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authorjulie <julielangou@users.noreply.github.com>2011-10-06 06:53:11 +0000
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+*> \brief <b> SSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+* Definition
+* ==========
+*
+* SUBROUTINE SSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
+* M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
+* LIWORK, INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER JOBZ, RANGE
+* INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
+* REAL ABSTOL, VL, VU
+* ..
+* .. Array Arguments ..
+* INTEGER ISUPPZ( * ), IWORK( * )
+* REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
+* ..
+*
+* Purpose
+* =======
+*
+*>\details \b Purpose:
+*>\verbatim
+*>
+*> SSTEVR computes selected eigenvalues and, optionally, eigenvectors
+*> of a real symmetric tridiagonal matrix T. Eigenvalues and
+*> eigenvectors can be selected by specifying either a range of values
+*> or a range of indices for the desired eigenvalues.
+*>
+*> Whenever possible, SSTEVR calls SSTEMR to compute the
+*> eigenspectrum using Relatively Robust Representations. SSTEMR
+*> computes eigenvalues by the dqds algorithm, while orthogonal
+*> eigenvectors are computed from various "good" L D L^T representations
+*> (also known as Relatively Robust Representations). Gram-Schmidt
+*> orthogonalization is avoided as far as possible. More specifically,
+*> the various steps of the algorithm are as follows. For the i-th
+*> unreduced block of T,
+*> (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
+*> is a relatively robust representation,
+*> (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
+*> relative accuracy by the dqds algorithm,
+*> (c) If there is a cluster of close eigenvalues, "choose" sigma_i
+*> close to the cluster, and go to step (a),
+*> (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
+*> compute the corresponding eigenvector by forming a
+*> rank-revealing twisted factorization.
+*> The desired accuracy of the output can be specified by the input
+*> parameter ABSTOL.
+*>
+*> For more details, see "A new O(n^2) algorithm for the symmetric
+*> tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon,
+*> Computer Science Division Technical Report No. UCB//CSD-97-971,
+*> UC Berkeley, May 1997.
+*>
+*>
+*> Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested
+*> on machines which conform to the ieee-754 floating point standard.
+*> SSTEVR calls SSTEBZ and SSTEIN on non-ieee machines and
+*> when partial spectrum requests are made.
+*>
+*> Normal execution of SSTEMR may create NaNs and infinities and
+*> hence may abort due to a floating point exception in environments
+*> which do not handle NaNs and infinities in the ieee standard default
+*> manner.
+*>
+*>\endverbatim
+*
+* Arguments
+* =========
+*
+*> \param[in] JOBZ
+*> \verbatim
+*> JOBZ is CHARACTER*1
+*> = 'N': Compute eigenvalues only;
+*> = 'V': Compute eigenvalues and eigenvectors.
+*> \endverbatim
+*>
+*> \param[in] RANGE
+*> \verbatim
+*> RANGE is CHARACTER*1
+*> = 'A': all eigenvalues will be found.
+*> = 'V': all eigenvalues in the half-open interval (VL,VU]
+*> will be found.
+*> = 'I': the IL-th through IU-th eigenvalues will be found.
+*> For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and
+*> SSTEIN are called
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The order of the matrix. N >= 0.
+*> \endverbatim
+*>
+*> \param[in,out] D
+*> \verbatim
+*> D is REAL array, dimension (N)
+*> On entry, the n diagonal elements of the tridiagonal matrix
+*> A.
+*> On exit, D may be multiplied by a constant factor chosen
+*> to avoid over/underflow in computing the eigenvalues.
+*> \endverbatim
+*>
+*> \param[in,out] E
+*> \verbatim
+*> E is REAL array, dimension (max(1,N-1))
+*> On entry, the (n-1) subdiagonal elements of the tridiagonal
+*> matrix A in elements 1 to N-1 of E.
+*> On exit, E may be multiplied by a constant factor chosen
+*> to avoid over/underflow in computing the eigenvalues.
+*> \endverbatim
+*>
+*> \param[in] VL
+*> \verbatim
+*> VL is REAL
+*> \endverbatim
+*>
+*> \param[in] VU
+*> \verbatim
+*> VU is REAL
+*> If RANGE='V', the lower and upper bounds of the interval to
+*> be searched for eigenvalues. VL < VU.
+*> Not referenced if RANGE = 'A' or 'I'.
