version 1.1, 2010/01/26 15:22:46
|
version 1.21, 2018/05/29 07:18:35
|
Line 1
|
Line 1
|
|
*> \brief \b ZSTEMR |
|
* |
|
* =========== DOCUMENTATION =========== |
|
* |
|
* Online html documentation available at |
|
* http://www.netlib.org/lapack/explore-html/ |
|
* |
|
*> \htmlonly |
|
*> Download ZSTEMR + dependencies |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zstemr.f"> |
|
*> [TGZ]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zstemr.f"> |
|
*> [ZIP]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zstemr.f"> |
|
*> [TXT]</a> |
|
*> \endhtmlonly |
|
* |
|
* Definition: |
|
* =========== |
|
* |
|
* SUBROUTINE ZSTEMR( 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 |
|
* DOUBLE PRECISION VL, VU |
|
* .. |
|
* .. Array Arguments .. |
|
* INTEGER ISUPPZ( * ), IWORK( * ) |
|
* DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ) |
|
* COMPLEX*16 Z( LDZ, * ) |
|
* .. |
|
* |
|
* |
|
*> \par Purpose: |
|
* ============= |
|
*> |
|
*> \verbatim |
|
*> |
|
*> ZSTEMR 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.ZSTEMR 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. |
|
*> |
|
*> 2. LAPACK routines can be used to reduce a complex Hermitean matrix to |
|
*> real symmetric tridiagonal form. |
|
*> |
|
*> (Any complex Hermitean tridiagonal matrix has real values on its diagonal |
|
*> and potentially complex numbers on its off-diagonals. By applying a |
|
*> similarity transform with an appropriate diagonal matrix |
|
*> diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean |
|
*> matrix can be transformed into a real symmetric matrix and complex |
|
*> arithmetic can be entirely avoided.) |
|
*> |
|
*> While the eigenvectors of the real symmetric tridiagonal matrix are real, |
|
*> the eigenvectors of original complex Hermitean matrix have complex entries |
|
*> in general. |
|
*> Since LAPACK drivers overwrite the matrix data with the eigenvectors, |
|
*> ZSTEMR accepts complex workspace to facilitate interoperability |
|
*> with ZUNMTR or ZUPMTR. |
|
*> \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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION |
|
*> |
|
*> If RANGE='V', the lower bound of the interval to |
|
*> be searched for eigenvalues. VL < VU. |
|
*> Not referenced if RANGE = 'A' or 'I'. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] VU |
|
*> \verbatim |
|
*> VU is DOUBLE PRECISION |
|
*> |
|
*> If RANGE='V', the upper bound 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 |
|
*> |
|
*> If RANGE='I', the index of the |
|
*> smallest eigenvalue to be returned. |
|
*> 1 <= IL <= IU <= N, if N > 0. |
|
*> Not referenced if RANGE = 'A' or 'V'. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] IU |
|
*> \verbatim |
|
*> IU is INTEGER |
|
*> |
|
*> If RANGE='I', the index of the |
|
*> largest eigenvalue to be returned. |
|
*> 1 <= IL <= IU <= N, if N > 0. |
|
*> Not referenced if RANGE = 'A' or 'V'. |
|
*> \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 DOUBLE PRECISION array, dimension (N) |
|
*> The first M elements contain the selected eigenvalues in |
|
*> ascending order. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] Z |
|
*> \verbatim |
|
*> Z is COMPLEX*16 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 DOUBLE PRECISION 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 DLARRE, |
|
*> if INFO = 2X, internal error in ZLARRV. |
|
*> Here, the digit X = ABS( IINFO ) < 10, where IINFO is |
|
*> the nonzero error code returned by DLARRE or |
|
*> ZLARRV, respectively. |
|
*> \endverbatim |
|
* |
|
* Authors: |
|
* ======== |
|
* |
|
*> \author Univ. of Tennessee |
|
*> \author Univ. of California Berkeley |
|
*> \author Univ. of Colorado Denver |
