by LWORK .EQ. -1 or
+*> LRWORK .EQ. -1), then on exit IWORK(1) contains the required length of
+*> IWORK for the job parameters used in the call.
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> INFO is INTEGER
+*> < 0 : if INFO = -i, then the i-th argument had an illegal value.
+*> = 0 : successful exit;
+*> > 0 : ZGEJSV did not converge in the maximal allowed number
+*> of sweeps. The computed values may be inaccurate.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date June 2016
+*
+*> \ingroup complex16GEsing
+*
+*> \par Further Details:
+* =====================
+*>
+*> \verbatim
+*>
+*> ZGEJSV implements a preconditioned Jacobi SVD algorithm. It uses ZGEQP3,
+*> ZGEQRF, and ZGELQF as preprocessors and preconditioners. Optionally, an
+*> additional row pivoting can be used as a preprocessor, which in some
+*> cases results in much higher accuracy. An example is matrix A with the
+*> structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
+*> diagonal matrices and C is well-conditioned matrix. In that case, complete
+*> pivoting in the first QR factorizations provides accuracy dependent on the
+*> condition number of C, and independent of D1, D2. Such higher accuracy is
+*> not completely understood theoretically, but it works well in practice.
+*> Further, if A can be written as A = B*D, with well-conditioned B and some
+*> diagonal D, then the high accuracy is guaranteed, both theoretically and
+*> in software, independent of D. For more details see [1], [2].
+*> The computational range for the singular values can be the full range
+*> ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
+*> & LAPACK routines called by ZGEJSV are implemented to work in that range.
+*> If that is not the case, then the restriction for safe computation with
+*> the singular values in the range of normalized IEEE numbers is that the
+*> spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
+*> overflow. This code (ZGEJSV) is best used in this restricted range,
+*> meaning that singular values of magnitude below ||A||_2 / DLAMCH('O') are
+*> returned as zeros. See JOBR for details on this.
+*> Further, this implementation is somewhat slower than the one described
+*> in [1,2] due to replacement of some non-LAPACK components, and because
+*> the choice of some tuning parameters in the iterative part (ZGESVJ) is
+*> left to the implementer on a particular machine.
+*> The rank revealing QR factorization (in this code: ZGEQP3) should be
+*> implemented as in [3]. We have a new version of ZGEQP3 under development
+*> that is more robust than the current one in LAPACK, with a cleaner cut in
+*> rank deficient cases. It will be available in the SIGMA library [4].
+*> If M is much larger than N, it is obvious that the initial QRF with
+*> column pivoting can be preprocessed by the QRF without pivoting. That
+*> well known trick is not used in ZGEJSV because in some cases heavy row
+*> weighting can be treated with complete pivoting. The overhead in cases
+*> M much larger than N is then only due to pivoting, but the benefits in
+*> terms of accuracy have prevailed. The implementer/user can incorporate
+*> this extra QRF step easily. The implementer can also improve data movement
+*> (matrix transpose, matrix copy, matrix transposed copy) - this
+*> implementation of ZGEJSV uses only the simplest, naive data movement.
+*> \endverbatim
+*
+*> \par Contributor:
+* ==================
+*>
+*> Zlatko Drmac, Department of Mathematics, Faculty of Science,
+*> University of Zagreb (Zagreb, Croatia); drmac@math.hr
+*
+*> \par References:
+* ================
+*>
+*> \verbatim
+*>
+*> [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
+*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
+*> LAPACK Working note 169.
+*> [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
+*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
+*> LAPACK Working note 170.
+*> [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
+*> factorization software - a case study.
+*> ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
+*> LAPACK Working note 176.
+*> [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
+*> QSVD, (H,K)-SVD computations.
+*> Department of Mathematics, University of Zagreb, 2008, 2016.
+*> \endverbatim
+*
+*> \par Bugs, examples and comments:
+* =================================
+*>
+*> Please report all bugs and send interesting examples and/or comments to
+*> drmac@math.hr. Thank you.
+*>
+* =====================================================================
+ SUBROUTINE ZGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
+ $ M, N, A, LDA, SVA, U, LDU, V, LDV,
+ $ CWORK, LWORK, RWORK, LRWORK, IWORK, INFO )
+*
+* -- LAPACK computational routine (version 3.7.0) --
+* -- LAPACK is a software package provided by Univ. of Tennessee, --
+* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
+* December 2016
+*
+* .. Scalar Arguments ..
+ IMPLICIT NONE
+ INTEGER INFO, LDA, LDU, LDV, LWORK, LRWORK, M, N
+* ..
+* .. Array Arguments ..
+ COMPLEX*16 A( LDA, * ), U( LDU, * ), V( LDV, * ),
+ $ CWORK( LWORK )
+ DOUBLE PRECISION SVA( N ), RWORK( LRWORK )
+ INTEGER IWORK( * )
+ CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
+* ..
+*
+* ===========================================================================
+*
+* .. Local Parameters ..
+ DOUBLE PRECISION ZERO, ONE
+ PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
+ COMPLEX*16 CZERO, CONE
+ PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ), CONE = ( 1.0D0, 0.0D0 ) )
+* ..
+* .. Local Scalars ..
+ COMPLEX*16 CTEMP
+ DOUBLE PRECISION AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1,
+ $ COND_OK, CONDR1, CONDR2, ENTRA, ENTRAT, EPSLN,
+ $ MAXPRJ, SCALEM, SCONDA, SFMIN, SMALL, TEMP1,
+ $ USCAL1, USCAL2, XSC
+ INTEGER IERR, N1, NR, NUMRANK, p, q, WARNING
+ LOGICAL ALMORT, DEFR, ERREST, GOSCAL, JRACC, KILL, LQUERY,
+ $ LSVEC, L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN, NOSCAL,
+ $ ROWPIV, RSVEC, TRANSP
+*
+ INTEGER OPTWRK, MINWRK, MINRWRK, MINIWRK
+ INTEGER LWCON, LWLQF, LWQP3, LWQRF, LWUNMLQ, LWUNMQR, LWUNMQRM,
+ $ LWSVDJ, LWSVDJV, LRWQP3, LRWCON, LRWSVDJ, IWOFF
+ INTEGER LWRK_ZGELQF, LWRK_ZGEQP3, LWRK_ZGEQP3N, LWRK_ZGEQRF,
+ $ LWRK_ZGESVJ, LWRK_ZGESVJV, LWRK_ZGESVJU, LWRK_ZUNMLQ,
+ $ LWRK_ZUNMQR, LWRK_ZUNMQRM
+* ..
+* .. Local Arrays
+ COMPLEX*16 CDUMMY(1)
+ DOUBLE PRECISION RDUMMY(1)
+*
+* .. Intrinsic Functions ..
+ INTRINSIC ABS, DCMPLX, CONJG, DLOG, MAX, MIN, DBLE, NINT, SQRT
+* ..
+* .. External Functions ..
+ DOUBLE PRECISION DLAMCH, DZNRM2
+ INTEGER IDAMAX, IZAMAX
+ LOGICAL LSAME
+ EXTERNAL IDAMAX, IZAMAX, LSAME, DLAMCH, DZNRM2
+* ..
+* .. External Subroutines ..
+ EXTERNAL DLASSQ, ZCOPY, ZGELQF, ZGEQP3, ZGEQRF, ZLACPY, ZLAPMR,
+ $ ZLASCL, DLASCL, ZLASET, ZLASSQ, ZLASWP, ZUNGQR, ZUNMLQ,
+ $ ZUNMQR, ZPOCON, DSCAL, ZDSCAL, ZSWAP, ZTRSM, ZLACGV,
+ $ XERBLA
+*
+ EXTERNAL ZGESVJ
+* ..
+*
+* Test the input arguments
+*
+ LSVEC = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' )
+ JRACC = LSAME( JOBV, 'J' )
+ RSVEC = LSAME( JOBV, 'V' ) .OR. JRACC
+ ROWPIV = LSAME( JOBA, 'F' ) .OR. LSAME( JOBA, 'G' )
+ L2RANK = LSAME( JOBA, 'R' )
+ L2ABER = LSAME( JOBA, 'A' )
+ ERREST = LSAME( JOBA, 'E' ) .OR. LSAME( JOBA, 'G' )
+ L2TRAN = LSAME( JOBT, 'T' ) .AND. ( M .EQ. N )
+ L2KILL = LSAME( JOBR, 'R' )
+ DEFR = LSAME( JOBR, 'N' )
+ L2PERT = LSAME( JOBP, 'P' )
+*
+ LQUERY = ( LWORK .EQ. -1 ) .OR. ( LRWORK .EQ. -1 )
+*
+ IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR.
+ $ ERREST .OR. LSAME( JOBA, 'C' ) )) THEN
+ INFO = - 1
+ ELSE IF ( .NOT.( LSVEC .OR. LSAME( JOBU, 'N' ) .OR.
+ $ ( LSAME( JOBU, 'W' ) .AND. RSVEC .AND. L2TRAN ) ) ) THEN
+ INFO = - 2
+ ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.
+ $ ( LSAME( JOBV, 'W' ) .AND. LSVEC .AND. L2TRAN ) ) ) THEN
+ INFO = - 3
+ ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) ) THEN
+ INFO = - 4
+ ELSE IF ( .NOT. ( LSAME(JOBT,'T') .OR. LSAME(JOBT,'N') ) ) THEN
+ INFO = - 5
+ ELSE IF ( .NOT. ( L2PERT .OR. LSAME( JOBP, 'N' ) ) ) THEN
+ INFO = - 6
+ ELSE IF ( M .LT. 0 ) THEN
+ INFO = - 7
+ ELSE IF ( ( N .LT. 0 ) .OR. ( N .GT. M ) ) THEN
+ INFO = - 8
+ ELSE IF ( LDA .LT. M ) THEN
+ INFO = - 10
+ ELSE IF ( LSVEC .AND. ( LDU .LT. M ) ) THEN
+ INFO = - 13
+ ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN
+ INFO = - 15
+ ELSE
+* #:)
+ INFO = 0
+ END IF
+*
+ IF ( INFO .EQ. 0 ) THEN
+* .. compute the minimal and the optimal workspace lengths
+* [[The expressions for computing the minimal and the optimal
+* values of LCWORK, LRWORK are written with a lot of redundancy and
+* can be simplified. However, this verbose form is useful for
+* maintenance and modifications of the code.]]
