zheevr - compute selected eigenvalues and, optionally, eigenvectors of a complex Hermitian tridiagonal matrix T
SUBROUTINE ZHEEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, * ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, * LIWORK, INFO) CHARACTER * 1 JOBZ, RANGE, UPLO DOUBLE COMPLEX A(LDA,*), Z(LDZ,*), WORK(*) INTEGER N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK, INFO INTEGER ISUPPZ(*), IWORK(*) DOUBLE PRECISION VL, VU, ABSTOL DOUBLE PRECISION W(*), RWORK(*)
SUBROUTINE ZHEEVR_64( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, * ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, * LIWORK, INFO) CHARACTER * 1 JOBZ, RANGE, UPLO DOUBLE COMPLEX A(LDA,*), Z(LDZ,*), WORK(*) INTEGER*8 N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK, INFO INTEGER*8 ISUPPZ(*), IWORK(*) DOUBLE PRECISION VL, VU, ABSTOL DOUBLE PRECISION W(*), RWORK(*)
SUBROUTINE HEEVR( JOBZ, RANGE, UPLO, [N], A, [LDA], VL, VU, IL, IU, * ABSTOL, M, W, Z, [LDZ], ISUPPZ, [WORK], [LWORK], [RWORK], [LRWORK], * [IWORK], [LIWORK], [INFO]) CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO COMPLEX(8), DIMENSION(:) :: WORK COMPLEX(8), DIMENSION(:,:) :: A, Z INTEGER :: N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK, INFO INTEGER, DIMENSION(:) :: ISUPPZ, IWORK REAL(8) :: VL, VU, ABSTOL REAL(8), DIMENSION(:) :: W, RWORK
SUBROUTINE HEEVR_64( JOBZ, RANGE, UPLO, [N], A, [LDA], VL, VU, IL, * IU, ABSTOL, M, W, Z, [LDZ], ISUPPZ, [WORK], [LWORK], [RWORK], * [LRWORK], [IWORK], [LIWORK], [INFO]) CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO COMPLEX(8), DIMENSION(:) :: WORK COMPLEX(8), DIMENSION(:,:) :: A, Z INTEGER(8) :: N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK, INFO INTEGER(8), DIMENSION(:) :: ISUPPZ, IWORK REAL(8) :: VL, VU, ABSTOL REAL(8), DIMENSION(:) :: W, RWORK
#include <sunperf.h>
void zheevr(char jobz, char range, char uplo, int n, doublecomplex *a, int lda, double vl, double vu, int il, int iu, double abstol, int *m, double *w, doublecomplex *z, int ldz, int *isuppz, int *info);
void zheevr_64(char jobz, char range, char uplo, long n, doublecomplex *a, long lda, double vl, double vu, long il, long iu, double abstol, long *m, double *w, doublecomplex *z, long ldz, long *isuppz, long *info);
zheevr computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian 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, CHEEVR calls CSTEGR to compute the
eigenspectrum using Relatively Robust Representations. CSTEGR 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 : CHEEVR calls CSTEGR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. CHEEVR calls SSTEBZ and CSTEIN on non-ieee machines and
when partial spectrum requests are made.
Normal execution of CSTEGR 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.
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
= '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.
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
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 furutre 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.
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.
WORK(1) returns the optimal LWORK.
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.
RWORK(1) returns the optimal
(and minimal) LRWORK.
If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the RWORK array, and no error message related to LRWORK is issued by XERBLA.
IWORK(1) returns the optimal
(and minimal) LIWORK.
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.
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: Internal error
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