+*> \endverbatim
+*>
+*> \param[in] IL
+*> \verbatim
+*> IL is INTEGER
+*> \endverbatim
+*>
+*> \param[in] IU
+*> \verbatim
+*> IU is INTEGER
+*> If RANGE='I', the indices (in ascending order) of the
+*> smallest and largest eigenvalues to be returned.
+*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
+*> Not referenced if RANGE = 'A' or 'V'.
+*> \endverbatim
+*>
+*> \param[in] ABSTOL
+*> \verbatim
+*> ABSTOL is REAL
+*> The absolute error tolerance for the eigenvalues.
+*> An approximate eigenvalue is accepted as converged
+*> when it is determined to lie in an interval [a,b]
+*> of width less than or equal to
+*> \endverbatim
+*> \verbatim
+*> ABSTOL + EPS * max( |a|,|b| ) ,
+*> \endverbatim
+*> \verbatim
+*> where EPS is the machine precision. If ABSTOL is less than
+*> or equal to zero, then EPS*|T| will be used in its place,
+*> where |T| is the 1-norm of the tridiagonal matrix obtained
+*> by reducing A to tridiagonal form.
+*> \endverbatim
+*> \verbatim
+*> See "Computing Small Singular Values of Bidiagonal Matrices
+*> with Guaranteed High Relative Accuracy," by Demmel and
+*> Kahan, LAPACK Working Note #3.
+*> \endverbatim
+*> \verbatim
+*> If high relative accuracy is important, set ABSTOL to
+*> SLAMCH( 'Safe minimum' ). Doing so will guarantee that
+*> eigenvalues are computed to high relative accuracy when
+*> possible in future releases. The current code does not
+*> make any guarantees about high relative accuracy, but
+*> future releases will. See J. Barlow and J. Demmel,
+*> "Computing Accurate Eigensystems of Scaled Diagonally
+*> Dominant Matrices", LAPACK Working Note #7, for a discussion
+*> of which matrices define their eigenvalues to high relative
+*> accuracy.
+*> \endverbatim
+*>
+*> \param[out] M
+*> \verbatim
+*> M is INTEGER
+*> The total number of eigenvalues found. 0 <= M <= N.
+*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
+*> \endverbatim
+*>
+*> \param[out] W
+*> \verbatim
+*> W is REAL array, dimension (N)
+*> The first M elements contain the selected eigenvalues in
+*> ascending order.
+*> \endverbatim
+*>
+*> \param[out] Z
+*> \verbatim
+*> Z is REAL array, dimension (LDZ, max(1,M) )
+*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
+*> contain the orthonormal eigenvectors of the matrix A
+*> corresponding to the selected eigenvalues, with the i-th
+*> column of Z holding the eigenvector associated with W(i).
+*> Note: the user must ensure that at least max(1,M) columns are
+*> supplied in the array Z; if RANGE = 'V', the exact value of M
+*> is not known in advance and an upper bound must be used.
+*> \endverbatim
+*>
+*> \param[in] LDZ
+*> \verbatim
+*> LDZ is INTEGER
+*> The leading dimension of the array Z. LDZ >= 1, and if
+*> JOBZ = 'V', LDZ >= max(1,N).
+*> \endverbatim
+*>
+*> \param[out] ISUPPZ
+*> \verbatim
+*> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
+*> The support of the eigenvectors in Z, i.e., the indices
+*> indicating the nonzero elements in Z. The i-th eigenvector
+*> is nonzero only in elements ISUPPZ( 2*i-1 ) through
+*> ISUPPZ( 2*i ).
+*> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
+*> \endverbatim
+*>
+*> \param[out] WORK
+*> \verbatim
+*> WORK is REAL array, dimension (MAX(1,LWORK))
+*> On exit, if INFO = 0, WORK(1) returns the optimal (and
+*> minimal) LWORK.
+*> \endverbatim
+*>
+*> \param[in] LWORK
+*> \verbatim
+*> LWORK is INTEGER
+*> The dimension of the array WORK. LWORK >= 20*N.
+*> \endverbatim
+*> \verbatim
+*> If LWORK = -1, then a workspace query is assumed; the routine
+*> only calculates the optimal sizes of the WORK and IWORK
+*> arrays, 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 optimal (and
+*> minimal) LIWORK.
+*> \endverbatim
+*>
+*> \param[in] LIWORK
+*> \verbatim
+*> LIWORK is INTEGER
+*> The dimension of the array IWORK. LIWORK >= 10*N.