+*
+* .. minimal workspace length for ZGEQP3 of an M x N matrix,
+* ZGEQRF of an N x N matrix, ZGELQF of an N x N matrix,
+* ZUNMLQ for computing N x N matrix, ZUNMQR for computing N x N
+* matrix, ZUNMQR for computing M x N matrix, respectively.
+ LWQP3 = N+1
+ LWQRF = MAX( 1, N )
+ LWLQF = MAX( 1, N )
+ LWUNMLQ = MAX( 1, N )
+ LWUNMQR = MAX( 1, N )
+ LWUNMQRM = MAX( 1, M )
+* .. minimal workspace length for ZPOCON of an N x N matrix
+ LWCON = 2 * N
+* .. minimal workspace length for ZGESVJ of an N x N matrix,
+* without and with explicit accumulation of Jacobi rotations
+ LWSVDJ = MAX( 2 * N, 1 )
+ LWSVDJV = MAX( 2 * N, 1 )
+* .. minimal REAL workspace length for ZGEQP3, ZPOCON, ZGESVJ
+ LRWQP3 = N
+ LRWCON = N
+ LRWSVDJ = N
+ IF ( LQUERY ) THEN
+ CALL ZGEQP3( M, N, A, LDA, IWORK, CDUMMY, CDUMMY, -1,
+ $ RDUMMY, IERR )
+ LWRK_ZGEQP3 = CDUMMY(1)
+ CALL ZGEQRF( N, N, A, LDA, CDUMMY, CDUMMY,-1, IERR )
+ LWRK_ZGEQRF = CDUMMY(1)
+ CALL ZGELQF( N, N, A, LDA, CDUMMY, CDUMMY,-1, IERR )
+ LWRK_ZGELQF = CDUMMY(1)
+ END IF
+ MINWRK = 2
+ OPTWRK = 2
+ MINIWRK = N
+ IF ( .NOT. (LSVEC .OR. RSVEC ) ) THEN
+* .. minimal and optimal sizes of the complex workspace if
+* only the singular values are requested
+ IF ( ERREST ) THEN
+ MINWRK = MAX( N+LWQP3, N**2+LWCON, N+LWQRF, LWSVDJ )
+ ELSE
+ MINWRK = MAX( N+LWQP3, N+LWQRF, LWSVDJ )
+ END IF
+ IF ( LQUERY ) THEN
+ CALL ZGESVJ( 'L', 'N', 'N', N, N, A, LDA, SVA, N, V,
+ $ LDV, CDUMMY, -1, RDUMMY, -1, IERR )
+ LWRK_ZGESVJ = CDUMMY(1)
+ IF ( ERREST ) THEN
+ OPTWRK = MAX( N+LWRK_ZGEQP3, N**2+LWCON,
+ $ N+LWRK_ZGEQRF, LWRK_ZGESVJ )
+ ELSE
+ OPTWRK = MAX( N+LWRK_ZGEQP3, N+LWRK_ZGEQRF,
+ $ LWRK_ZGESVJ )
+ END IF
+ END IF
+ IF ( L2TRAN .OR. ROWPIV ) THEN
+ IF ( ERREST ) THEN
+ MINRWRK = MAX( 7, 2*M, LRWQP3, LRWCON, LRWSVDJ )
+ ELSE
+ MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
+ END IF
+ ELSE
+ IF ( ERREST ) THEN
+ MINRWRK = MAX( 7, LRWQP3, LRWCON, LRWSVDJ )
+ ELSE
+ MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
+ END IF
+ END IF
+ IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
+ ELSE IF ( RSVEC .AND. (.NOT.LSVEC) ) THEN
+* .. minimal and optimal sizes of the complex workspace if the
+* singular values and the right singular vectors are requested
+ IF ( ERREST ) THEN
+ MINWRK = MAX( N+LWQP3, LWCON, LWSVDJ, N+LWLQF,
+ $ 2*N+LWQRF, N+LWSVDJ, N+LWUNMLQ )
+ ELSE
+ MINWRK = MAX( N+LWQP3, LWSVDJ, N+LWLQF, 2*N+LWQRF,
+ $ N+LWSVDJ, N+LWUNMLQ )
+ END IF
+ IF ( LQUERY ) THEN
+ CALL ZGESVJ( 'L', 'U', 'N', N,N, U, LDU, SVA, N, A,
+ $ LDA, CDUMMY, -1, RDUMMY, -1, IERR )
+ LWRK_ZGESVJ = CDUMMY(1)
+ CALL ZUNMLQ( 'L', 'C', N, N, N, A, LDA, CDUMMY,
+ $ V, LDV, CDUMMY, -1, IERR )
+ LWRK_ZUNMLQ = CDUMMY(1)
+ IF ( ERREST ) THEN
+ OPTWRK = MAX( N+LWRK_ZGEQP3, LWCON, LWRK_ZGESVJ,
+ $ N+LWRK_ZGELQF, 2*N+LWRK_ZGEQRF,
+ $ N+LWRK_ZGESVJ, N+LWRK_ZUNMLQ )
+ ELSE
+ OPTWRK = MAX( N+LWRK_ZGEQP3, LWRK_ZGESVJ,N+LWRK_ZGELQF,
+ $ 2*N+LWRK_ZGEQRF, N+LWRK_ZGESVJ,
+ $ N+LWRK_ZUNMLQ )
+ END IF
+ END IF
+ IF ( L2TRAN .OR. ROWPIV ) THEN
+ IF ( ERREST ) THEN
+ MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
+ ELSE
+ MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
+ END IF
+ ELSE
+ IF ( ERREST ) THEN
+ MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
+ ELSE
+ MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
+ END IF
+ END IF
+ IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
+ ELSE IF ( LSVEC .AND. (.NOT.RSVEC) ) THEN
+* .. minimal and optimal sizes of the complex workspace if the
+* singular values and the left singular vectors are requested
+ IF ( ERREST ) THEN
+ MINWRK = N + MAX( LWQP3,LWCON,N+LWQRF,LWSVDJ,LWUNMQRM )
+ ELSE
+ MINWRK = N + MAX( LWQP3, N+LWQRF, LWSVDJ, LWUNMQRM )
+ END IF
+ IF ( LQUERY ) THEN
+ CALL ZGESVJ( 'L', 'U', 'N', N,N, U, LDU, SVA, N, A,
+ $ LDA, CDUMMY, -1, RDUMMY, -1, IERR )
+ LWRK_ZGESVJ = CDUMMY(1)
+ CALL ZUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
+ $ LDU, CDUMMY, -1, IERR )
+ LWRK_ZUNMQRM = CDUMMY(1)
+ IF ( ERREST ) THEN
+ OPTWRK = N + MAX( LWRK_ZGEQP3, LWCON, N+LWRK_ZGEQRF,
+ $ LWRK_ZGESVJ, LWRK_ZUNMQRM )
+ ELSE
+ OPTWRK = N + MAX( LWRK_ZGEQP3, N+LWRK_ZGEQRF,
+ $ LWRK_ZGESVJ, LWRK_ZUNMQRM )
+ END IF
+ END IF
+ IF ( L2TRAN .OR. ROWPIV ) THEN
+ IF ( ERREST ) THEN
+ MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
+ ELSE
+ MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
+ END IF
+ ELSE
+ IF ( ERREST ) THEN
+ MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
+ ELSE
+ MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
+ END IF
+ END IF
+ IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
+ ELSE
+* .. minimal and optimal sizes of the complex workspace if the
+* full SVD is requested
+ IF ( .NOT. JRACC ) THEN
+ IF ( ERREST ) THEN
+ MINWRK = MAX( N+LWQP3, N+LWCON, 2*N+N**2+LWCON,
+ $ 2*N+LWQRF, 2*N+LWQP3,
+ $ 2*N+N**2+N+LWLQF, 2*N+N**2+N+N**2+LWCON,
+ $ 2*N+N**2+N+LWSVDJ, 2*N+N**2+N+LWSVDJV,
+ $ 2*N+N**2+N+LWUNMQR,2*N+N**2+N+LWUNMLQ,
+ $ N+N**2+LWSVDJ, N+LWUNMQRM )
+ ELSE
+ MINWRK = MAX( N+LWQP3, 2*N+N**2+LWCON,
+ $ 2*N+LWQRF, 2*N+LWQP3,
+ $ 2*N+N**2+N+LWLQF, 2*N+N**2+N+N**2+LWCON,
+ $ 2*N+N**2+N+LWSVDJ, 2*N+N**2+N+LWSVDJV,
+ $ 2*N+N**2+N+LWUNMQR,2*N+N**2+N+LWUNMLQ,
+ $ N+N**2+LWSVDJ, N+LWUNMQRM )
+ END IF
+ MINIWRK = MINIWRK + N
+ IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
+ ELSE
+ IF ( ERREST ) THEN
+ MINWRK = MAX( N+LWQP3, N+LWCON, 2*N+LWQRF,
+ $ 2*N+N**2+LWSVDJV, 2*N+N**2+N+LWUNMQR,
+ $ N+LWUNMQRM )
+ ELSE
+ MINWRK = MAX( N+LWQP3, 2*N+LWQRF,
+ $ 2*N+N**2+LWSVDJV, 2*N+N**2+N+LWUNMQR,
+ $ N+LWUNMQRM )
+ END IF
+ IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
+ END IF
+ IF ( LQUERY ) THEN
+ CALL ZUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
+ $ LDU, CDUMMY, -1, IERR )
+ LWRK_ZUNMQRM = CDUMMY(1)
+ CALL ZUNMQR( 'L', 'N', N, N, N, A, LDA, CDUMMY, U,
+ $ LDU, CDUMMY, -1, IERR )
+ LWRK_ZUNMQR = CDUMMY(1)
+ IF ( .NOT. JRACC ) THEN
