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author | julie <julielangou@users.noreply.github.com> | 2011-10-06 06:53:11 +0000 |
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committer | julie <julielangou@users.noreply.github.com> | 2011-10-06 06:53:11 +0000 |
commit | e1d39294aee16fa6db9ba079b14442358217db71 (patch) | |
tree | 30e5aa04c1f6596991fda5334f63dfb9b8027849 /SRC/sggesx.f | |
parent | 5fe0466a14e395641f4f8a300ecc9dcb8058081b (diff) | |
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diff --git a/SRC/sggesx.f b/SRC/sggesx.f index 09776ffa..d033e984 100644 --- a/SRC/sggesx.f +++ b/SRC/sggesx.f @@ -1,12 +1,371 @@ +*> \brief <b> SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b> +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +* Definition +* ========== +* +* SUBROUTINE SGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, +* B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, +* VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK, +* LIWORK, BWORK, INFO ) +* +* .. Scalar Arguments .. +* CHARACTER JOBVSL, JOBVSR, SENSE, SORT +* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N, +* $ SDIM +* .. +* .. Array Arguments .. +* LOGICAL BWORK( * ) +* INTEGER IWORK( * ) +* REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), +* $ B( LDB, * ), BETA( * ), RCONDE( 2 ), +* $ RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ), +* $ WORK( * ) +* .. +* .. Function Arguments .. +* LOGICAL SELCTG +* EXTERNAL SELCTG +* .. +* +* Purpose +* ======= +* +*>\details \b Purpose: +*>\verbatim +*> +*> SGGESX computes for a pair of N-by-N real nonsymmetric matrices +*> (A,B), the generalized eigenvalues, the real Schur form (S,T), and, +*> optionally, the left and/or right matrices of Schur vectors (VSL and +*> VSR). This gives the generalized Schur factorization +*> +*> (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T ) +*> +*> Optionally, it also orders the eigenvalues so that a selected cluster +*> of eigenvalues appears in the leading diagonal blocks of the upper +*> quasi-triangular matrix S and the upper triangular matrix T; computes +*> a reciprocal condition number for the average of the selected +*> eigenvalues (RCONDE); and computes a reciprocal condition number for +*> the right and left deflating subspaces corresponding to the selected +*> eigenvalues (RCONDV). The leading columns of VSL and VSR then form +*> an orthonormal basis for the corresponding left and right eigenspaces +*> (deflating subspaces). +*> +*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w +*> or a ratio alpha/beta = w, such that A - w*B is singular. It is +*> usually represented as the pair (alpha,beta), as there is a +*> reasonable interpretation for beta=0 or for both being zero. +*> +*> A pair of matrices (S,T) is in generalized real Schur form if T is +*> upper triangular with non-negative diagonal and S is block upper +*> triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond +*> to real generalized eigenvalues, while 2-by-2 blocks of S will be +*> "standardized" by making the corresponding elements of T have the +*> form: +*> [ a 0 ] +*> [ 0 b ] +*> +*> and the pair of corresponding 2-by-2 blocks in S and T will have a +*> complex conjugate pair of generalized eigenvalues. +*> +*> +*>\endverbatim +* +* Arguments +* ========= +* +*> \param[in] JOBVSL +*> \verbatim +*> JOBVSL is CHARACTER*1 +*> = 'N': do not compute the left Schur vectors; +*> = 'V': compute the left Schur vectors. +*> \endverbatim +*> +*> \param[in] JOBVSR +*> \verbatim +*> JOBVSR is CHARACTER*1 +*> = 'N': do not compute the right Schur vectors; +*> = 'V': compute the right Schur vectors. +*> \endverbatim +*> +*> \param[in] SORT +*> \verbatim +*> SORT is CHARACTER*1 +*> Specifies whether or not to order the eigenvalues on the +*> diagonal of the generalized Schur form. +*> = 'N': Eigenvalues are not ordered; +*> = 'S': Eigenvalues are ordered (see SELCTG). +*> \endverbatim +*> +*> \param[in] SELCTG +*> \verbatim +*> SELCTG is procedure) LOGICAL FUNCTION of three REAL arguments +*> SELCTG must be declared EXTERNAL in the calling subroutine. +*> If SORT = 'N', SELCTG is not referenced. +*> If SORT = 'S', SELCTG is used to select eigenvalues to sort +*> to the top left of the Schur form. +*> An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if +*> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either +*> one of a complex conjugate pair of eigenvalues is selected, +*> then both complex eigenvalues are selected. +*> Note that a selected complex eigenvalue may no longer satisfy +*> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering, +*> since ordering may change the value of complex eigenvalues +*> (especially if the eigenvalue is ill-conditioned), in this +*> case INFO is set to N+3. +*> \endverbatim +*> +*> \param[in] SENSE +*> \verbatim +*> SENSE is CHARACTER*1 +*> Determines which reciprocal condition numbers are computed. +*> = 'N' : None are computed; +*> = 'E' : Computed for average of selected eigenvalues only; +*> = 'V' : Computed for selected deflating subspaces only; +*> = 'B' : Computed for both. +*> If SENSE = 'E', 'V', or 'B', SORT must equal 'S'. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The order of the matrices A, B, VSL, and VSR. N >= 0. +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is REAL array, dimension (LDA, N) +*> On entry, the first of the pair of matrices. +*> On exit, A has been overwritten by its generalized Schur +*> form S. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of A. LDA >= max(1,N). +*> \endverbatim +*> +*> \param[in,out] B +*> \verbatim +*> B is REAL array, dimension (LDB, N) +*> On entry, the second of the pair of matrices. +*> On exit, B has been overwritten by its generalized Schur +*> form T. +*> \endverbatim +*> +*> \param[in] LDB +*> \verbatim +*> LDB is INTEGER +*> The leading dimension of B. LDB >= max(1,N). +*> \endverbatim +*> +*> \param[out] SDIM +*> \verbatim +*> SDIM is INTEGER +*> If SORT = 'N', SDIM = 0. +*> If SORT = 'S', SDIM = number of eigenvalues (after sorting) +*> for which SELCTG is true. (Complex conjugate pairs for which +*> SELCTG is true for either eigenvalue count as 2.) +*> \endverbatim +*> +*> \param[out] ALPHAR +*> \verbatim +*> ALPHAR is REAL array, dimension (N) +*> \endverbatim +*> +*> \param[out] ALPHAI +*> \verbatim +*> ALPHAI is REAL array, dimension (N) +*> \endverbatim +*> +*> \param[out] BETA +*> \verbatim +*> BETA is REAL array, dimension (N) +*> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will +*> be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i +*> and BETA(j),j=1,...,N are the diagonals of the complex Schur +*> form (S,T) that would result if the 2-by-2 diagonal blocks of +*> the real Schur form of (A,B) were further reduced to +*> triangular form using 2-by-2 complex unitary transformations. +*> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if +*> positive, then the j-th and (j+1)-st eigenvalues are a +*> complex conjugate pair, with ALPHAI(j+1) negative. +*> \endverbatim +*> \verbatim +*> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) +*> may easily over- or underflow, and BETA(j) may even be zero. +*> Thus, the user should avoid naively computing the ratio. +*> However, ALPHAR and ALPHAI will be always less than and +*> usually comparable with norm(A) in magnitude, and BETA always +*> less than and usually comparable with norm(B). +*> \endverbatim +*> +*> \param[out] VSL +*> \verbatim +*> VSL is REAL array, dimension (LDVSL,N) +*> If JOBVSL = 'V', VSL will contain the left Schur vectors. +*> Not referenced if JOBVSL = 'N'. +*> \endverbatim +*> +*> \param[in] LDVSL +*> \verbatim +*> LDVSL is INTEGER +*> The leading dimension of the matrix VSL. LDVSL >=1, and +*> if JOBVSL = 'V', LDVSL >= N. +*> \endverbatim +*> +*> \param[out] VSR +*> \verbatim +*> VSR is REAL array, dimension (LDVSR,N) +*> If JOBVSR = 'V', VSR will contain the right Schur vectors. +*> Not referenced if JOBVSR = 'N'. +*> \endverbatim +*> +*> \param[in] LDVSR +*> \verbatim +*> LDVSR is INTEGER +*> The leading dimension of the matrix VSR. LDVSR >= 1, and +*> if JOBVSR = 'V', LDVSR >= N. +*> \endverbatim +*> +*> \param[out] RCONDE +*> \verbatim +*> RCONDE is REAL array, dimension ( 2 ) +*> If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the +*> reciprocal condition numbers for the average of the selected +*> eigenvalues. +*> Not referenced if SENSE = 'N' or 'V'. +*> \endverbatim +*> +*> \param[out] RCONDV +*> \verbatim +*> RCONDV is REAL array, dimension ( 2 ) +*> If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the +*> reciprocal condition numbers for the selected deflating +*> subspaces. +*> Not referenced if SENSE = 'N' or 'E'. +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is REAL array, dimension (MAX(1,LWORK)) +*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. +*> \endverbatim +*> +*> \param[in] LWORK +*> \verbatim +*> LWORK is INTEGER +*> The dimension of the array WORK. +*> If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B', +*> LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else +*> LWORK >= max( 8*N, 6*N+16 ). +*> Note that 2*SDIM*(N-SDIM) <= N*N/2. +*> Note also that an error is only returned if +*> LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B' +*> this may not be large enough. +*> \endverbatim +*> \verbatim +*> If LWORK = -1, then a workspace query is assumed; the routine +*> only calculates the bound on the optimal size of the WORK +*> array and the minimum size of the IWORK array, returns these +*> values as the first entries of the WORK and IWORK arrays, and +*> no error message related to LWORK or LIWORK is issued by +*> XERBLA. +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) +*> On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK. +*> \endverbatim +*> +*> \param[in] LIWORK +*> \verbatim +*> LIWORK is INTEGER +*> The dimension of the array IWORK. +*> If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise +*> LIWORK >= N+6. +*> \endverbatim +*> \verbatim +*> If LIWORK = -1, then a workspace query is assumed; the +*> routine only calculates the bound on the optimal size of the +*> WORK array and the minimum size of the IWORK array, returns +*> these values as the first entries of the WORK and IWORK +*> arrays, and no error message related to LWORK or LIWORK is +*> issued by XERBLA. +*> \endverbatim +*> +*> \param[out] BWORK +*> \verbatim +*> BWORK is LOGICAL array, dimension (N) +*> Not referenced if SORT = 'N'. +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> = 0: successful exit +*> < 0: if INFO = -i, the i-th argument had an illegal value. +*> = 1,...,N: +*> The QZ iteration failed. (A,B) are not in Schur +*> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should +*> be correct for j=INFO+1,...,N. +*> > N: =N+1: other than QZ iteration failed in SHGEQZ +*> =N+2: after reordering, roundoff changed values of +*> some complex eigenvalues so that leading +*> eigenvalues in the Generalized Schur form no +*> longer satisfy SELCTG=.TRUE. This could also +*> be caused due to scaling. +*> =N+3: reordering failed in STGSEN. +*> \endverbatim +*> +* +* Authors +* ======= +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \date November 2011 +* +*> \ingroup realGEeigen +* +* +* Further Details +* =============== +*>\details \b Further \b Details +*> \verbatim +*> +*> An approximate (asymptotic) bound on the average absolute error of +*> the selected eigenvalues is +*> +*> EPS * norm((A, B)) / RCONDE( 1 ). +*> +*> An approximate (asymptotic) bound on the maximum angular error in +*> the computed deflating subspaces is +*> +*> EPS * norm((A, B)) / RCONDV( 2 ). +*> +*> See LAPACK User's Guide, section 4.11 for more information. +*> +*> \endverbatim +*> +* ===================================================================== SUBROUTINE SGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, $ B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, $ VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK, $ LIWORK, BWORK, INFO ) * -* -- LAPACK driver routine (version 3.2.1) -- +* -- LAPACK eigen routine (version 3.2.