+*> \endverbatim
+*> \verbatim
+*> If LIWORK = -1, then a workspace query is assumed; the
+*> routine only calculates the optimal sizes of the WORK and
+*> IWORK arrays, 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] INFO
+*> \verbatim
+*> INFO is INTEGER
+*> = 0: successful exit
+*> < 0: if INFO = -i, the i-th argument had an illegal value
+*> > 0: Internal error
+*> \endverbatim
+*>
+*
+* Authors
+* =======
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date November 2011
+*
+*> \ingroup realOTHEReigen
+*
+*
+* Further Details
+* ===============
+*>\details \b Further \b Details
+*> \verbatim
+*>
+*> Based on contributions by
+*> Inderjit Dhillon, IBM Almaden, USA
+*> Osni Marques, LBNL/NERSC, USA
+*> Ken Stanley, Computer Science Division, University of
+*> California at Berkeley, USA
+*> Jason Riedy, Computer Science Division, University of
+*> California at Berkeley, USA
+*>
+*> \endverbatim
+*>
+* =====================================================================
SUBROUTINE SSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
$ M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
$ LIWORK, INFO )
*
-* -- LAPACK driver routine (version 3.2) --
+* -- LAPACK eigen routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* November 2006
+* November 2011
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE
@@ -17,193 +318,6 @@
REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
-* Purpose
-* =======
-*
-* SSTEVR computes selected eigenvalues and, optionally, eigenvectors
-* of a real symmetric tridiagonal matrix T. Eigenvalues and
-* eigenvectors can be selected by specifying either a range of values
-* or a range of indices for the desired eigenvalues.
-*
-* Whenever possible, SSTEVR calls SSTEMR to compute the
-* eigenspectrum using Relatively Robust Representations. SSTEMR
-* computes eigenvalues by the dqds algorithm, while orthogonal
-* eigenvectors are computed from various "good" L D L^T representations
-* (also known as Relatively Robust Representations). Gram-Schmidt
-* orthogonalization is avoided as far as possible. More specifically,
-* the various steps of the algorithm are as follows. For the i-th
-* unreduced block of T,
-* (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
-* is a relatively robust representation,
-* (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
-* relative accuracy by the dqds algorithm,
-* (c) If there is a cluster of close eigenvalues, "choose" sigma_i
-* close to the cluster, and go to step (a),
-* (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
-* compute the corresponding eigenvector by forming a
-* rank-revealing twisted factorization.
-* The desired accuracy of the output can be specified by the input
-* parameter ABSTOL.
-*
-* For more details, see "A new O(n^2) algorithm for the symmetric
-* tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon,
-* Computer Science Division Technical Report No. UCB//CSD-97-971,
-* UC Berkeley, May 1997.
-*
-*
-* Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested
-* on machines which conform to the ieee-754 floating point standard.
-* SSTEVR calls SSTEBZ and SSTEIN on non-ieee machines and
-* when partial spectrum requests are made.
-*
-* Normal execution of SSTEMR may create NaNs and infinities and
-* hence may abort due to a floating point exception in environments
-* which do not handle NaNs and infinities in the ieee standard default
-* manner.
-*
-* Arguments
-* =========
-*
-* JOBZ (input) CHARACTER*1
-* = 'N': Compute eigenvalues only;
-* = 'V': Compute eigenvalues and eigenvectors.
-*
-* RANGE (input) CHARACTER*1
-* = 'A': all eigenvalues will be found.
-* = 'V': all eigenvalues in the half-open interval (VL,VU]
-* will be found.
-* = 'I': the IL-th through IU-th eigenvalues will be found.
-* For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and
-* SSTEIN are called
-*
-* N (input) INTEGER
-* The order of the matrix. N >= 0.
-*
-* D (input/output) REAL array, dimension (N)
-* On entry, the n diagonal elements of the tridiagonal matrix
-* A.
-* On exit, D may be multiplied by a constant factor chosen
-* to avoid over/underflow in computing the eigenvalues.
-*
-* E (input/output) REAL array, dimension (max(1,N-1))
-* On entry, the (n-1) subdiagonal elements of the tridiagonal
-* matrix A in elements 1 to N-1 of E.
-* On exit, E may be multiplied by a constant factor chosen
-* to avoid over/underflow in computing the eigenvalues.
