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authorjulie <julielangou@users.noreply.github.com>2011-10-06 06:53:11 +0000
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+*> \brief \b SSTEMR
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+* Definition
+* ==========
+*
+* SUBROUTINE SSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
+* M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
+* IWORK, LIWORK, INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER JOBZ, RANGE
+* LOGICAL TRYRAC
+* INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
+* REAL VL, VU
+* ..
+* .. Array Arguments ..
+* INTEGER ISUPPZ( * ), IWORK( * )
+* REAL D( * ), E( * ), W( * ), WORK( * )
+* REAL Z( LDZ, * )
+* ..
+*
+* Purpose
+* =======
+*
+*>\details \b Purpose:
+*>\verbatim
+*>
+*> SSTEMR computes selected eigenvalues and, optionally, eigenvectors
+*> of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
+*> a well defined set of pairwise different real eigenvalues, the corresponding
+*> real eigenvectors are pairwise orthogonal.
+*>
+*> The spectrum may be computed either completely or partially by specifying
+*> either an interval (VL,VU] or a range of indices IL:IU for the desired
+*> eigenvalues.
+*>
+*> Depending on the number of desired eigenvalues, these are computed either
+*> by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
+*> computed by the use of various suitable L D L^T factorizations near clusters
+*> of close eigenvalues (referred to as RRRs, Relatively Robust
+*> Representations). An informal sketch of the algorithm follows.
+*>
+*> For each unreduced block (submatrix) of T,
+*> (a) Compute T - sigma I = L D L^T, so that L and D
+*> define all the wanted eigenvalues to high relative accuracy.
+*> This means that small relative changes in the entries of D and L
+*> cause only small relative changes in the eigenvalues and
+*> eigenvectors. The standard (unfactored) representation of the
+*> tridiagonal matrix T does not have this property in general.
+*> (b) Compute the eigenvalues to suitable accuracy.
+*> If the eigenvectors are desired, the algorithm attains full
+*> accuracy of the computed eigenvalues only right before
+*> the corresponding vectors have to be computed, see steps c) and d).
+*> (c) For each cluster of close eigenvalues, select a new
+*> shift close to the cluster, find a new factorization, and refine
+*> the shifted eigenvalues to suitable accuracy.
+*> (d) For each eigenvalue with a large enough relative separation compute
+*> the corresponding eigenvector by forming a rank revealing twisted
+*> factorization. Go back to (c) for any clusters that remain.
+*>
+*> For more details, see:
+*> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
+*> to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
+*> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
+*> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
+*> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
+*> 2004. Also LAPACK Working Note 154.
+*> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
+*> tridiagonal eigenvalue/eigenvector problem",
+*> Computer Science Division Technical Report No. UCB/CSD-97-971,
+*> UC Berkeley, May 1997.
+*>
+*> Further Details
+*> 1.SSTEMR works only on machines which follow IEEE-754
+*> floating-point standard in their handling of infinities and NaNs.
+*> This permits the use of efficient inner loops avoiding a check for
+*> zero divisors.
+*>
+*>\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.
+*> \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
+*> T. On exit, D is overwritten.
+*> \endverbatim
+*>
+*> \param[in,out] E
+*> \verbatim
+*> E is REAL array, dimension (N)
+*> On entry, the (N-1) subdiagonal elements of the tridiagonal
+*> matrix T in elements 1 to N-1 of E. E(N) need not be set on
+*> input, but is used internally as workspace.
+*> On exit, E is overwritten.
+*> \endverbatim
+*>
+*> \param[in] VL
+*> \verbatim
+*> VL is REAL
+*> \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
+*> \endverbatim
+*>
+*> \param[in] IL
+*> \verbatim
+*> IL is INTEGER
+*> \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.
+*> Not referenced if RANGE = 'A' or 'V'.
+*> \endverbatim
+*> \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', and if INFO = 0, then the first M columns of Z
+*> contain the orthonormal eigenvectors of the matrix T
+*> corresponding to the selected eigenvalues, with the i-th
+*> column of Z holding the eigenvector associated with W(i).
+*> If JOBZ = 'N', then Z is not referenced.
+*> 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 can be computed with a workspace
+*> query by setting NZC = -1, see below.
+*> \endverbatim
+*>
+*> \param[in] LDZ
+*> \verbatim
+*> LDZ is INTEGER
+*> The leading dimension of the array Z. LDZ >= 1, and if
+*> JOBZ = 'V', then LDZ >= max(1,N).
+*> \endverbatim
+*>
+*> \param[in] NZC
+*> \verbatim
+*> NZC is INTEGER
+*> The number of eigenvectors to be held in the array Z.
+*> If RANGE = 'A', then NZC >= max(1,N).
+*> If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
+*> If RANGE = 'I', then NZC >= IU-IL+1.
+*> If NZC = -1, then a workspace query is assumed; the
+*> routine calculates the number of columns of the array Z that
+*> are needed to hold the eigenvectors.
+*> This value is returned as the first entry of the Z array, and
+*> no error message related to NZC is issued by XERBLA.