+ CALL ZGEQP3( N,N, A, LDA, IWORK, CDUMMY,CDUMMY, -1,
+ $ RDUMMY, IERR )
+ LWRK_ZGEQP3N = CDUMMY(1)
+ CALL ZGESVJ( 'L', 'U', 'N', N, N, U, LDU, SVA,
+ $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
+ LWRK_ZGESVJ = CDUMMY(1)
+ CALL ZGESVJ( 'U', 'U', 'N', N, N, U, LDU, SVA,
+ $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
+ LWRK_ZGESVJU = CDUMMY(1)
+ CALL ZGESVJ( 'L', 'U', 'V', N, N, U, LDU, SVA,
+ $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
+ LWRK_ZGESVJV = CDUMMY(1)
+ CALL ZUNMLQ( 'L', 'C', N, N, N, A, LDA, CDUMMY,
+ $ V, LDV, CDUMMY, -1, IERR )
+ LWRK_ZUNMLQ = CDUMMY(1)
+ IF ( ERREST ) THEN
+ OPTWRK = MAX( N+LWRK_ZGEQP3, N+LWCON,
+ $ 2*N+N**2+LWCON, 2*N+LWRK_ZGEQRF,
+ $ 2*N+LWRK_ZGEQP3N,
+ $ 2*N+N**2+N+LWRK_ZGELQF,
+ $ 2*N+N**2+N+N**2+LWCON,
+ $ 2*N+N**2+N+LWRK_ZGESVJ,
+ $ 2*N+N**2+N+LWRK_ZGESVJV,
+ $ 2*N+N**2+N+LWRK_ZUNMQR,
+ $ 2*N+N**2+N+LWRK_ZUNMLQ,
+ $ N+N**2+LWRK_ZGESVJU,
+ $ N+LWRK_ZUNMQRM )
+ ELSE
+ OPTWRK = MAX( N+LWRK_ZGEQP3,
+ $ 2*N+N**2+LWCON, 2*N+LWRK_ZGEQRF,
+ $ 2*N+LWRK_ZGEQP3N,
+ $ 2*N+N**2+N+LWRK_ZGELQF,
+ $ 2*N+N**2+N+N**2+LWCON,
+ $ 2*N+N**2+N+LWRK_ZGESVJ,
+ $ 2*N+N**2+N+LWRK_ZGESVJV,
+ $ 2*N+N**2+N+LWRK_ZUNMQR,
+ $ 2*N+N**2+N+LWRK_ZUNMLQ,
+ $ N+N**2+LWRK_ZGESVJU,
+ $ N+LWRK_ZUNMQRM )
+ END IF
+ ELSE
+ CALL ZGESVJ( 'L', 'U', 'V', N, N, U, LDU, SVA,
+ $ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
+ LWRK_ZGESVJV = CDUMMY(1)
+ CALL ZUNMQR( 'L', 'N', N, N, N, CDUMMY, N, CDUMMY,
+ $ V, LDV, CDUMMY, -1, IERR )
+ LWRK_ZUNMQR = CDUMMY(1)
+ CALL ZUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
+ $ LDU, CDUMMY, -1, IERR )
+ LWRK_ZUNMQRM = CDUMMY(1)
+ IF ( ERREST ) THEN
+ OPTWRK = MAX( N+LWRK_ZGEQP3, N+LWCON,
+ $ 2*N+LWRK_ZGEQRF, 2*N+N**2,
+ $ 2*N+N**2+LWRK_ZGESVJV,
+ $ 2*N+N**2+N+LWRK_ZUNMQR,N+LWRK_ZUNMQRM )
+ ELSE
+ OPTWRK = MAX( N+LWRK_ZGEQP3, 2*N+LWRK_ZGEQRF,
+ $ 2*N+N**2, 2*N+N**2+LWRK_ZGESVJV,
+ $ 2*N+N**2+N+LWRK_ZUNMQR,
+ $ N+LWRK_ZUNMQRM )
+ END IF
+ END IF
+ END IF
+ IF ( L2TRAN .OR. ROWPIV ) THEN
+ MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
+ ELSE
+ MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
+ END IF
+ END IF
+ MINWRK = MAX( 2, MINWRK )
+ OPTWRK = MAX( 2, OPTWRK )
+ IF ( LWORK .LT. MINWRK .AND. (.NOT.LQUERY) ) INFO = - 17
+ IF ( LRWORK .LT. MINRWRK .AND. (.NOT.LQUERY) ) INFO = - 19
+ END IF
+*
+ IF ( INFO .NE. 0 ) THEN
+* #:(
+ CALL XERBLA( 'ZGEJSV', - INFO )
+ RETURN
+ ELSE IF ( LQUERY ) THEN
+ CWORK(1) = OPTWRK
+ CWORK(2) = MINWRK
+ RWORK(1) = MINRWRK
+ IWORK(1) = MAX( 4, MINIWRK )
+ RETURN
+ END IF
+*
+* Quick return for void matrix (Y3K safe)
+* #:)
+ IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) THEN
+ IWORK(1:4) = 0
+ RWORK(1:7) = 0
+ RETURN
+ ENDIF
+*
+* Determine whether the matrix U should be M x N or M x M
+*
+ IF ( LSVEC ) THEN
+ N1 = N
+ IF ( LSAME( JOBU, 'F' ) ) N1 = M
+ END IF
+*
+* Set numerical parameters
+*
+*! NOTE: Make sure DLAMCH() does not fail on the target architecture.
+*
+ EPSLN = DLAMCH('Epsilon')
+ SFMIN = DLAMCH('SafeMinimum')
+ SMALL = SFMIN / EPSLN
+ BIG = DLAMCH('O')
+* BIG = ONE / SFMIN
+*
+* Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N
+*
+*(!) If necessary, scale SVA() to protect the largest norm from
+* overflow. It is possible that this scaling pushes the smallest
+* column norm left from the underflow threshold (extreme case).
+*
+ SCALEM = ONE / SQRT(DBLE(M)*DBLE(N))
+ NOSCAL = .TRUE.
+ GOSCAL = .TRUE.
+ DO 1874 p = 1, N
+ AAPP = ZERO
+ AAQQ = ONE
+ CALL ZLASSQ( M, A(1,p), 1, AAPP, AAQQ )
+ IF ( AAPP .GT. BIG ) THEN
+ INFO = - 9
+ CALL XERBLA( 'ZGEJSV', -INFO )
+ RETURN
+ END IF
+ AAQQ = SQRT(AAQQ)
+ IF ( ( AAPP .LT. (BIG / AAQQ) ) .AND. NOSCAL ) THEN
+ SVA(p) = AAPP * AAQQ
+ ELSE
+ NOSCAL = .FALSE.
+ SVA(p) = AAPP * ( AAQQ * SCALEM )
+ IF ( GOSCAL ) THEN
+ GOSCAL = .FALSE.
+ CALL DSCAL( p-1, SCALEM, SVA, 1 )
+ END IF
+ END IF
+ 1874 CONTINUE
+*
+ IF ( NOSCAL ) SCALEM = ONE
+*
+ AAPP = ZERO
+ AAQQ = BIG
+ DO 4781 p = 1, N
+ AAPP = MAX( AAPP, SVA(p) )
+ IF ( SVA(p) .NE. ZERO ) AAQQ = MIN( AAQQ, SVA(p) )
+ 4781 CONTINUE
+*
+* Quick return for zero M x N matrix
+* #:)
+ IF ( AAPP .EQ. ZERO ) THEN
+ IF ( LSVEC ) CALL ZLASET( 'G', M, N1, CZERO, CONE, U, LDU )
+ IF ( RSVEC ) CALL ZLASET( 'G', N, N, CZERO, CONE, V, LDV )
+ RWORK(1) = ONE
+ RWORK(2) = ONE
+ IF ( ERREST ) RWORK(3) = ONE
+ IF ( LSVEC .AND. RSVEC ) THEN
+ RWORK(4) = ONE
+ RWORK(5) = ONE
+ END IF
+ IF ( L2TRAN ) THEN
+ RWORK(6) = ZERO
+ RWORK(7) = ZERO
+ END IF
+ IWORK(1) = 0
+ IWORK(2) = 0
+ IWORK(3) = 0
+ IWORK(4) = -1
+ RETURN
+ END IF
+*
+* Issue warning if denormalized column norms detected. Override the
+* high relative accuracy request. Issue licence to kill nonzero columns
+* (set them to zero) whose norm is less than sigma_max / BIG (roughly).
+* #:(
+ WARNING = 0
+ IF ( AAQQ .LE. SFMIN ) THEN
+ L2RANK = .TRUE.
+ L2KILL = .TRUE.
+ WARNING = 1
+ END IF
+*
+* Quick return for one-column matrix
+* #:)
+ IF ( N .EQ. 1 ) THEN
+*
+ IF ( LSVEC ) THEN
+ CALL ZLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR )
+ CALL ZLACPY( 'A', M, 1, A, LDA, U, LDU )
+* computing all M left singular vectors of the M x 1 matrix
+ IF ( N1 .NE. N ) THEN
+ CALL ZGEQRF( M, N, U,LDU, CWORK, CWORK(N+1),LWORK-N,IERR )
+ CALL ZUNGQR( M,N1,1, U,LDU,CWORK,CWORK(N+1),LWORK-N,IERR )
+ CALL ZCOPY( M, A(1,1), 1, U(1,1), 1 )
+ END IF
+ END IF
+ IF ( RSVEC ) THEN
+ V(1,1) = CONE
+ END IF
+ IF ( SVA(1) .LT. (BIG*SCALEM) ) THEN
+ SVA(1) = SVA(1) / SCALEM
+ SCALEM = ONE
+ END IF
+ RWORK(1) = ONE / SCALEM
+ RWORK(2) = ONE
+ IF ( SVA(1) .NE. ZERO ) THEN
+ IWORK(1) = 1
+ IF ( ( SVA(1) / SCALEM) .GE. SFMIN ) THEN
+ IWORK(2) = 1
+ ELSE
+ IWORK(2) = 0
+ END IF
+ ELSE
+ IWORK(1) = 0
+ IWORK(2) = 0
+ END IF
+ IWORK(3) = 0
+ IWORK(4) = -1
+ IF ( ERREST ) RWORK(3) = ONE
+ IF ( LSVEC .AND. RSVEC ) THEN
+ RWORK(4) = ONE
+ RWORK(5) = ONE
+ END IF
+ IF ( L2TRAN ) THEN
+ RWORK(6) = ZERO
+ RWORK(7) = ZERO
+ END IF
+ RETURN
+*
+ END IF
+*
+ TRANSP = .FALSE.