-*
-* VL (input) REAL
-*
-* VU (input) REAL
-* If RANGE='V', the lower and upper bounds of the interval to
-* be searched for eigenvalues. VL < VU.
-* Not referenced if RANGE = 'A' or 'I'.
-*
-* IL (input) INTEGER
-*
-* IU (input) INTEGER
-* If RANGE='I', the indices (in ascending order) of the
-* smallest and largest eigenvalues to be returned.
-* 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
-* Not referenced if RANGE = 'A' or 'V'.
-*
-* ABSTOL (input) REAL
-* The absolute error tolerance for the eigenvalues.
-* An approximate eigenvalue is accepted as converged
-* when it is determined to lie in an interval [a,b]
-* of width less than or equal to
-*
-* ABSTOL + EPS * max( |a|,|b| ) ,
-*
-* where EPS is the machine precision. If ABSTOL is less than
-* or equal to zero, then EPS*|T| will be used in its place,
-* where |T| is the 1-norm of the tridiagonal matrix obtained
-* by reducing A to tridiagonal form.
-*
-* See "Computing Small Singular Values of Bidiagonal Matrices
-* with Guaranteed High Relative Accuracy," by Demmel and
-* Kahan, LAPACK Working Note #3.
-*
-* If high relative accuracy is important, set ABSTOL to
-* SLAMCH( 'Safe minimum' ). Doing so will guarantee that
-* eigenvalues are computed to high relative accuracy when
-* possible in future releases. The current code does not
-* make any guarantees about high relative accuracy, but
-* future releases will. See J. Barlow and J. Demmel,
-* "Computing Accurate Eigensystems of Scaled Diagonally
-* Dominant Matrices", LAPACK Working Note #7, for a discussion
-* of which matrices define their eigenvalues to high relative
-* accuracy.
-*
-* M (output) INTEGER
-* The total number of eigenvalues found. 0 <= M <= N.
-* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
-*
-* W (output) REAL array, dimension (N)
-* The first M elements contain the selected eigenvalues in
-* ascending order.
-*
-* Z (output) REAL array, dimension (LDZ, max(1,M) )
-* If JOBZ = 'V', then if INFO = 0, the first M columns of Z
-* contain the orthonormal eigenvectors of the matrix A
-* corresponding to the selected eigenvalues, with the i-th
-* column of Z holding the eigenvector associated with W(i).
-* Note: the user must ensure that at least max(1,M) columns are
-* supplied in the array Z; if RANGE = 'V', the exact value of M
-* is not known in advance and an upper bound must be used.
-*
-* LDZ (input) INTEGER
-* The leading dimension of the array Z. LDZ >= 1, and if
-* JOBZ = 'V', LDZ >= max(1,N).
-*
-* ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
-* The support of the eigenvectors in Z, i.e., the indices
-* indicating the nonzero elements in Z. The i-th eigenvector
-* is nonzero only in elements ISUPPZ( 2*i-1 ) through
-* ISUPPZ( 2*i ).
-* Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
-*
-* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
-* On exit, if INFO = 0, WORK(1) returns the optimal (and
-* minimal) LWORK.
-*
-* LWORK (input) INTEGER
-* The dimension of the array WORK. LWORK >= 20*N.
-*
-* If LWORK = -1, then a workspace query is assumed; the routine
-* only calculates the optimal sizes of the WORK and IWORK
-* arrays, 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/output) INTEGER array, dimension (MAX(1,LIWORK))
-* On exit, if INFO = 0, IWORK(1) returns the optimal (and
-* minimal) LIWORK.
-*
-* LIWORK (input) INTEGER
-* The dimension of the array IWORK. LIWORK >= 10*N.
-*
-* If LIWORK = -1, then a workspace query is assumed; the
-* routine only calculates the optimal sizes of the WORK and
-* IWORK arrays, 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.
-*
-* INFO (output) INTEGER
-* = 0: successful exit
-* < 0: if INFO = -i, the i-th argument had an illegal value
-* > 0: Internal error
-*
-* Further Details
-* ===============
-*
-* Based on contributions by
-* Inderjit Dhillon, IBM Almaden, USA
-* Osni Marques, LBNL/NERSC, USA
-* Ken Stanley, Computer Science Division, University of
-* California at Berkeley, USA
-* Jason Riedy, Computer Science Division, University of
-* California at Berkeley, USA
-*
* =====================================================================
*
* .. Parameters ..