|
*> \author NAG Ltd. |
|
* |
|
*> \date June 2016 |
|
* |
|
*> \ingroup complex16OTHERcomputational |
|
* |
|
*> \par Contributors: |
|
* ================== |
|
*> |
|
*> Beresford Parlett, University of California, Berkeley, USA \n |
|
*> Jim Demmel, University of California, Berkeley, USA \n |
|
*> Inderjit Dhillon, University of Texas, Austin, USA \n |
|
*> Osni Marques, LBNL/NERSC, USA \n |
|
*> Christof Voemel, University of California, Berkeley, USA \n |
|
* |
|
* ===================================================================== |
SUBROUTINE ZSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, |
SUBROUTINE ZSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, |
$ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, |
$ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, |
$ IWORK, LIWORK, INFO ) |
$ IWORK, LIWORK, INFO ) |
IMPLICIT NONE |
|
* |
|
* -- LAPACK computational routine (version 3.2.1) -- |
|
* |
|
* -- April 2009 -- |
|
* |
* |
|
* -- LAPACK computational routine (version 3.7.1) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
|
* June 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER JOBZ, RANGE |
CHARACTER JOBZ, RANGE |
Line 22
|
Line 355
|
COMPLEX*16 Z( LDZ, * ) |
COMPLEX*16 Z( LDZ, * ) |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
|
* |
|
* ZSTEMR 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.ZSTEMR 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. |
|
* |
|
* 2. LAPACK routines can be used to reduce a complex Hermitean matrix to |
|
* real symmetric tridiagonal form. |
|
* |
|
* (Any complex Hermitean tridiagonal matrix has real values on its diagonal |
|
* and potentially complex numbers on its off-diagonals. By applying a |
|
* similarity transform with an appropriate diagonal matrix |
|
* diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean |
|
* matrix can be transformed into a real symmetric matrix and complex |
|
* arithmetic can be entirely avoided.) |
|
* |
|
* While the eigenvectors of the real symmetric tridiagonal matrix are real, |
|
* the eigenvectors of original complex Hermitean matrix have complex entries |
|
* in general. |
|
* Since LAPACK drivers overwrite the matrix data with the eigenvectors, |
|
* ZSTEMR accepts complex workspace to facilitate interoperability |
|
* with ZUNMTR or ZUPMTR. |
|
* |
|
* 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) DOUBLE PRECISION array, dimension (N) |
|
* On entry, the N diagonal elements of the tridiagonal matrix |
|
* T. On exit, D is overwritten. |
|
* |
|
* E (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION |
|
* VU (input) DOUBLE PRECISION |
|
* 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) DOUBLE PRECISION array, dimension (N) |
|
* The first M elements contain the selected eigenvalues in |
|
* ascending order. |
|
* |
|
* Z (output) COMPLEX*16 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) DOUBLE PRECISION 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 DLARRE, |
|
* if INFO = 2X, internal error in ZLARRV. |
|
* Here, the digit X = ABS( IINFO ) < 10, where IINFO is |
|
* the nonzero error code returned by DLARRE or |
|
* ZLARRV, 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 .. |
* .. Parameters .. |
Line 293
|
Line 417
|
WU = ZERO |
WU = ZERO |
IIL = 0 |
IIL = 0 |
IIU = 0 |
IIU = 0 |
|
NSPLIT = 0 |
|
|
IF( VALEIG ) THEN |
IF( VALEIG ) THEN |
* We do not reference VL, VU in the cases RANGE = 'I','A' |
* We do not reference VL, VU in the cases RANGE = 'I','A' |
Line 410
|
Line 535
|
IF (SN.NE.ZERO) THEN |
IF (SN.NE.ZERO) THEN |
IF (CS.NE.ZERO) THEN |
IF (CS.NE.ZERO) THEN |
ISUPPZ(2*M-1) = 1 |
ISUPPZ(2*M-1) = 1 |
ISUPPZ(2*M-1) = 2 |