+*
+ AATMAX = -ONE
+ AATMIN = BIG
+ IF ( ROWPIV .OR. L2TRAN ) THEN
+*
+* Compute the row norms, needed to determine row pivoting sequence
+* (in the case of heavily row weighted A, row pivoting is strongly
+* advised) and to collect information needed to compare the
+* structures of A * A^* and A^* * A (in the case L2TRAN.EQ..TRUE.).
+*
+ IF ( L2TRAN ) THEN
+ DO 1950 p = 1, M
+ XSC = ZERO
+ TEMP1 = ONE
+ CALL ZLASSQ( N, A(p,1), LDA, XSC, TEMP1 )
+* ZLASSQ gets both the ell_2 and the ell_infinity norm
+* in one pass through the vector
+ RWORK(M+p) = XSC * SCALEM
+ RWORK(p) = XSC * (SCALEM*SQRT(TEMP1))
+ AATMAX = MAX( AATMAX, RWORK(p) )
+ IF (RWORK(p) .NE. ZERO)
+ $ AATMIN = MIN(AATMIN,RWORK(p))
+ 1950 CONTINUE
+ ELSE
+ DO 1904 p = 1, M
+ RWORK(M+p) = SCALEM*ABS( A(p,IZAMAX(N,A(p,1),LDA)) )
+ AATMAX = MAX( AATMAX, RWORK(M+p) )
+ AATMIN = MIN( AATMIN, RWORK(M+p) )
+ 1904 CONTINUE
+ END IF
+*
+ END IF
+*
+* For square matrix A try to determine whether A^* would be better
+* input for the preconditioned Jacobi SVD, with faster convergence.
+* The decision is based on an O(N) function of the vector of column
+* and row norms of A, based on the Shannon entropy. This should give
+* the right choice in most cases when the difference actually matters.
+* It may fail and pick the slower converging side.
+*
+ ENTRA = ZERO
+ ENTRAT = ZERO
+ IF ( L2TRAN ) THEN
+*
+ XSC = ZERO
+ TEMP1 = ONE
+ CALL DLASSQ( N, SVA, 1, XSC, TEMP1 )
+ TEMP1 = ONE / TEMP1
+*
+ ENTRA = ZERO
+ DO 1113 p = 1, N
+ BIG1 = ( ( SVA(p) / XSC )**2 ) * TEMP1
+ IF ( BIG1 .NE. ZERO ) ENTRA = ENTRA + BIG1 * DLOG(BIG1)
+ 1113 CONTINUE
+ ENTRA = - ENTRA / DLOG(DBLE(N))
+*
+* Now, SVA().^2/Trace(A^* * A) is a point in the probability simplex.
+* It is derived from the diagonal of A^* * A. Do the same with the
+* diagonal of A * A^*, compute the entropy of the corresponding
+* probability distribution. Note that A * A^* and A^* * A have the
+* same trace.
+*
+ ENTRAT = ZERO
+ DO 1114 p = 1, M
+ BIG1 = ( ( RWORK(p) / XSC )**2 ) * TEMP1
+ IF ( BIG1 .NE. ZERO ) ENTRAT = ENTRAT + BIG1 * DLOG(BIG1)
+ 1114 CONTINUE
+ ENTRAT = - ENTRAT / DLOG(DBLE(M))
+*
+* Analyze the entropies and decide A or A^*. Smaller entropy
+* usually means better input for the algorithm.
+*
+ TRANSP = ( ENTRAT .LT. ENTRA )
+*
+* If A^* is better than A, take the adjoint of A. This is allowed
+* only for square matrices, M=N.
+ IF ( TRANSP ) THEN
+* In an optimal implementation, this trivial transpose
+* should be replaced with faster transpose.
+ DO 1115 p = 1, N - 1
+ A(p,p) = CONJG(A(p,p))
+ DO 1116 q = p + 1, N
+ CTEMP = CONJG(A(q,p))
+ A(q,p) = CONJG(A(p,q))
+ A(p,q) = CTEMP
+ 1116 CONTINUE
+ 1115 CONTINUE
+ A(N,N) = CONJG(A(N,N))
+ DO 1117 p = 1, N
+ RWORK(M+p) = SVA(p)
+ SVA(p) = RWORK(p)
+* previously computed row 2-norms are now column 2-norms
+* of the transposed matrix
+ 1117 CONTINUE
+ TEMP1 = AAPP
+ AAPP = AATMAX
+ AATMAX = TEMP1
+ TEMP1 = AAQQ
+ AAQQ = AATMIN
+ AATMIN = TEMP1
+ KILL = LSVEC
+ LSVEC = RSVEC
+ RSVEC = KILL
+ IF ( LSVEC ) N1 = N
+*
+ ROWPIV = .TRUE.
+ END IF
+*
+ END IF
+* END IF L2TRAN
+*
+* Scale the matrix so that its maximal singular value remains less
+* than SQRT(BIG) -- the matrix is scaled so that its maximal column
+* has Euclidean norm equal to SQRT(BIG/N). The only reason to keep
+* SQRT(BIG) instead of BIG is the fact that ZGEJSV uses LAPACK and
+* BLAS routines that, in some implementations, are not capable of
+* working in the full interval [SFMIN,BIG] and that they may provoke
+* overflows in the intermediate results. If the singular values spread
+* from SFMIN to BIG, then ZGESVJ will compute them. So, in that case,
+* one should use ZGESVJ instead of ZGEJSV.
+* >> change in the April 2016 update: allow bigger range, i.e. the
+* largest column is allowed up to BIG/N and ZGESVJ will do the rest.
+ BIG1 = SQRT( BIG )
+ TEMP1 = SQRT( BIG / DBLE(N) )
+* TEMP1 = BIG/DBLE(N)
+*
+ CALL DLASCL( 'G', 0, 0, AAPP, TEMP1, N, 1, SVA, N, IERR )
+ IF ( AAQQ .GT. (AAPP * SFMIN) ) THEN
+ AAQQ = ( AAQQ / AAPP ) * TEMP1
+ ELSE
+ AAQQ = ( AAQQ * TEMP1 ) / AAPP
+ END IF
+ TEMP1 = TEMP1 * SCALEM
+ CALL ZLASCL( 'G', 0, 0, AAPP, TEMP1, M, N, A, LDA, IERR )
+*
+* To undo scaling at the end of this procedure, multiply the
+* computed singular values with USCAL2 / USCAL1.
+*
+ USCAL1 = TEMP1
+ USCAL2 = AAPP
+*
+ IF ( L2KILL ) THEN
+* L2KILL enforces computation of nonzero singular values in
+* the restricted range of condition number of the initial A,
+* sigma_max(A) / sigma_min(A) approx. SQRT(BIG)/SQRT(SFMIN).
+ XSC = SQRT( SFMIN )
+ ELSE
+ XSC = SMALL
+*
+* Now, if the condition number of A is too big,
+* sigma_max(A) / sigma_min(A) .GT. SQRT(BIG/N) * EPSLN / SFMIN,
+* as a precaution measure, the full SVD is computed using ZGESVJ
+* with accumulated Jacobi rotations. This provides numerically
+* more robust computation, at the cost of slightly increased run
+* time. Depending on the concrete implementation of BLAS and LAPACK
+* (i.e. how they behave in presence of extreme ill-conditioning) the
+* implementor may decide to remove this switch.
+ IF ( ( AAQQ.LT.SQRT(SFMIN) ) .AND. LSVEC .AND. RSVEC ) THEN
+ JRACC = .TRUE.
+ END IF
+*
+ END IF
+ IF ( AAQQ .LT. XSC ) THEN
+ DO 700 p = 1, N
+ IF ( SVA(p) .LT. XSC ) THEN
+ CALL ZLASET( 'A', M, 1, CZERO, CZERO, A(1,p), LDA )
+ SVA(p) = ZERO
+ END IF
+ 700 CONTINUE
+ END IF
+*
+* Preconditioning using QR factorization with pivoting
+*
+ IF ( ROWPIV ) THEN
+* Optional row permutation (Bjoerck row pivoting):
+* A result by Cox and Higham shows that the Bjoerck's
+* row pivoting combined with standard column pivoting
+* has similar effect as Powell-Reid complete pivoting.
+* The ell-infinity norms of A are made nonincreasing.
+ IF ( ( LSVEC .AND. RSVEC ) .AND. .NOT.( JRACC ) ) THEN
+ IWOFF = 2*N
+ ELSE
+ IWOFF = N
+ END IF
+ DO 1952 p = 1, M - 1
+ q = IDAMAX( M-p+1, RWORK(M+p), 1 ) + p - 1
+ IWORK(IWOFF+p) = q
+ IF ( p .NE. q ) THEN
+ TEMP1 = RWORK(M+p)
+ RWORK(M+p) = RWORK(M+q)
+ RWORK(M+q) = TEMP1
+ END IF
+ 1952 CONTINUE
+ CALL ZLASWP( N, A, LDA, 1, M-1, IWORK(IWOFF+1), 1 )
+ END IF
+*
+* End of the preparation phase (scaling, optional sorting and
+* transposing, optional flushing of small columns).
+*
+* Preconditioning
+*
+* If the full SVD is needed, the right singular vectors are computed
+* from a matrix equation, and for that we need theoretical analysis
+* of the Businger-Golub pivoting. So we use ZGEQP3 as the first RR QRF.
+* In all other cases the first RR QRF can be chosen by other criteria
+* (eg speed by replacing global with restricted window pivoting, such
+* as in xGEQPX from TOMS # 782). Good results will be obtained using
+* xGEQPX with properly (!) chosen numerical parameters.
