zheevr(3P) Sun Performance Library zheevr(3P)NAMEzheevr - compute selected eigenvalues and, optionally, eigenvectors of
a complex Hermitian tridiagonal matrix T
SYNOPSIS
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(*)
F95 INTERFACE
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
C INTERFACE
#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);
PURPOSEzheevr 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, ZHEEVR 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 param‐
eter ABSTOL.
For more details, see "A new O(n^2) algorithm for the symmetric tridi‐
agonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer
Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May
1997.
Note 1 : ZHEEVR calls CSTEGR when the full spectrum is requested on
machines which conform to the ieee-754 floating point standard. ZHEEVR
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.
ARGUMENTS
JOBZ (input)
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
RANGE (input)
= '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.
UPLO (input)
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) The order of the matrix A. N >= 0.
A (input/output)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper triangu‐
lar part of the matrix A. If UPLO = 'L', the leading N-by-N
lower triangular part of A contains the lower triangular part
of the matrix A. On exit, the lower triangle (if UPLO='L')
or the upper triangle (if UPLO='U') of A, including the diag‐
onal, is destroyed.
LDA (input)
The leading dimension of the array A. LDA >= max(1,N).
VL (input)
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'.
VU (input)
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)
If RANGE='I', the indices (in ascending order) of the small‐
est 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'.
IU (input)
If RANGE='I', the indices (in ascending order) of the small‐
est 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)
The absolute error tolerance for the eigenvalues. An approx‐
imate eigenvalue is accepted as converged when it is deter‐
mined 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 DLAMCH(
'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 guaran‐
tees 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 eigenval‐
ues to high relative accuracy.
M (output)
The total number of eigenvalues found. 0 <= M <= N. If
RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
W (output)
The first M elements contain the selected eigenvalues in
ascending order.
Z (output)
If JOBZ = 'V', then if INFO = 0, the first M columns of Z
contain the orthonormal eigenvectors of the matrix A corre‐
sponding 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 an upper bound must be used.
LDZ (input)
The leading dimension of the array Z. LDZ >= 1, and if JOBZ
= 'V', LDZ >= max(1,N).
ISUPPZ (output)
The support of the eigenvectors in Z, i.e., the indices indi‐
cating the nonzero elements in Z. The i-th eigenvector is
nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i
).
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The length of the array WORK. LWORK >= max(1,2*N). For
optimal efficiency, LWORK >= (NB+1)*N, where NB is the max of
the blocksize for ZHETRD and for ZUNMTR as returned by
ILAENV.
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 (workspace)
On exit, if INFO = 0, RWORK(1) returns the optimal (and mini‐
mal) LRWORK.
LRWORK (input)
The length of the array RWORK. LRWORK >= max(1,24*N).
If LRWORK = -1, then a workspace query is assumed; the rou‐
tine 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 (workspace/output)
On exit, if INFO = 0, IWORK(1) returns the optimal (and mini‐
mal) LIWORK.
LIWORK (input)
The dimension of the array IWORK. LIWORK >= max(1,10*N).
If LIWORK = -1, then a workspace query is assumed; the rou‐
tine 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)
= 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
6 Mar 2009 zheevr(3P)