ISUPPZ(2*M) = 2 |
ELSE |
ELSE |
ISUPPZ(2*M-1) = 1 |
ISUPPZ(2*M-1) = 1 |
ISUPPZ(2*M-1) = 1 |
ISUPPZ(2*M) = 1 |
END IF |
END IF |
ELSE |
ELSE |
ISUPPZ(2*M-1) = 2 |
ISUPPZ(2*M-1) = 2 |
Line 434
|
Line 559
|
IF (SN.NE.ZERO) THEN |
IF (SN.NE.ZERO) THEN |
IF (CS.NE.ZERO) THEN |
IF (CS.NE.ZERO) THEN |
ISUPPZ(2*M-1) = 1 |
ISUPPZ(2*M-1) = 1 |
ISUPPZ(2*M-1) = 2 |
ISUPPZ(2*M) = 2 |
ELSE |
ELSE |
ISUPPZ(2*M-1) = 1 |
ISUPPZ(2*M-1) = 1 |
ISUPPZ(2*M-1) = 1 |
ISUPPZ(2*M) = 1 |
END IF |
END IF |
ELSE |
ELSE |
ISUPPZ(2*M-1) = 2 |
ISUPPZ(2*M-1) = 2 |
Line 445
|
Line 570
|
END IF |
END IF |
ENDIF |
ENDIF |
ENDIF |
ENDIF |
RETURN |
ELSE |
END IF |
|
|
|
* Continue with general N |
* Continue with general N |
|
|
INDGRS = 1 |
INDGRS = 1 |
INDERR = 2*N + 1 |
INDERR = 2*N + 1 |
INDGP = 3*N + 1 |
INDGP = 3*N + 1 |
INDD = 4*N + 1 |
INDD = 4*N + 1 |
INDE2 = 5*N + 1 |
INDE2 = 5*N + 1 |
INDWRK = 6*N + 1 |
INDWRK = 6*N + 1 |
* |
* |
IINSPL = 1 |
IINSPL = 1 |
IINDBL = N + 1 |
IINDBL = N + 1 |
IINDW = 2*N + 1 |
IINDW = 2*N + 1 |
IINDWK = 3*N + 1 |
IINDWK = 3*N + 1 |
* |
* |
* Scale matrix to allowable range, if necessary. |
* Scale matrix to allowable range, if necessary. |
* The allowable range is related to the PIVMIN parameter; see the |
* The allowable range is related to the PIVMIN parameter; see the |
* comments in DLARRD. The preference for scaling small values |
* comments in DLARRD. The preference for scaling small values |
* up is heuristic; we expect users' matrices not to be close to the |
* up is heuristic; we expect users' matrices not to be close to the |
* RMAX threshold. |
* RMAX threshold. |
* |
* |
SCALE = ONE |
SCALE = ONE |
TNRM = DLANST( 'M', N, D, E ) |
TNRM = DLANST( 'M', N, D, E ) |
IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN |
IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN |
SCALE = RMIN / TNRM |
SCALE = RMIN / TNRM |
ELSE IF( TNRM.GT.RMAX ) THEN |
ELSE IF( TNRM.GT.RMAX ) THEN |
SCALE = RMAX / TNRM |
SCALE = RMAX / TNRM |
END IF |
END IF |
IF( SCALE.NE.ONE ) THEN |
IF( SCALE.NE.ONE ) THEN |
CALL DSCAL( N, SCALE, D, 1 ) |
CALL DSCAL( N, SCALE, D, 1 ) |
CALL DSCAL( N-1, SCALE, E, 1 ) |
CALL DSCAL( N-1, SCALE, E, 1 ) |
TNRM = TNRM*SCALE |
TNRM = TNRM*SCALE |
IF( VALEIG ) THEN |
IF( VALEIG ) THEN |
* If eigenvalues in interval have to be found, |
* If eigenvalues in interval have to be found, |
* scale (WL, WU] accordingly |
* scale (WL, WU] accordingly |
WL = WL*SCALE |
WL = WL*SCALE |
WU = WU*SCALE |
WU = WU*SCALE |
ENDIF |
ENDIF |
END IF |
END IF |
* |
* |
* Compute the desired eigenvalues of the tridiagonal after splitting |
* Compute the desired eigenvalues of the tridiagonal after splitting |
* into smaller subblocks if the corresponding off-diagonal elements |
* into smaller subblocks if the corresponding off-diagonal elements |
* are small |
* are small |
* THRESH is the splitting parameter for DLARRE |
* THRESH is the splitting parameter for DLARRE |
* A negative THRESH forces the old splitting criterion based on the |
* A negative THRESH forces the old splitting criterion based on the |
* size of the off-diagonal. A positive THRESH switches to splitting |
* size of the off-diagonal. A positive THRESH switches to splitting |
* which preserves relative accuracy. |
* which preserves relative accuracy. |
* |
* |
IF( TRYRAC ) THEN |
IF( TRYRAC ) THEN |
* Test whether the matrix warrants the more expensive relative approach. |
* Test whether the matrix warrants the more expensive relative approach. |
CALL DLARRR( N, D, E, IINFO ) |
CALL DLARRR( N, D, E, IINFO ) |
ELSE |
ELSE |
* The user does not care about relative accurately eigenvalues |
* The user does not care about relative accurately eigenvalues |
IINFO = -1 |
IINFO = -1 |
