zhbgvx(3P) Sun Performance Library zhbgvx(3P)NAMEzhbgvx - compute all the eigenvalues, and optionally, the eigenvectors
of a complex generalized Hermitian-definite banded eigenproblem, of the
form A*x=(lambda)*B*x
SYNOPSIS
SUBROUTINE ZHBGVX(JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB,
Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK,
IFAIL, INFO)
CHARACTER * 1 JOBZ, RANGE, UPLO
DOUBLE COMPLEX AB(LDAB,*), BB(LDBB,*), Q(LDQ,*), Z(LDZ,*), WORK(*)
INTEGER N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
INTEGER IWORK(*), IFAIL(*)
DOUBLE PRECISION VL, VU, ABSTOL
DOUBLE PRECISION W(*), RWORK(*)
SUBROUTINE ZHBGVX_64(JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
IWORK, IFAIL, INFO)
CHARACTER * 1 JOBZ, RANGE, UPLO
DOUBLE COMPLEX AB(LDAB,*), BB(LDBB,*), Q(LDQ,*), Z(LDZ,*), WORK(*)
INTEGER*8 N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
INTEGER*8 IWORK(*), IFAIL(*)
DOUBLE PRECISION VL, VU, ABSTOL
DOUBLE PRECISION W(*), RWORK(*)
F95 INTERFACE
SUBROUTINE HBGVX(JOBZ, RANGE, UPLO, [N], KA, KB, AB, [LDAB], BB,
[LDBB], Q, [LDQ], VL, VU, IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK],
[RWORK], [IWORK], IFAIL, [INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: AB, BB, Q, Z
INTEGER :: N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
INTEGER, DIMENSION(:) :: IWORK, IFAIL
REAL(8) :: VL, VU, ABSTOL
REAL(8), DIMENSION(:) :: W, RWORK
SUBROUTINE HBGVX_64(JOBZ, RANGE, UPLO, [N], KA, KB, AB, [LDAB], BB,
[LDBB], Q, [LDQ], VL, VU, IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK],
[RWORK], [IWORK], IFAIL, [INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: AB, BB, Q, Z
INTEGER(8) :: N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
INTEGER(8), DIMENSION(:) :: IWORK, IFAIL
REAL(8) :: VL, VU, ABSTOL
REAL(8), DIMENSION(:) :: W, RWORK
C INTERFACE
#include <sunperf.h>
void zhbgvx(char jobz, char range, char uplo, int n, int ka, int kb,
doublecomplex *ab, int ldab, doublecomplex *bb, int ldbb,
doublecomplex *q, int ldq, double vl, double vu, int il, int
iu, double abstol, int *m, double *w, doublecomplex *z, int
ldz, int *ifail, int *info);
void zhbgvx_64(char jobz, char range, char uplo, long n, long ka, long
kb, doublecomplex *ab, long ldab, doublecomplex *bb, long
ldbb, doublecomplex *q, long ldq, double vl, double vu, long
il, long iu, double abstol, long *m, double *w, doublecomplex
*z, long ldz, long *ifail, long *info);
PURPOSEzhbgvx computes all the eigenvalues, and optionally, the eigenvectors
of a complex generalized Hermitian-definite banded eigenproblem, of the
form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian and
banded, and B is also positive definite. Eigenvalues and eigenvectors
can be selected by specifying either all eigenvalues, a range of values
or a range of indices for the desired eigenvalues.
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 triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored.
N (input) The order of the matrices A and B. N >= 0.
KA (input)
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KA >= 0.
KB (input)
The number of superdiagonals of the matrix B if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KB >= 0.
AB (input/output)
On entry, the upper or lower triangle of the Hermitian band
matrix A, stored in the first ka+1 rows of the array. The j-
th column of A is stored in the j-th column of the array AB
as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for
max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
j<=i<=min(n,j+ka).
On exit, the contents of AB are destroyed.
LDAB (input)
The leading dimension of the array AB. LDAB >= KA+1.
BB (input/output)
On entry, the upper or lower triangle of the Hermitian band
matrix B, stored in the first kb+1 rows of the array. The j-
th column of B is stored in the j-th column of the array BB
as follows: if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for
max(1,j-kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for
j<=i<=min(n,j+kb).
On exit, the factor S from the split Cholesky factorization B
= S**H*S, as returned by CPBSTF.
LDBB (input)
The leading dimension of the array BB. LDBB >= KB+1.
Q (output)
If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x
= (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and
consequently C to tridiagonal form. If JOBZ = 'N', the array
Q is not referenced.
LDQ (input)
The leading dimension of the array Q. If JOBZ = 'N', LDQ >=
1. If JOBZ = 'V', LDQ >= 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 AP to tridiagonal form.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*DLAMCH('S'), not zero.
If this routine returns with INFO>0, indicating that some
eigenvectors did not converge, try setting ABSTOL to
2*DLAMCH('S').
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)
If INFO = 0, the eigenvalues in ascending order.
Z (output)
If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
eigenvectors, with the i-th column of Z holding the eigenvec‐
tor associated with W(i). The eigenvectors are normalized so
that Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced.
LDZ (input)
The leading dimension of the array Z. LDZ >= 1, and if JOBZ
= 'V', LDZ >= N.
WORK (workspace)
dimension(N)
RWORK (workspace)
dimension(7*N)
IWORK (workspace)
dimension(5*N)
IFAIL (output)
If JOBZ = 'V', then if INFO = 0, the first M elements of
IFAIL are zero. If INFO > 0, then IFAIL contains the indices
of the eigenvectors that failed to converge. If JOBZ = 'N',
then IFAIL is not referenced.
INFO (output)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is:
<= N: then i eigenvectors failed to converge. Their indices
are stored in array IFAIL. > N: if INFO = N + i, for 1 <=
i <= N, then CPBSTF
returned INFO = i: B is not positive definite. The factor‐
ization of B could not be completed and no eigenvalues or
eigenvectors were computed.
FURTHER DETAILS
Based on contributions by
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
6 Mar 2009 zhbgvx(3P)
