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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/sstevr.f | |
parent | 5fe0466a14e395641f4f8a300ecc9dcb8058081b (diff) | |
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diff --git a/SRC/sstevr.f b/SRC/sstevr.f index 9e65739e..7cb4bf45 100644 --- a/SRC/sstevr.f +++ b/SRC/sstevr.f @@ -1,11 +1,312 @@ +*> \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 .. |