+* Any improvement of ZGEQP3 improves overal performance of ZGEJSV.
+*
+* A * P1 = Q1 * [ R1^* 0]^*:
+ DO 1963 p = 1, N
+* .. all columns are free columns
+ IWORK(p) = 0
+ 1963 CONTINUE
+ CALL ZGEQP3( M, N, A, LDA, IWORK, CWORK, CWORK(N+1), LWORK-N,
+ $ RWORK, IERR )
+*
+* The upper triangular matrix R1 from the first QRF is inspected for
+* rank deficiency and possibilities for deflation, or possible
+* ill-conditioning. Depending on the user specified flag L2RANK,
+* the procedure explores possibilities to reduce the numerical
+* rank by inspecting the computed upper triangular factor. If
+* L2RANK or L2ABER are up, then ZGEJSV will compute the SVD of
+* A + dA, where ||dA|| <= f(M,N)*EPSLN.
+*
+ NR = 1
+ IF ( L2ABER ) THEN
+* Standard absolute error bound suffices. All sigma_i with
+* sigma_i < N*EPSLN*||A|| are flushed to zero. This is an
+* agressive enforcement of lower numerical rank by introducing a
+* backward error of the order of N*EPSLN*||A||.
+ TEMP1 = SQRT(DBLE(N))*EPSLN
+ DO 3001 p = 2, N
+ IF ( ABS(A(p,p)) .GE. (TEMP1*ABS(A(1,1))) ) THEN
+ NR = NR + 1
+ ELSE
+ GO TO 3002
+ END IF
+ 3001 CONTINUE
+ 3002 CONTINUE
+ ELSE IF ( L2RANK ) THEN
+* .. similarly as above, only slightly more gentle (less agressive).
+* Sudden drop on the diagonal of R1 is used as the criterion for
+* close-to-rank-deficient.
+ TEMP1 = SQRT(SFMIN)
+ DO 3401 p = 2, N
+ IF ( ( ABS(A(p,p)) .LT. (EPSLN*ABS(A(p-1,p-1))) ) .OR.
+ $ ( ABS(A(p,p)) .LT. SMALL ) .OR.
+ $ ( L2KILL .AND. (ABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3402
+ NR = NR + 1
+ 3401 CONTINUE
+ 3402 CONTINUE
+*
+ ELSE
+* The goal is high relative accuracy. However, if the matrix
+* has high scaled condition number the relative accuracy is in
+* general not feasible. Later on, a condition number estimator
+* will be deployed to estimate the scaled condition number.
+* Here we just remove the underflowed part of the triangular
+* factor. This prevents the situation in which the code is
+* working hard to get the accuracy not warranted by the data.
+ TEMP1 = SQRT(SFMIN)
+ DO 3301 p = 2, N
+ IF ( ( ABS(A(p,p)) .LT. SMALL ) .OR.
+ $ ( L2KILL .AND. (ABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3302
+ NR = NR + 1
+ 3301 CONTINUE
+ 3302 CONTINUE
+*
+ END IF
+*
+ ALMORT = .FALSE.
+ IF ( NR .EQ. N ) THEN
+ MAXPRJ = ONE
+ DO 3051 p = 2, N
+ TEMP1 = ABS(A(p,p)) / SVA(IWORK(p))
+ MAXPRJ = MIN( MAXPRJ, TEMP1 )
+ 3051 CONTINUE
+ IF ( MAXPRJ**2 .GE. ONE - DBLE(N)*EPSLN ) ALMORT = .TRUE.
+ END IF
+*
+*
+ SCONDA = - ONE
+ CONDR1 = - ONE
+ CONDR2 = - ONE
+*
+ IF ( ERREST ) THEN
+ IF ( N .EQ. NR ) THEN
+ IF ( RSVEC ) THEN
+* .. V is available as workspace
+ CALL ZLACPY( 'U', N, N, A, LDA, V, LDV )
+ DO 3053 p = 1, N
+ TEMP1 = SVA(IWORK(p))
+ CALL ZDSCAL( p, ONE/TEMP1, V(1,p), 1 )
+ 3053 CONTINUE
+ IF ( LSVEC )THEN
+ CALL ZPOCON( 'U', N, V, LDV, ONE, TEMP1,
+ $ CWORK(N+1), RWORK, IERR )
+ ELSE
+ CALL ZPOCON( 'U', N, V, LDV, ONE, TEMP1,
+ $ CWORK, RWORK, IERR )
+ END IF
+*
+ ELSE IF ( LSVEC ) THEN
+* .. U is available as workspace
+ CALL ZLACPY( 'U', N, N, A, LDA, U, LDU )
+ DO 3054 p = 1, N
+ TEMP1 = SVA(IWORK(p))
+ CALL ZDSCAL( p, ONE/TEMP1, U(1,p), 1 )
+ 3054 CONTINUE
+ CALL ZPOCON( 'U', N, U, LDU, ONE, TEMP1,
+ $ CWORK(N+1), RWORK, IERR )
+ ELSE
+ CALL ZLACPY( 'U', N, N, A, LDA, CWORK, N )
+*[] CALL ZLACPY( 'U', N, N, A, LDA, CWORK(N+1), N )
+* Change: here index shifted by N to the left, CWORK(1:N)
+* not needed for SIGMA only computation
+ DO 3052 p = 1, N
+ TEMP1 = SVA(IWORK(p))
+*[] CALL ZDSCAL( p, ONE/TEMP1, CWORK(N+(p-1)*N+1), 1 )
+ CALL ZDSCAL( p, ONE/TEMP1, CWORK((p-1)*N+1), 1 )
+ 3052 CONTINUE
+* .. the columns of R are scaled to have unit Euclidean lengths.
+*[] CALL ZPOCON( 'U', N, CWORK(N+1), N, ONE, TEMP1,
+*[] $ CWORK(N+N*N+1), RWORK, IERR )
+ CALL ZPOCON( 'U', N, CWORK, N, ONE, TEMP1,
+ $ CWORK(N*N+1), RWORK, IERR )
+*
+ END IF
+ IF ( TEMP1 .NE. ZERO ) THEN
+ SCONDA = ONE / SQRT(TEMP1)
+ ELSE
+ SCONDA = - ONE
+ END IF
+* SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
+* N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
+ ELSE
+ SCONDA = - ONE
+ END IF
+ END IF
+*
+ L2PERT = L2PERT .AND. ( ABS( A(1,1)/A(NR,NR) ) .GT. SQRT(BIG1) )
+* If there is no violent scaling, artificial perturbation is not needed.
+*
+* Phase 3:
+*
+ IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN
+*
+* Singular Values only
+*
+* .. transpose A(1:NR,1:N)
+ DO 1946 p = 1, MIN( N-1, NR )
+ CALL ZCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )
+ CALL ZLACGV( N-p+1, A(p,p), 1 )
+ 1946 CONTINUE
+ IF ( NR .EQ. N ) A(N,N) = CONJG(A(N,N))
+*
+* The following two DO-loops introduce small relative perturbation
+* into the strict upper triangle of the lower triangular matrix.
+* Small entries below the main diagonal are also changed.
+* This modification is useful if the computing environment does not
+* provide/allow FLUSH TO ZERO underflow, for it prevents many
+* annoying denormalized numbers in case of strongly scaled matrices.
+* The perturbation is structured so that it does not introduce any
+* new perturbation of the singular values, and it does not destroy
+* the job done by the preconditioner.
+* The licence for this perturbation is in the variable L2PERT, which
+* should be .FALSE. if FLUSH TO ZERO underflow is active.
+*
+ IF ( .NOT. ALMORT ) THEN
+*
+ IF ( L2PERT ) THEN
+* XSC = SQRT(SMALL)
+ XSC = EPSLN / DBLE(N)
+ DO 4947 q = 1, NR
+ CTEMP = DCMPLX(XSC*ABS(A(q,q)),ZERO)
+ DO 4949 p = 1, N
+ IF ( ( (p.GT.q) .AND. (ABS(A(p,q)).LE.TEMP1) )
+ $ .OR. ( p .LT. q ) )
+* $ A(p,q) = TEMP1 * ( A(p,q) / ABS(A(p,q)) )
+ $ A(p,q) = CTEMP
+ 4949 CONTINUE
+ 4947 CONTINUE
+ ELSE
+ CALL ZLASET( 'U', NR-1,NR-1, CZERO,CZERO, A(1,2),LDA )
+ END IF
+*
+* .. second preconditioning using the QR factorization
+*
+ CALL ZGEQRF( N,NR, A,LDA, CWORK, CWORK(N+1),LWORK-N, IERR )
+*
+* .. and transpose upper to lower triangular
+ DO 1948 p = 1, NR - 1
+ CALL ZCOPY( NR-p, A(p,p+1), LDA, A(p+1,p), 1 )
+ CALL ZLACGV( NR-p+1, A(p,p), 1 )
+ 1948 CONTINUE
+*
+ END IF
+*
+* Row-cyclic Jacobi SVD algorithm with column pivoting
+*
+* .. again some perturbation (a "background noise") is added
+* to drown denormals
+ IF ( L2PERT ) THEN
+* XSC = SQRT(SMALL)
+ XSC = EPSLN / DBLE(N)
+ DO 1947 q = 1, NR
+ CTEMP = DCMPLX(XSC*ABS(A(q,q)),ZERO)
+ DO 1949 p = 1, NR
+ IF ( ( (p.GT.q) .AND. (ABS(A(p,q)).LE.TEMP1) )
+ $ .OR. ( p .LT. q ) )
+* $ A(p,q) = TEMP1 * ( A(p,q) / ABS(A(p,q)) )
+ $ A(p,q) = CTEMP
+ 1949 CONTINUE
+ 1947 CONTINUE
+ ELSE
+ CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, A(1,2), LDA )
+ END IF
+*
+* .. and one-sided Jacobi rotations are started on a lower
+* triangular matrix (plus perturbation which is ignored in
+* the part which destroys triangular form (confusing?!))
