Contents
cheevr - compute selected eigenvalues and, optionally,
eigenvectors of a complex Hermitian tridiagonal matrix T
SUBROUTINE CHEEVR(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
COMPLEX A(LDA,*), Z(LDZ,*), WORK(*)
INTEGER N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK, INFO
INTEGER ISUPPZ(*), IWORK(*)
REAL VL, VU, ABSTOL
REAL W(*), RWORK(*)
SUBROUTINE CHEEVR_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
COMPLEX A(LDA,*), Z(LDZ,*), WORK(*)
INTEGER*8 N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK,
INFO
INTEGER*8 ISUPPZ(*), IWORK(*)
REAL VL, VU, ABSTOL
REAL 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, DIMENSION(:) :: WORK
COMPLEX, DIMENSION(:,:) :: A, Z
INTEGER :: N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK,
INFO
INTEGER, DIMENSION(:) :: ISUPPZ, IWORK
REAL :: VL, VU, ABSTOL
REAL, 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, DIMENSION(:) :: WORK
COMPLEX, DIMENSION(:,:) :: A, Z
INTEGER(8) :: N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK,
INFO
INTEGER(8), DIMENSION(:) :: ISUPPZ, IWORK
REAL :: VL, VU, ABSTOL
REAL, DIMENSION(:) :: W, RWORK
C INTERFACE
#include <sunperf.h>
void cheevr(char jobz, char range, char uplo, int n, complex
*a, int lda, float vl, float vu, int il, int iu,
float abstol, int *m, float *w, complex *z, int
ldz, int *isuppz, int *info);
void cheevr_64(char jobz, char range, char uplo, long n,
complex *a, long lda, float vl, float vu, long il,
long iu, float abstol, long *m, float *w, complex
*z, long ldz, long *isuppz, long *info);
cheevr computes selected eigenvalues and, optionally, eigen-
vectors of a complex Hermitian tridiagonal matrix T. Eigen-
values 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 sym-
metric 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.
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 con-
tains the upper triangular 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 diagonal, 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 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'.
IU (input)
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)
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 tri-
diagonal 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 furu-
tre releases will. See J. Barlow and J. Demmel,
"Computing Accurate Eigensystems of Scaled Diago-
nally Dominant Matrices", LAPACK Working Note #7,
for a discussion of which matrices define their
eigenvalues 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 eigen-
values 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 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 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 indicating 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
CHETRD and for CUNMTR 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 minimal) LRWORK.
LRWORK (input)
The length of the array RWORK. LRWORK >=
max(1,24*N).
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 (workspace/output)
On exit, if INFO = 0, IWORK(1) returns the optimal
(and minimal) LIWORK.
LIWORK (input)
The dimension of the array IWORK. LIWORK >=
max(1,10*N).
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)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal 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