+*> \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 computed eigenvector
+*> is nonzero only in elements ISUPPZ( 2*i-1 ) through
+*> ISUPPZ( 2*i ). This is relevant in the case when the matrix
+*> is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
+*> \endverbatim
+*>
+*> \param[in,out] TRYRAC
+*> \verbatim
+*> TRYRAC is LOGICAL
+*> If TRYRAC.EQ..TRUE., indicates that the code should check whether
+*> the tridiagonal matrix defines its eigenvalues to high relative
+*> accuracy. If so, the code uses relative-accuracy preserving
+*> algorithms that might be (a bit) slower depending on the matrix.
+*> If the matrix does not define its eigenvalues to high relative
+*> accuracy, the code can uses possibly faster algorithms.
+*> If TRYRAC.EQ..FALSE., the code is not required to guarantee
+*> relatively accurate eigenvalues and can use the fastest possible
+*> techniques.
+*> On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
+*> does not define its eigenvalues to high relative accuracy.
+*> \endverbatim
+*>
+*> \param[out] WORK
+*> \verbatim
+*> WORK is REAL array, dimension (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 >= max(1,18*N)
+*> if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
+*> If LWORK = -1, then a workspace query is assumed; the routine
+*> only calculates the optimal size of the WORK array, returns
+*> this value as the first entry of the WORK array, and no error
+*> message related to LWORK is issued by XERBLA.
+*> \endverbatim
+*>
+*> \param[out] IWORK
+*> \verbatim
+*> IWORK is INTEGER array, dimension (LIWORK)
+*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
+*> \endverbatim
+*>
+*> \param[in] LIWORK
+*> \verbatim
+*> LIWORK is INTEGER
+*> The dimension of the array IWORK. LIWORK >= max(1,10*N)
+*> if the eigenvectors are desired, and LIWORK >= max(1,8*N)
+*> if only the eigenvalues are to be computed.
+*> If LIWORK = -1, then a workspace query is assumed; the
+*> routine only calculates the optimal size of the IWORK array,
+*> returns this value as the first entry of the IWORK array, and
+*> no error message related to LIWORK is issued by XERBLA.
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> INFO is INTEGER
+*> On exit, INFO
+*> = 0: successful exit
+*> < 0: if INFO = -i, the i-th argument had an illegal value
+*> > 0: if INFO = 1X, internal error in SLARRE,
+*> if INFO = 2X, internal error in SLARRV.
+*> Here, the digit X = ABS( IINFO ) < 10, where IINFO is
+*> the nonzero error code returned by SLARRE or
+*> SLARRV, respectively.
+*> \endverbatim
+*>
+*
+* Authors
+* =======
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date November 2011
+*
+*> \ingroup realOTHERcomputational
+*
+*
+* Further Details
+* ===============
+*>\details \b Further \b Details
+*> \verbatim
+*>
+*> Based on contributions by
+*> Beresford Parlett, University of California, Berkeley, USA
+*> Jim Demmel, University of California, Berkeley, USA
+*> Inderjit Dhillon, University of Texas, Austin, USA
+*> Osni Marques, LBNL/NERSC, USA
+*> Christof Voemel, University of California, Berkeley, USA
+*>
+*> \endverbatim
+*>
+* =====================================================================
SUBROUTINE SSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
$ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
$ IWORK, LIWORK, INFO )
- IMPLICIT NONE
*
-* -- LAPACK computational routine (version 3.2.2) --
+* -- LAPACK computational routine (version 3.2.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* -- June 2010 --
+* November 2011
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE
@@ -20,198 +323,6 @@
REAL Z( LDZ, * )
* ..
*
-* Purpose
-* =======
-*
-* SSTEMR computes selected eigenvalues and, optionally, eigenvectors
-* of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
-* a well defined set of pairwise different real eigenvalues, the corresponding
-* real eigenvectors are pairwise orthogonal.
-*
-* The spectrum may be computed either completely or partially by specifying
-* either an interval (VL,VU] or a range of indices IL:IU for the desired
-* eigenvalues.
-*
-* Depending on the number of desired eigenvalues, these are computed either
-* by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
-* computed by the use of various suitable L D L^T factorizations near clusters
-* of close eigenvalues (referred to as RRRs, Relatively Robust
-* Representations). An informal sketch of the algorithm follows.
-*
-* For each unreduced block (submatrix) of T,
-* (a) Compute T - sigma I = L D L^T, so that L and D
-* define all the wanted eigenvalues to high relative accuracy.
-* This means that small relative changes in the entries of D and L
-* cause only small relative changes in the eigenvalues and
-* eigenvectors. The standard (unfactored) representation of the
-* tridiagonal matrix T does not have this property in general.
-* (b) Compute the eigenvalues to suitable accuracy.
-* If the eigenvectors are desired, the algorithm attains full
-* accuracy of the computed eigenvalues only right before
-* the corresponding vectors have to be computed, see steps c) and d).
-* (c) For each cluster of close eigenvalues, select a new
-* shift close to the cluster, find a new factorization, and refine
-* the shifted eigenvalues to suitable accuracy.