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- -* -- April 2009 -- +* November 2011 * * .. Scalar Arguments .. CHARACTER JOBVSL, JOBVSR, SENSE, SORT @@ -26,227 +385,6 @@ EXTERNAL SELCTG * .. * -* Purpose -* ======= -* -* SGGESX computes for a pair of N-by-N real nonsymmetric matrices -* (A,B), the generalized eigenvalues, the real Schur form (S,T), and, -* optionally, the left and/or right matrices of Schur vectors (VSL and -* VSR). This gives the generalized Schur factorization -* -* (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T ) -* -* Optionally, it also orders the eigenvalues so that a selected cluster -* of eigenvalues appears in the leading diagonal blocks of the upper -* quasi-triangular matrix S and the upper triangular matrix T; computes -* a reciprocal condition number for the average of the selected -* eigenvalues (RCONDE); and computes a reciprocal condition number for -* the right and left deflating subspaces corresponding to the selected -* eigenvalues (RCONDV). The leading columns of VSL and VSR then form -* an orthonormal basis for the corresponding left and right eigenspaces -* (deflating subspaces). -* -* A generalized eigenvalue for a pair of matrices (A,B) is a scalar w -* or a ratio alpha/beta = w, such that A - w*B is singular. It is -* usually represented as the pair (alpha,beta), as there is a -* reasonable interpretation for beta=0 or for both being zero. -* -* A pair of matrices (S,T) is in generalized real Schur form if T is -* upper triangular with non-negative diagonal and S is block upper -* triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond -* to real generalized eigenvalues, while 2-by-2 blocks of S will be -* "standardized" by making the corresponding elements of T have the -* form: -* [ a 0 ] -* [ 0 b ] -* -* and the pair of corresponding 2-by-2 blocks in S and T will have a -* complex conjugate pair of generalized eigenvalues. -* -* -* Arguments -* ========= -* -* JOBVSL (input) CHARACTER*1 -* = 'N': do not compute the left Schur vectors; -* = 'V': compute the left Schur vectors. -* -* JOBVSR (input) CHARACTER*1 -* = 'N': do not compute the right Schur vectors; -* = 'V': compute the right Schur vectors. -* -* SORT (input) CHARACTER*1 -* Specifies whether or not to order the eigenvalues on the -* diagonal of the generalized Schur form. -* = 'N': Eigenvalues are not ordered; -* = 'S': Eigenvalues are ordered (see SELCTG). -* -* SELCTG (external procedure) LOGICAL FUNCTION of three REAL arguments -* SELCTG must be declared EXTERNAL in the calling subroutine. -* If SORT = 'N', SELCTG is not referenced. -* If SORT = 'S', SELCTG is used to select eigenvalues to sort -* to the top left of the Schur form. -* An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if -* SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either -* one of a complex conjugate pair of eigenvalues is selected, -* then both complex eigenvalues are selected. -* Note that a selected complex eigenvalue may no longer satisfy -* SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering, -* since ordering may change the value of complex eigenvalues -* (especially if the eigenvalue is ill-conditioned), in this -* case INFO is set to N+3. -* -* SENSE (input) CHARACTER*1 -* Determines which reciprocal condition numbers are computed. -* = 'N' : None are computed; -* = 'E' : Computed for average of selected eigenvalues only; -* = 'V' : Computed for selected deflating subspaces only; -* = 'B' : Computed for both. -* If SENSE = 'E', 'V', or 'B', SORT must equal 'S'. -* -* N (input) INTEGER -* The order of the matrices A, B, VSL, and VSR. N >= 0. -* -* A (input/output) REAL array, dimension (LDA, N) -* On entry, the first of the pair of matrices. -* On exit, A has been overwritten by its generalized Schur -* form S. -* -* LDA (input) INTEGER -* The leading dimension of A. LDA >= max(1,N). -* -* B (input/output) REAL array, dimension (LDB, N) -* On entry, the second of the pair of matrices. -* On exit, B has been overwritten by its generalized Schur -* form T. -* -* LDB (input) INTEGER -* The leading dimension of B. LDB >= max(1,N). -* -* SDIM (output) INTEGER -* If SORT = 'N', SDIM = 0. -* If SORT = 'S', SDIM = number of eigenvalues (after sorting) -* for which SELCTG is true. (Complex conjugate pairs for which -* SELCTG is true for either eigenvalue count as 2.) -* -* ALPHAR (output) REAL array, dimension (N) -* -* ALPHAI (output) REAL array, dimension (N) -* -* BETA (output) REAL array, dimension (N) -* On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will -* be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i -* and BETA(j),j=1,...,N are the diagonals of the complex Schur -* form (S,T) that would result if the 2-by-2 diagonal blocks of -* the real Schur form of (A,B) were further reduced to -* triangular form using 2-by-2 complex unitary transformations. -* If ALPHAI(j) is zero, then the j-th eigenvalue is real; if -* positive, then the j-th and (j+1)-st eigenvalues are a -* complex conjugate pair, with ALPHAI(j+1) negative. -* -* Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) -* may easily over- or underflow, and BETA(j) may even be zero. -* Thus, the user should avoid naively computing the ratio. -* However, ALPHAR and ALPHAI will be always less than and -* usually comparable with norm(A) in magnitude, and BETA always -* less than and usually comparable with norm(B). -* -* VSL (output) REAL array, dimension (LDVSL,N) -* If JOBVSL = 'V', VSL will contain the left Schur vectors. -* Not referenced if JOBVSL = 'N'. -* -* LDVSL (input) INTEGER -* The leading dimension of the matrix VSL. LDVSL >=1, and -* if JOBVSL = 'V', LDVSL >= N. -* -* VSR (output) REAL array, dimension (LDVSR,N) -* If JOBVSR = 'V', VSR will contain the right Schur vectors. -* Not referenced if JOBVSR = 'N'. -* -* LDVSR (input) INTEGER -* The leading dimension of the matrix VSR. LDVSR >= 1, and -* if JOBVSR = 'V', LDVSR >= N. -* -* RCONDE (output) REAL array, dimension ( 2 ) -* If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the -* reciprocal condition numbers for the average of the selected -* eigenvalues. -* Not referenced if SENSE = 'N' or 'V'. -* -* RCONDV (output) REAL array, dimension ( 2 ) -* If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the -* reciprocal condition numbers for the selected deflating -* subspaces. -* Not referenced if SENSE = 'N' or 'E'. -* -* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK)) -* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. -* -* LWORK (input) INTEGER -* The dimension of the array WORK. -* If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B', -* LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else -* LWORK >= max( 8*N, 6*N+16 ). -* Note that 2*SDIM*(N-SDIM) <= N*N/2. -* Note also that an error is only returned if -* LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B' -* this may not be large enough. -* -* If LWORK = -1, then a workspace query is assumed; the routine -* only calculates the bound on the optimal size of the WORK -* array and the minimum size of the IWORK array, returns these -* values as the first entries of the WORK and IWORK arrays, and -* no error message related to LWORK or LIWORK is issued by -* XERBLA. -* -* IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK)) -* On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK. -* -* LIWORK (input) INTEGER -* The dimension of the array IWORK. -* If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise -* LIWORK >= N+6. -* -* If LIWORK = -1, then a workspace query is assumed; the -* routine only calculates the bound on the optimal size of the -* WORK array and the minimum size of the IWORK array, returns -* these values as the first entries of the WORK and IWORK -* arrays, and no error message related to LWORK or LIWORK is -* issued by XERBLA. -* -* BWORK (workspace) LOGICAL array, dimension (N) -* Not referenced if SORT = 'N'. -* -* INFO (output) INTEGER -* = 0: successful exit -* < 0: if INFO = -i, the i-th argument had an illegal value. -* = 1,...,N: -* The QZ iteration failed. (A,B) are not in Schur -* form, but ALPHAR(j), ALPHAI(j), and BETA(j) should -* be correct for j=INFO+1,...,N. -* > N: =N+1: other than QZ iteration failed in SHGEQZ -* =N+2: after reordering, roundoff changed values of -* some complex eigenvalues so that leading -* eigenvalues in the Generalized Schur form no -* longer satisfy SELCTG=.TRUE. This could also -* be caused due to scaling. -* =N+3: reordering failed in STGSEN. -* -* Further Details -* =============== -* -* An approximate (asymptotic) bound on the average absolute error of -* the selected eigenvalues is -* -* EPS * norm((A, B)) / RCONDE( 1 ). -* -* An approximate (asymptotic) bound on the maximum angular error in -* the computed deflating subspaces is -* -* EPS * norm((A, B)) / RCONDV( 2 ). -* -* See LAPACK User's Guide, section 4.11 for more information. -* * ===================================================================== * * .. Parameters .. |