+*
+ CALL ZGESVJ( 'L', 'N', 'N', NR, NR, A, LDA, SVA,
+ $ N, V, LDV, CWORK, LWORK, RWORK, LRWORK, INFO )
+*
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+*
+*
+ ELSE IF ( ( RSVEC .AND. ( .NOT. LSVEC ) .AND. ( .NOT. JRACC ) )
+ $ .OR.
+ $ ( JRACC .AND. ( .NOT. LSVEC ) .AND. ( NR .NE. N ) ) ) THEN
+*
+* -> Singular Values and Right Singular Vectors <-
+*
+ IF ( ALMORT ) THEN
+*
+* .. in this case NR equals N
+ DO 1998 p = 1, NR
+ CALL ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
+ CALL ZLACGV( N-p+1, V(p,p), 1 )
+ 1998 CONTINUE
+ CALL ZLASET( 'U', NR-1,NR-1, CZERO, CZERO, V(1,2), LDV )
+*
+ CALL ZGESVJ( 'L','U','N', N, NR, V, LDV, SVA, NR, A, LDA,
+ $ CWORK, LWORK, RWORK, LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+
+ ELSE
+*
+* .. two more QR factorizations ( one QRF is not enough, two require
+* accumulated product of Jacobi rotations, three are perfect )
+*
+ CALL ZLASET( 'L', NR-1,NR-1, CZERO, CZERO, A(2,1), LDA )
+ CALL ZGELQF( NR,N, A, LDA, CWORK, CWORK(N+1), LWORK-N, IERR)
+ CALL ZLACPY( 'L', NR, NR, A, LDA, V, LDV )
+ CALL ZLASET( 'U', NR-1,NR-1, CZERO, CZERO, V(1,2), LDV )
+ CALL ZGEQRF( NR, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
+ $ LWORK-2*N, IERR )
+ DO 8998 p = 1, NR
+ CALL ZCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )
+ CALL ZLACGV( NR-p+1, V(p,p), 1 )
+ 8998 CONTINUE
+ CALL ZLASET('U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV)
+*
+ CALL ZGESVJ( 'L', 'U','N', NR, NR, V,LDV, SVA, NR, U,
+ $ LDU, CWORK(N+1), LWORK-N, RWORK, LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+ IF ( NR .LT. N ) THEN
+ CALL ZLASET( 'A',N-NR, NR, CZERO,CZERO, V(NR+1,1), LDV )
+ CALL ZLASET( 'A',NR, N-NR, CZERO,CZERO, V(1,NR+1), LDV )
+ CALL ZLASET( 'A',N-NR,N-NR,CZERO,CONE, V(NR+1,NR+1),LDV )
+ END IF
+*
+ CALL ZUNMLQ( 'L', 'C', N, N, NR, A, LDA, CWORK,
+ $ V, LDV, CWORK(N+1), LWORK-N, IERR )
+*
+ END IF
+* .. permute the rows of V
+* DO 8991 p = 1, N
+* CALL ZCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA )
+* 8991 CONTINUE
+* CALL ZLACPY( 'All', N, N, A, LDA, V, LDV )
+ CALL ZLAPMR( .FALSE., N, N, V, LDV, IWORK )
+*
+ IF ( TRANSP ) THEN
+ CALL ZLACPY( 'A', N, N, V, LDV, U, LDU )
+ END IF
+*
+ ELSE IF ( JRACC .AND. (.NOT. LSVEC) .AND. ( NR.EQ. N ) ) THEN
+*
+ CALL ZLASET( 'L', N-1,N-1, CZERO, CZERO, A(2,1), LDA )
+*
+ CALL ZGESVJ( 'U','N','V', N, N, A, LDA, SVA, N, V, LDV,
+ $ CWORK, LWORK, RWORK, LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+ CALL ZLAPMR( .FALSE., N, N, V, LDV, IWORK )
+*
+ ELSE IF ( LSVEC .AND. ( .NOT. RSVEC ) ) THEN
+*
+* .. Singular Values and Left Singular Vectors ..
+*
+* .. second preconditioning step to avoid need to accumulate
+* Jacobi rotations in the Jacobi iterations.
+ DO 1965 p = 1, NR
+ CALL ZCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 )
+ CALL ZLACGV( N-p+1, U(p,p), 1 )
+ 1965 CONTINUE
+ CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
+*
+ CALL ZGEQRF( N, NR, U, LDU, CWORK(N+1), CWORK(2*N+1),
+ $ LWORK-2*N, IERR )
+*
+ DO 1967 p = 1, NR - 1
+ CALL ZCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )
+ CALL ZLACGV( N-p+1, U(p,p), 1 )
+ 1967 CONTINUE
+ CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
+*
+ CALL ZGESVJ( 'L', 'U', 'N', NR,NR, U, LDU, SVA, NR, A,
+ $ LDA, CWORK(N+1), LWORK-N, RWORK, LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+*
+ IF ( NR .LT. M ) THEN
+ CALL ZLASET( 'A', M-NR, NR,CZERO, CZERO, U(NR+1,1), LDU )
+ IF ( NR .LT. N1 ) THEN
+ CALL ZLASET( 'A',NR, N1-NR, CZERO, CZERO, U(1,NR+1),LDU )
+ CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,U(NR+1,NR+1),LDU )
+ END IF
+ END IF
+*
+ CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
+ $ LDU, CWORK(N+1), LWORK-N, IERR )
+*
+ IF ( ROWPIV )
+ $ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
+*
+ DO 1974 p = 1, N1
+ XSC = ONE / DZNRM2( M, U(1,p), 1 )
+ CALL ZDSCAL( M, XSC, U(1,p), 1 )
+ 1974 CONTINUE
+*
+ IF ( TRANSP ) THEN
+ CALL ZLACPY( 'A', N, N, U, LDU, V, LDV )
+ END IF
+*
+ ELSE
+*
+* .. Full SVD ..
+*
+ IF ( .NOT. JRACC ) THEN
+*
+ IF ( .NOT. ALMORT ) THEN
+*
+* Second Preconditioning Step (QRF [with pivoting])
+* Note that the composition of TRANSPOSE, QRF and TRANSPOSE is
+* equivalent to an LQF CALL. Since in many libraries the QRF
+* seems to be better optimized than the LQF, we do explicit
+* transpose and use the QRF. This is subject to changes in an
+* optimized implementation of ZGEJSV.
+*
+ DO 1968 p = 1, NR
+ CALL ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
+ CALL ZLACGV( N-p+1, V(p,p), 1 )
+ 1968 CONTINUE
+*
+* .. the following two loops perturb small entries to avoid
+* denormals in the second QR factorization, where they are
+* as good as zeros. This is done to avoid painfully slow
+* computation with denormals. The relative size of the perturbation
+* is a parameter that can be changed by the implementer.
+* This perturbation device will be obsolete on machines with
+* properly implemented arithmetic.
+* To switch it off, set L2PERT=.FALSE. To remove it from the
+* code, remove the action under L2PERT=.TRUE., leave the ELSE part.
+* The following two loops should be blocked and fused with the
+* transposed copy above.
+*
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL)
+ DO 2969 q = 1, NR
+ CTEMP = DCMPLX(XSC*ABS( V(q,q) ),ZERO)
+ DO 2968 p = 1, N
+ IF ( ( p .GT. q ) .AND. ( ABS(V(p,q)) .LE. TEMP1 )
+ $ .OR. ( p .LT. q ) )
+* $ V(p,q) = TEMP1 * ( V(p,q) / ABS(V(p,q)) )
+ $ V(p,q) = CTEMP
+ IF ( p .LT. q ) V(p,q) = - V(p,q)
+ 2968 CONTINUE
+ 2969 CONTINUE
+ ELSE
+ CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV )
+ END IF
+*
+* Estimate the row scaled condition number of R1
+* (If R1 is rectangular, N > NR, then the condition number
+* of the leading NR x NR submatrix is estimated.)
+*
+ CALL ZLACPY( 'L', NR, NR, V, LDV, CWORK(2*N+1), NR )
+ DO 3950 p = 1, NR
+ TEMP1 = DZNRM2(NR-p+1,CWORK(2*N+(p-1)*NR+p),1)
+ CALL ZDSCAL(NR-p+1,ONE/TEMP1,CWORK(2*N+(p-1)*NR+p),1)
+ 3950 CONTINUE
+ CALL ZPOCON('L',NR,CWORK(2*N+1),NR,ONE,TEMP1,
+ $ CWORK(2*N+NR*NR+1),RWORK,IERR)
+ CONDR1 = ONE / SQRT(TEMP1)
+* .. here need a second oppinion on the condition number
+* .. then assume worst case scenario
+* R1 is OK for inverse <=> CONDR1 .LT. DBLE(N)
+* more conservative <=> CONDR1 .LT. SQRT(DBLE(N))
+*
+ COND_OK = SQRT(SQRT(DBLE(NR)))
+*[TP] COND_OK is a tuning parameter.
+*
+ IF ( CONDR1 .LT. COND_OK ) THEN
+* .. the second QRF without pivoting. Note: in an optimized
+* implementation, this QRF should be implemented as the QRF
+* of a lower triangular matrix.
+* R1^* = Q2 * R2
+ CALL ZGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
+ $ LWORK-2*N, IERR )
+*
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL)/EPSLN
+ DO 3959 p = 2, NR
+ DO 3958 q = 1, p - 1
+ CTEMP=DCMPLX(XSC*MIN(ABS(V(p,p)),ABS(V(q,q))),
+ $ ZERO)
+ IF ( ABS(V(q,p)) .LE. TEMP1 )
+* $ V(q,p) = TEMP1 * ( V(q,p) / ABS(V(q,p)) )
+ $ V(q,p) = CTEMP
+ 3958 CONTINUE
+ 3959 CONTINUE
+ END IF
+*
+ IF ( NR .NE. N )
+ $ CALL ZLACPY( 'A', N, NR, V, LDV, CWORK(2*N+1), N )
+* .. save ...