-* (d) For each eigenvalue with a large enough relative separation compute
-* the corresponding eigenvector by forming a rank revealing twisted
-* factorization. Go back to (c) for any clusters that remain.
-*
-* For more details, see:
-* - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
-* to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
-* Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
-* - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
-* Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
-* 2004. Also LAPACK Working Note 154.
-* - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
-* tridiagonal eigenvalue/eigenvector problem",
-* Computer Science Division Technical Report No. UCB/CSD-97-971,
-* UC Berkeley, May 1997.
-*
-* Further Details
-* 1.SSTEMR works only on machines which follow IEEE-754
-* floating-point standard in their handling of infinities and NaNs.
-* This permits the use of efficient inner loops avoiding a check for
-* zero divisors.
-*
-* 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.
-*
-* 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
-* T. On exit, D is overwritten.
-*
-* E (input/output) REAL array, dimension (N)
-* On entry, the (N-1) subdiagonal elements of the tridiagonal
-* matrix T in elements 1 to N-1 of E. E(N) need not be set on
-* input, but is used internally as workspace.
-* On exit, E is overwritten.
-*
-* 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.
-* Not referenced if RANGE = 'A' or 'V'.
-*
-* 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', and if INFO = 0, then the first M columns of Z
-* contain the orthonormal eigenvectors of the matrix T
-* corresponding to the selected eigenvalues, with the i-th
-* column of Z holding the eigenvector associated with W(i).
-* If JOBZ = 'N', then Z is not referenced.
-* 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 can be computed with a workspace
-* query by setting NZC = -1, see below.
-*
-* LDZ (input) INTEGER
-* The leading dimension of the array Z. LDZ >= 1, and if
-* JOBZ = 'V', then LDZ >= max(1,N).
-*
-* NZC (input) INTEGER
-* The number of eigenvectors to be held in the array Z.
-* If RANGE = 'A', then NZC >= max(1,N).
-* If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
-* If RANGE = 'I', then NZC >= IU-IL+1.
-* If NZC = -1, then a workspace query is assumed; the
-* routine calculates the number of columns of the array Z that
-* are needed to hold the eigenvectors.
-* This value is returned as the first entry of the Z array, and
-* no error message related to NZC is issued by XERBLA.
-*
-* 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 computed eigenvector
-* is nonzero only in elements ISUPPZ( 2*i-1 ) through
-* ISUPPZ( 2*i ). This is relevant in the case when the matrix
-* is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
-*
-* TRYRAC (input/output) LOGICAL
-* If TRYRAC.EQ..TRUE., indicates that the code should check whether
-* the tridiagonal matrix defines its eigenvalues to high relative
-* accuracy. If so, the code uses relative-accuracy preserving
-* algorithms that might be (a bit) slower depending on the matrix.
-* If the matrix does not define its eigenvalues to high relative
-* accuracy, the code can uses possibly faster algorithms.
-* If TRYRAC.EQ..FALSE., the code is not required to guarantee
-* relatively accurate eigenvalues and can use the fastest possible
-* techniques.
-* On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
-* does not define its eigenvalues to high relative accuracy.
-*
-* WORK (workspace/output) REAL array, dimension (LWORK)
-* On exit, if INFO = 0, WORK(1) returns the optimal
-* (and minimal) LWORK.
-*
-* LWORK (input) INTEGER
-* The dimension of the array WORK. LWORK >= max(1,18*N)
-* if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
-* If LWORK = -1, then a workspace query is assumed; the routine
-* only calculates the optimal size of the WORK array, returns
-* this value as the first entry of the WORK array, and no error
-* message related to LWORK is issued by XERBLA.
-*
-* IWORK (workspace/output) INTEGER array, dimension (LIWORK)
-* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
-*
-* LIWORK (input) INTEGER
-* The dimension of the array IWORK. LIWORK >= max(1,10*N)
-* if the eigenvectors are desired, and LIWORK >= max(1,8*N)
-* if only the eigenvalues are to be computed.
-* If LIWORK = -1, then a workspace query is assumed; the
-* routine only calculates the optimal size of the IWORK array,
-* returns this value as the first entry of the IWORK array, and
-* no error message related to LIWORK is issued by XERBLA.
-*
-* INFO (output) INTEGER
-* On exit, INFO
-* = 0: successful exit
-* < 0: if INFO = -i, the i-th argument had an illegal value
-* > 0: if INFO = 1X, internal error in SLARRE,
-* if INFO = 2X, internal error in SLARRV.
-* Here, the digit X = ABS( IINFO ) < 10, where IINFO is
-* the nonzero error code returned by SLARRE or
-* SLARRV, respectively.
-*
-*
-* Further Details
-* ===============
-*
-* Based on contributions by
-* Beresford Parlett, University of California, Berkeley, USA
-* Jim Demmel, University of California, Berkeley, USA
-* Inderjit Dhillon, University of Texas, Austin, USA
-* Osni Marques, LBNL/NERSC, USA
-* Christof Voemel, University of California, Berkeley, USA
-*
* =====================================================================
*
* .. Parameters ..