ENDIF |
ENDIF |
* Set the splitting criterion |
* Set the splitting criterion |
IF (IINFO.EQ.0) THEN |
IF (IINFO.EQ.0) THEN |
THRESH = EPS |
THRESH = EPS |
ELSE |
ELSE |
THRESH = -EPS |
THRESH = -EPS |
* relative accuracy is desired but T does not guarantee it |
* relative accuracy is desired but T does not guarantee it |
TRYRAC = .FALSE. |
TRYRAC = .FALSE. |
ENDIF |
ENDIF |
* |
* |
IF( TRYRAC ) THEN |
IF( TRYRAC ) THEN |
* Copy original diagonal, needed to guarantee relative accuracy |
* Copy original diagonal, needed to guarantee relative accuracy |
CALL DCOPY(N,D,1,WORK(INDD),1) |
CALL DCOPY(N,D,1,WORK(INDD),1) |
ENDIF |
ENDIF |
* Store the squares of the offdiagonal values of T |
* Store the squares of the offdiagonal values of T |
DO 5 J = 1, N-1 |
DO 5 J = 1, N-1 |
WORK( INDE2+J-1 ) = E(J)**2 |
WORK( INDE2+J-1 ) = E(J)**2 |
5 CONTINUE |
5 CONTINUE |
|
|
* Set the tolerance parameters for bisection |
* Set the tolerance parameters for bisection |
IF( .NOT.WANTZ ) THEN |
IF( .NOT.WANTZ ) THEN |
* DLARRE computes the eigenvalues to full precision. |
* DLARRE computes the eigenvalues to full precision. |
RTOL1 = FOUR * EPS |
RTOL1 = FOUR * EPS |
RTOL2 = FOUR * EPS |
RTOL2 = FOUR * EPS |
ELSE |
ELSE |
* DLARRE computes the eigenvalues to less than full precision. |
* DLARRE computes the eigenvalues to less than full precision. |
* ZLARRV will refine the eigenvalue approximations, and we only |
* ZLARRV will refine the eigenvalue approximations, and we only |
* need less accurate initial bisection in DLARRE. |
* need less accurate initial bisection in DLARRE. |
* Note: these settings do only affect the subset case and DLARRE |
* Note: these settings do only affect the subset case and DLARRE |
RTOL1 = SQRT(EPS) |
RTOL1 = SQRT(EPS) |
RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS ) |
RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS ) |
ENDIF |
ENDIF |
CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E, |
CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E, |
$ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT, |
$ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT, |
$ IWORK( IINSPL ), M, W, WORK( INDERR ), |
$ IWORK( IINSPL ), M, W, WORK( INDERR ), |
$ WORK( INDGP ), IWORK( IINDBL ), |
$ WORK( INDGP ), IWORK( IINDBL ), |
$ IWORK( IINDW ), WORK( INDGRS ), PIVMIN, |
$ IWORK( IINDW ), WORK( INDGRS ), PIVMIN, |
$ WORK( INDWRK ), IWORK( IINDWK ), IINFO ) |
$ WORK( INDWRK ), IWORK( IINDWK ), IINFO ) |
IF( IINFO.NE.0 ) THEN |
IF( IINFO.NE.0 ) THEN |
INFO = 10 + ABS( IINFO ) |
INFO = 10 + ABS( IINFO ) |
RETURN |
RETURN |
END IF |
END IF |
* Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired |
* Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired |
* part of the spectrum. All desired eigenvalues are contained in |
* part of the spectrum. All desired eigenvalues are contained in |
* (WL,WU] |
* (WL,WU] |
|
|
|
|
IF( WANTZ ) THEN |
IF( WANTZ ) THEN |
* |
* |
* Compute the desired eigenvectors corresponding to the computed |
* Compute the desired eigenvectors corresponding to the computed |
* eigenvalues |
* eigenvalues |
* |
* |
CALL ZLARRV( N, WL, WU, D, E, |
CALL ZLARRV( N, WL, WU, D, E, |
$ PIVMIN, IWORK( IINSPL ), M, |
$ PIVMIN, IWORK( IINSPL ), M, |
$ 1, M, MINRGP, RTOL1, RTOL2, |
$ 1, M, MINRGP, RTOL1, RTOL2, |
$ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ), |
$ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ), |
$ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ, |
$ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ, |
$ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO ) |
$ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO ) |
IF( IINFO.NE.0 ) THEN |
IF( IINFO.NE.0 ) THEN |