+*
+* .. this transposed copy should be better than naive
+ DO 1969 p = 1, NR - 1
+ CALL ZCOPY( NR-p, V(p,p+1), LDV, V(p+1,p), 1 )
+ CALL ZLACGV(NR-p+1, V(p,p), 1 )
+ 1969 CONTINUE
+ V(NR,NR)=CONJG(V(NR,NR))
+*
+ CONDR2 = CONDR1
+*
+ ELSE
+*
+* .. ill-conditioned case: second QRF with pivoting
+* Note that windowed pivoting would be equaly good
+* numerically, and more run-time efficient. So, in
+* an optimal implementation, the next call to ZGEQP3
+* should be replaced with eg. CALL ZGEQPX (ACM TOMS #782)
+* with properly (carefully) chosen parameters.
+*
+* R1^* * P2 = Q2 * R2
+ DO 3003 p = 1, NR
+ IWORK(N+p) = 0
+ 3003 CONTINUE
+ CALL ZGEQP3( N, NR, V, LDV, IWORK(N+1), CWORK(N+1),
+ $ CWORK(2*N+1), LWORK-2*N, RWORK, IERR )
+** CALL ZGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
+** $ LWORK-2*N, IERR )
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL)
+ DO 3969 p = 2, NR
+ DO 3968 q = 1, p - 1
+ CTEMP=DCMPLX(XSC*MIN(ABS(V(p,p)),ABS(V(q,q))),
+ $ ZERO)
+ IF ( ABS(V(q,p)) .LE. TEMP1 )
+* $ V(q,p) = TEMP1 * ( V(q,p) / ABS(V(q,p)) )
+ $ V(q,p) = CTEMP
+ 3968 CONTINUE
+ 3969 CONTINUE
+ END IF
+*
+ CALL ZLACPY( 'A', N, NR, V, LDV, CWORK(2*N+1), N )
+*
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL)
+ DO 8970 p = 2, NR
+ DO 8971 q = 1, p - 1
+ CTEMP=DCMPLX(XSC*MIN(ABS(V(p,p)),ABS(V(q,q))),
+ $ ZERO)
+* V(p,q) = - TEMP1*( V(q,p) / ABS(V(q,p)) )
+ V(p,q) = - CTEMP
+ 8971 CONTINUE
+ 8970 CONTINUE
+ ELSE
+ CALL ZLASET( 'L',NR-1,NR-1,CZERO,CZERO,V(2,1),LDV )
+ END IF
+* Now, compute R2 = L3 * Q3, the LQ factorization.
+ CALL ZGELQF( NR, NR, V, LDV, CWORK(2*N+N*NR+1),
+ $ CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, IERR )
+* .. and estimate the condition number
+ CALL ZLACPY( 'L',NR,NR,V,LDV,CWORK(2*N+N*NR+NR+1),NR )
+ DO 4950 p = 1, NR
+ TEMP1 = DZNRM2( p, CWORK(2*N+N*NR+NR+p), NR )
+ CALL ZDSCAL( p, ONE/TEMP1, CWORK(2*N+N*NR+NR+p), NR )
+ 4950 CONTINUE
+ CALL ZPOCON( 'L',NR,CWORK(2*N+N*NR+NR+1),NR,ONE,TEMP1,
+ $ CWORK(2*N+N*NR+NR+NR*NR+1),RWORK,IERR )
+ CONDR2 = ONE / SQRT(TEMP1)
+*
+*
+ IF ( CONDR2 .GE. COND_OK ) THEN
+* .. save the Householder vectors used for Q3
+* (this overwrittes the copy of R2, as it will not be
+* needed in this branch, but it does not overwritte the
+* Huseholder vectors of Q2.).
+ CALL ZLACPY( 'U', NR, NR, V, LDV, CWORK(2*N+1), N )
+* .. and the rest of the information on Q3 is in
+* WORK(2*N+N*NR+1:2*N+N*NR+N)
+ END IF
+*
+ END IF
+*
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL)
+ DO 4968 q = 2, NR
+ CTEMP = XSC * V(q,q)
+ DO 4969 p = 1, q - 1
+* V(p,q) = - TEMP1*( V(p,q) / ABS(V(p,q)) )
+ V(p,q) = - CTEMP
+ 4969 CONTINUE
+ 4968 CONTINUE
+ ELSE
+ CALL ZLASET( 'U', NR-1,NR-1, CZERO,CZERO, V(1,2), LDV )
+ END IF
+*
+* Second preconditioning finished; continue with Jacobi SVD
+* The input matrix is lower trinagular.
+*
+* Recover the right singular vectors as solution of a well
+* conditioned triangular matrix equation.
+*
+ IF ( CONDR1 .LT. COND_OK ) THEN
+*
+ CALL ZGESVJ( 'L','U','N',NR,NR,V,LDV,SVA,NR,U, LDU,
+ $ CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,RWORK,
+ $ LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+ DO 3970 p = 1, NR
+ CALL ZCOPY( NR, V(1,p), 1, U(1,p), 1 )
+ CALL ZDSCAL( NR, SVA(p), V(1,p), 1 )
+ 3970 CONTINUE
+
+* .. pick the right matrix equation and solve it
+*
+ IF ( NR .EQ. N ) THEN
+* :)) .. best case, R1 is inverted. The solution of this matrix
+* equation is Q2*V2 = the product of the Jacobi rotations
+* used in ZGESVJ, premultiplied with the orthogonal matrix
+* from the second QR factorization.
+ CALL ZTRSM('L','U','N','N', NR,NR,CONE, A,LDA, V,LDV)
+ ELSE
+* .. R1 is well conditioned, but non-square. Adjoint of R2
+* is inverted to get the product of the Jacobi rotations
+* used in ZGESVJ. The Q-factor from the second QR
+* factorization is then built in explicitly.
+ CALL ZTRSM('L','U','C','N',NR,NR,CONE,CWORK(2*N+1),
+ $ N,V,LDV)
+ IF ( NR .LT. N ) THEN
+ CALL ZLASET('A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV)
+ CALL ZLASET('A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV)
+ CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
+ END IF
+ CALL ZUNMQR('L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
+ $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR)
+ END IF
+*
+ ELSE IF ( CONDR2 .LT. COND_OK ) THEN
+*
+* The matrix R2 is inverted. The solution of the matrix equation
+* is Q3^* * V3 = the product of the Jacobi rotations (appplied to
+* the lower triangular L3 from the LQ factorization of
+* R2=L3*Q3), pre-multiplied with the transposed Q3.
+ CALL ZGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR, U,
+ $ LDU, CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR,
+ $ RWORK, LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+ DO 3870 p = 1, NR
+ CALL ZCOPY( NR, V(1,p), 1, U(1,p), 1 )
+ CALL ZDSCAL( NR, SVA(p), U(1,p), 1 )
+ 3870 CONTINUE
+ CALL ZTRSM('L','U','N','N',NR,NR,CONE,CWORK(2*N+1),N,
+ $ U,LDU)
+* .. apply the permutation from the second QR factorization
+ DO 873 q = 1, NR
+ DO 872 p = 1, NR
+ CWORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
+ 872 CONTINUE
+ DO 874 p = 1, NR
+ U(p,q) = CWORK(2*N+N*NR+NR+p)
+ 874 CONTINUE
+ 873 CONTINUE
+ IF ( NR .LT. N ) THEN
+ CALL ZLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
+ CALL ZLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
+ CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
+ END IF
+ CALL ZUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
+ $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
+ ELSE
+* Last line of defense.
+* #:( This is a rather pathological case: no scaled condition
+* improvement after two pivoted QR factorizations. Other
+* possibility is that the rank revealing QR factorization
+* or the condition estimator has failed, or the COND_OK
+* is set very close to ONE (which is unnecessary). Normally,
+* this branch should never be executed, but in rare cases of
+* failure of the RRQR or condition estimator, the last line of
+* defense ensures that ZGEJSV completes the task.
+* Compute the full SVD of L3 using ZGESVJ with explicit
+* accumulation of Jacobi rotations.
+ CALL ZGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR, U,
+ $ LDU, CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR,
+ $ RWORK, LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+ IF ( NR .LT. N ) THEN
+ CALL ZLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
+ CALL ZLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
+ CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
+ END IF
+ CALL ZUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
+ $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
+*
+ CALL ZUNMLQ( 'L', 'C', NR, NR, NR, CWORK(2*N+1), N,
+ $ CWORK(2*N+N*NR+1), U, LDU, CWORK(2*N+N*NR+NR+1),
+ $ LWORK-2*N-N*NR-NR, IERR )
+ DO 773 q = 1, NR
+ DO 772 p = 1, NR
+ CWORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
+ 772 CONTINUE
+ DO 774 p = 1, NR
+ U(p,q) = CWORK(2*N+N*NR+NR+p)
+ 774 CONTINUE
+ 773 CONTINUE
+*
+ END IF
+*
+* Permute the rows of V using the (column) permutation from the
+* first QRF. Also, scale the columns to make them unit in
+* Euclidean norm. This applies to all cases.
+*
+ TEMP1 = SQRT(DBLE(N)) * EPSLN
+ DO 1972 q = 1, N
+ DO 972 p = 1, N
+ CWORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
+ 972 CONTINUE
+ DO 973 p = 1, N
+ V(p,q) = CWORK(2*N+N*NR+NR+p)
+ 973 CONTINUE
+ XSC = ONE / DZNRM2( N, V(1,q), 1 )
+ IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
+ $ CALL ZDSCAL( N, XSC, V(1,q), 1 )
+ 1972 CONTINUE
+* At this moment, V contains the right singular vectors of A.
+* Next, assemble the left singular vector matrix U (M x N).
+ IF ( NR .LT. M ) THEN
+ CALL ZLASET('A', M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU)
+ IF ( NR .LT. N1 ) THEN
+ CALL ZLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)
+ CALL ZLASET('A',M-NR,N1-NR,CZERO,CONE,
+ $ U(NR+1,NR+1),LDU)
+ END IF
+ END IF
+*
+* The Q matrix from the first QRF is built into the left singular
+* matrix U. This applies to all cases.
+*
+ CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
+ $ LDU, CWORK(N+1), LWORK-N, IERR )
+
+* The columns of U are normalized. The cost is O(M*N) flops.