INFO = 20 + ABS( IINFO ) |
INFO = 20 + ABS( IINFO ) |
RETURN |
RETURN |
END IF |
END IF |
ELSE |
ELSE |
* DLARRE computes eigenvalues of the (shifted) root representation |
* DLARRE computes eigenvalues of the (shifted) root representation |
* ZLARRV returns the eigenvalues of the unshifted matrix. |
* ZLARRV returns the eigenvalues of the unshifted matrix. |
* However, if the eigenvectors are not desired by the user, we need |
* However, if the eigenvectors are not desired by the user, we need |
* to apply the corresponding shifts from DLARRE to obtain the |
* to apply the corresponding shifts from DLARRE to obtain the |
* eigenvalues of the original matrix. |
* eigenvalues of the original matrix. |
DO 20 J = 1, M |
DO 20 J = 1, M |
ITMP = IWORK( IINDBL+J-1 ) |
ITMP = IWORK( IINDBL+J-1 ) |
W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) ) |
W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) ) |
20 CONTINUE |
20 CONTINUE |
END IF |
END IF |
* |
* |
|
|
IF ( TRYRAC ) THEN |
IF ( TRYRAC ) THEN |
* Refine computed eigenvalues so that they are relatively accurate |
* Refine computed eigenvalues so that they are relatively accurate |
* with respect to the original matrix T. |
* with respect to the original matrix T. |
IBEGIN = 1 |
IBEGIN = 1 |
WBEGIN = 1 |
WBEGIN = 1 |
DO 39 JBLK = 1, IWORK( IINDBL+M-1 ) |
DO 39 JBLK = 1, IWORK( IINDBL+M-1 ) |
IEND = IWORK( IINSPL+JBLK-1 ) |
IEND = IWORK( IINSPL+JBLK-1 ) |
IN = IEND - IBEGIN + 1 |
IN = IEND - IBEGIN + 1 |
WEND = WBEGIN - 1 |
WEND = WBEGIN - 1 |
* check if any eigenvalues have to be refined in this block |
* check if any eigenvalues have to be refined in this block |
36 CONTINUE |
36 CONTINUE |
IF( WEND.LT.M ) THEN |
IF( WEND.LT.M ) THEN |
IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN |
IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN |
WEND = WEND + 1 |
WEND = WEND + 1 |
GO TO 36 |
GO TO 36 |
|
END IF |
|
END IF |
|
IF( WEND.LT.WBEGIN ) THEN |
|
IBEGIN = IEND + 1 |
|
GO TO 39 |
END IF |
END IF |
END IF |
|
IF( WEND.LT.WBEGIN ) THEN |
|
IBEGIN = IEND + 1 |
|
GO TO 39 |
|
END IF |
|
|
|
OFFSET = IWORK(IINDW+WBEGIN-1)-1 |
OFFSET = IWORK(IINDW+WBEGIN-1)-1 |
IFIRST = IWORK(IINDW+WBEGIN-1) |
IFIRST = IWORK(IINDW+WBEGIN-1) |
ILAST = IWORK(IINDW+WEND-1) |
ILAST = IWORK(IINDW+WEND-1) |
RTOL2 = FOUR * EPS |
RTOL2 = FOUR * EPS |
CALL DLARRJ( IN, |
CALL DLARRJ( IN, |
$ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1), |
$ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1), |
$ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN), |
$ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN), |
$ WORK( INDERR+WBEGIN-1 ), |
$ WORK( INDERR+WBEGIN-1 ), |
$ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN, |
$ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN, |
$ TNRM, IINFO ) |
$ TNRM, IINFO ) |
IBEGIN = IEND + 1 |
IBEGIN = IEND + 1 |
WBEGIN = WEND + 1 |
WBEGIN = WEND + 1 |
39 CONTINUE |
39 CONTINUE |
ENDIF |
ENDIF |
* |
* |
* If matrix was scaled, then rescale eigenvalues appropriately. |
* If matrix was scaled, then rescale eigenvalues appropriately. |
* |
* |
IF( SCALE.NE.ONE ) THEN |
IF( SCALE.NE.ONE ) THEN |
CALL DSCAL( M, ONE / SCALE, W, 1 ) |
CALL DSCAL( M, ONE / SCALE, W, 1 ) |
|
END IF |
END IF |
END IF |
* |
* |
* If eigenvalues are not in increasing order, then sort them, |
* If eigenvalues are not in increasing order, then sort them, |
* possibly along with eigenvectors. |
* possibly along with eigenvectors. |
* |
* |
IF( NSPLIT.GT.1 ) THEN |
IF( NSPLIT.GT.1 .OR. N.EQ.2 ) THEN |
IF( .NOT. WANTZ ) THEN |
IF( .NOT. WANTZ ) THEN |
CALL DLASRT( 'I', M, W, IINFO ) |
CALL DLASRT( 'I', M, W, IINFO ) |
IF( IINFO.NE.0 ) THEN |
IF( IINFO.NE.0 ) THEN |