+ TEMP1 = SQRT(DBLE(M)) * EPSLN
+ DO 1973 p = 1, NR
+ XSC = ONE / DZNRM2( M, U(1,p), 1 )
+ IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
+ $ CALL ZDSCAL( M, XSC, U(1,p), 1 )
+ 1973 CONTINUE
+*
+* If the initial QRF is computed with row pivoting, the left
+* singular vectors must be adjusted.
+*
+ IF ( ROWPIV )
+ $ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
+*
+ ELSE
+*
+* .. the initial matrix A has almost orthogonal columns and
+* the second QRF is not needed
+*
+ CALL ZLACPY( 'U', N, N, A, LDA, CWORK(N+1), N )
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL)
+ DO 5970 p = 2, N
+ CTEMP = XSC * CWORK( N + (p-1)*N + p )
+ DO 5971 q = 1, p - 1
+* CWORK(N+(q-1)*N+p)=-TEMP1 * ( CWORK(N+(p-1)*N+q) /
+* $ ABS(CWORK(N+(p-1)*N+q)) )
+ CWORK(N+(q-1)*N+p)=-CTEMP
+ 5971 CONTINUE
+ 5970 CONTINUE
+ ELSE
+ CALL ZLASET( 'L',N-1,N-1,CZERO,CZERO,CWORK(N+2),N )
+ END IF
+*
+ CALL ZGESVJ( 'U', 'U', 'N', N, N, CWORK(N+1), N, SVA,
+ $ N, U, LDU, CWORK(N+N*N+1), LWORK-N-N*N, RWORK, LRWORK,
+ $ INFO )
+*
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+ DO 6970 p = 1, N
+ CALL ZCOPY( N, CWORK(N+(p-1)*N+1), 1, U(1,p), 1 )
+ CALL ZDSCAL( N, SVA(p), CWORK(N+(p-1)*N+1), 1 )
+ 6970 CONTINUE
+*
+ CALL ZTRSM( 'L', 'U', 'N', 'N', N, N,
+ $ CONE, A, LDA, CWORK(N+1), N )
+ DO 6972 p = 1, N
+ CALL ZCOPY( N, CWORK(N+p), N, V(IWORK(p),1), LDV )
+ 6972 CONTINUE
+ TEMP1 = SQRT(DBLE(N))*EPSLN
+ DO 6971 p = 1, N
+ XSC = ONE / DZNRM2( N, V(1,p), 1 )
+ IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
+ $ CALL ZDSCAL( N, XSC, V(1,p), 1 )
+ 6971 CONTINUE
+*
+* Assemble the left singular vector matrix U (M x N).
+*
+ IF ( N .LT. M ) THEN
+ CALL ZLASET( 'A', M-N, N, CZERO, CZERO, U(N+1,1), LDU )
+ IF ( N .LT. N1 ) THEN
+ CALL ZLASET('A',N, N1-N, CZERO, CZERO, U(1,N+1),LDU)
+ CALL ZLASET( 'A',M-N,N1-N, CZERO, CONE,U(N+1,N+1),LDU)
+ END IF
+ END IF
+ CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
+ $ LDU, CWORK(N+1), LWORK-N, IERR )
+ TEMP1 = SQRT(DBLE(M))*EPSLN
+ DO 6973 p = 1, N1
+ XSC = ONE / DZNRM2( M, U(1,p), 1 )
+ IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
+ $ CALL ZDSCAL( M, XSC, U(1,p), 1 )
+ 6973 CONTINUE
+*
+ IF ( ROWPIV )
+ $ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
+*
+ END IF
+*
+* end of the >> almost orthogonal case << in the full SVD
+*
+ ELSE
+*
+* This branch deploys a preconditioned Jacobi SVD with explicitly
+* accumulated rotations. It is included as optional, mainly for
+* experimental purposes. It does perfom well, and can also be used.
+* In this implementation, this branch will be automatically activated
+* if the condition number sigma_max(A) / sigma_min(A) is predicted
+* to be greater than the overflow threshold. This is because the
+* a posteriori computation of the singular vectors assumes robust
+* implementation of BLAS and some LAPACK procedures, capable of working
+* in presence of extreme values, e.g. when the singular values spread from
+* the underflow to the overflow threshold.
+*
+ DO 7968 p = 1, NR
+ CALL ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
+ CALL ZLACGV( N-p+1, V(p,p), 1 )
+ 7968 CONTINUE
+*
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL/EPSLN)
+ DO 5969 q = 1, NR
+ CTEMP = DCMPLX(XSC*ABS( V(q,q) ),ZERO)
+ DO 5968 p = 1, N
+ IF ( ( p .GT. q ) .AND. ( ABS(V(p,q)) .LE. TEMP1 )
+ $ .OR. ( p .LT. q ) )
+* $ V(p,q) = TEMP1 * ( V(p,q) / ABS(V(p,q)) )
+ $ V(p,q) = CTEMP
+ IF ( p .LT. q ) V(p,q) = - V(p,q)
+ 5968 CONTINUE
+ 5969 CONTINUE
+ ELSE
+ CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV )
+ END IF
+
+ CALL ZGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
+ $ LWORK-2*N, IERR )
+ CALL ZLACPY( 'L', N, NR, V, LDV, CWORK(2*N+1), N )
+*
+ DO 7969 p = 1, NR
+ CALL ZCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )
+ CALL ZLACGV( NR-p+1, U(p,p), 1 )
+ 7969 CONTINUE
+
+ IF ( L2PERT ) THEN
+ XSC = SQRT(SMALL/EPSLN)
+ DO 9970 q = 2, NR
+ DO 9971 p = 1, q - 1
+ CTEMP = DCMPLX(XSC * MIN(ABS(U(p,p)),ABS(U(q,q))),
+ $ ZERO)
+* U(p,q) = - TEMP1 * ( U(q,p) / ABS(U(q,p)) )
+ U(p,q) = - CTEMP
+ 9971 CONTINUE
+ 9970 CONTINUE
+ ELSE
+ CALL ZLASET('U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
+ END IF
+
+ CALL ZGESVJ( 'L', 'U', 'V', NR, NR, U, LDU, SVA,
+ $ N, V, LDV, CWORK(2*N+N*NR+1), LWORK-2*N-N*NR,
+ $ RWORK, LRWORK, INFO )
+ SCALEM = RWORK(1)
+ NUMRANK = NINT(RWORK(2))
+
+ IF ( NR .LT. N ) THEN
+ CALL ZLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
+ CALL ZLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
+ CALL ZLASET( 'A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV )
+ END IF
+
+ CALL ZUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
+ $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
+*
+* Permute the rows of V using the (column) permutation from the
+* first QRF. Also, scale the columns to make them unit in
+* Euclidean norm. This applies to all cases.
+*
+ TEMP1 = SQRT(DBLE(N)) * EPSLN
+ DO 7972 q = 1, N
+ DO 8972 p = 1, N
+ CWORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
+ 8972 CONTINUE
+ DO 8973 p = 1, N
+ V(p,q) = CWORK(2*N+N*NR+NR+p)
+ 8973 CONTINUE
+ XSC = ONE / DZNRM2( N, V(1,q), 1 )
+ IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
+ $ CALL ZDSCAL( N, XSC, V(1,q), 1 )
+ 7972 CONTINUE
+*
+* At this moment, V contains the right singular vectors of A.
+* Next, assemble the left singular vector matrix U (M x N).
+*
+ IF ( NR .LT. M ) THEN
+ CALL ZLASET( 'A', M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU )
+ IF ( NR .LT. N1 ) THEN
+ CALL ZLASET('A',NR, N1-NR, CZERO, CZERO, U(1,NR+1),LDU)
+ CALL ZLASET('A',M-NR,N1-NR, CZERO, CONE,U(NR+1,NR+1),LDU)
+ END IF
+ END IF
+*
+ CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
+ $ LDU, CWORK(N+1), LWORK-N, IERR )
+*
+ IF ( ROWPIV )
+ $ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )
+*
+*
+ END IF
+ IF ( TRANSP ) THEN
+* .. swap U and V because the procedure worked on A^*
+ DO 6974 p = 1, N
+ CALL ZSWAP( N, U(1,p), 1, V(1,p), 1 )
+ 6974 CONTINUE
+ END IF
+*
+ END IF
+* end of the full SVD
+*
+* Undo scaling, if necessary (and possible)
+*
+ IF ( USCAL2 .LE. (BIG/SVA(1))*USCAL1 ) THEN
+ CALL DLASCL( 'G', 0, 0, USCAL1, USCAL2, NR, 1, SVA, N, IERR )
+ USCAL1 = ONE
+ USCAL2 = ONE
+ END IF
+*
+ IF ( NR .LT. N ) THEN
+ DO 3004 p = NR+1, N
+ SVA(p) = ZERO
+ 3004 CONTINUE
+ END IF
+*
+ RWORK(1) = USCAL2 * SCALEM
+ RWORK(2) = USCAL1
+ IF ( ERREST ) RWORK(3) = SCONDA
+ IF ( LSVEC .AND. RSVEC ) THEN
+ RWORK(4) = CONDR1
+ RWORK(5) = CONDR2
+ END IF
+ IF ( L2TRAN ) THEN
+ RWORK(6) = ENTRA
+ RWORK(7) = ENTRAT
+ END IF
+*
+ IWORK(1) = NR
+ IWORK(2) = NUMRANK
+ IWORK(3) = WARNING
+ IF ( TRANSP ) THEN
+ IWORK(4) = 1
+ ELSE
+ IWORK(4) = -1
+ END IF
+
+*
+ RETURN
+* ..
+* .. END OF ZGEJSV
+* ..
+ END
+*
diff --git a/SRC/zlarrv.f b/SRC/zlarrv.f
index c765681..690e5b6 100644
--- a/SRC/zlarrv.f
+++ b/SRC/zlarrv.f
@@ -348,6 +348,13 @@
* ..
INFO = 0
+*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ RETURN
+ END IF
+*
* The first N entries of WORK are reserved for the eigenvalues
INDLD = N+1
INDLLD= 2*N+1