Contents
zhbevx - compute selected eigenvalues and, optionally,
eigenvectors of a complex Hermitian band matrix A
SUBROUTINE ZHBEVX(JOBZ, RANGE, UPLO, N, KD, A, LDA, Q, LDQ, VL,
VU, IL, IU, ABTOL, NFOUND, W, Z, LDZ, WORK, WORK2, IWORK3, IFAIL,
INFO)
CHARACTER * 1 JOBZ, RANGE, UPLO
DOUBLE COMPLEX A(LDA,*), Q(LDQ,*), Z(LDZ,*), WORK(*)
INTEGER N, KD, LDA, LDQ, IL, IU, NFOUND, LDZ, INFO
INTEGER IWORK3(*), IFAIL(*)
DOUBLE PRECISION VL, VU, ABTOL
DOUBLE PRECISION W(*), WORK2(*)
SUBROUTINE ZHBEVX_64(JOBZ, RANGE, UPLO, N, KD, A, LDA, Q, LDQ, VL,
VU, IL, IU, ABTOL, NFOUND, W, Z, LDZ, WORK, WORK2, IWORK3, IFAIL,
INFO)
CHARACTER * 1 JOBZ, RANGE, UPLO
DOUBLE COMPLEX A(LDA,*), Q(LDQ,*), Z(LDZ,*), WORK(*)
INTEGER*8 N, KD, LDA, LDQ, IL, IU, NFOUND, LDZ, INFO
INTEGER*8 IWORK3(*), IFAIL(*)
DOUBLE PRECISION VL, VU, ABTOL
DOUBLE PRECISION W(*), WORK2(*)
F95 INTERFACE
SUBROUTINE HBEVX(JOBZ, RANGE, UPLO, [N], KD, A, [LDA], Q, [LDQ],
VL, VU, IL, IU, ABTOL, [NFOUND], W, Z, [LDZ], [WORK], [WORK2],
[IWORK3], IFAIL, [INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, Q, Z
INTEGER :: N, KD, LDA, LDQ, IL, IU, NFOUND, LDZ, INFO
INTEGER, DIMENSION(:) :: IWORK3, IFAIL
REAL(8) :: VL, VU, ABTOL
REAL(8), DIMENSION(:) :: W, WORK2
SUBROUTINE HBEVX_64(JOBZ, RANGE, UPLO, [N], KD, A, [LDA], Q, [LDQ],
VL, VU, IL, IU, ABTOL, [NFOUND], W, Z, [LDZ], [WORK], [WORK2],
[IWORK3], IFAIL, [INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, Q, Z
INTEGER(8) :: N, KD, LDA, LDQ, IL, IU, NFOUND, LDZ, INFO
INTEGER(8), DIMENSION(:) :: IWORK3, IFAIL
REAL(8) :: VL, VU, ABTOL
REAL(8), DIMENSION(:) :: W, WORK2
C INTERFACE
#include <sunperf.h>
void zhbevx(char jobz, char range, char uplo, int n, int kd,
doublecomplex *a, int lda, doublecomplex *q, int
ldq, double vl, double vu, int il, int iu, double
abtol, int *nfound, double *w, doublecomplex *z,
int ldz, int *ifail, int *info);
void zhbevx_64(char jobz, char range, char uplo, long n,
long kd, doublecomplex *a, long lda, doublecomplex
*q, long ldq, double vl, double vu, long il, long
iu, double abtol, long *nfound, double *w, doub-
lecomplex *z, long ldz, long *ifail, long *info);
zhbevx computes selected eigenvalues and, optionally, eigen-
vectors of a complex Hermitian band matrix A. Eigenvalues
and eigenvectors can be selected by specifying either a
range of values or a range of indices for the desired eigen-
values.
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.
KD (input)
The number of superdiagonals of the matrix A if
UPLO = 'U', or the number of subdiagonals if UPLO
= 'L'. KD >= 0.
A (input/output)
On entry, the upper or lower triangle of the Her-
mitian band matrix A, stored in the first KD+1
rows of the array. The j-th column of A is stored
in the j-th column of the array A as follows: if
UPLO = 'U', A(kd+1+i-j,j) = A(i,j) for max(1,j-
kd)<=i<=j; if UPLO = 'L', A(1+i-j,j) = A(i,j)
for j<=i<=min(n,j+kd).
On exit, A is overwritten by values generated dur-
ing the reduction to tridiagonal form.
LDA (input)
The leading dimension of the array A. LDA >= KD +
1.
Q (output)
If JOBZ = 'V', the N-by-N unitary matrix used in
the reduction to tridiagonal form. If JOBZ = 'N',
the array Q is not referenced.
LDQ (input)
The leading dimension of the array Q. If JOBZ =
'V', then 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 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'.
ABTOL (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
ABTOL + EPS * max( |a|,|b| ) ,
where EPS is the machine precision. If ABTOL 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.
Eigenvalues will be computed most accurately when
ABTOL is set to twice the underflow threshold
2*SLAMCH('S'), not zero. If this routine returns
with INFO>0, indicating that some eigenvectors did
not converge, try setting ABTOL to 2*SLAMCH('S').
See "Computing Small Singular Values of Bidiagonal
Matrices with Guaranteed High Relative Accuracy,"
by Demmel and Kahan, LAPACK Working Note #3.
NFOUND (output)
The total number of eigenvalues found. 0 <=
NFOUND <= N. If RANGE = 'A', NFOUND = N, and if
RANGE = 'I', NFOUND = IU-IL+1.
W (output)
The first NFOUND elements contain the selected
eigenvalues in ascending order.
Z (input) If JOBZ = 'V', then if INFO = 0, the first NFOUND
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 an eigenvec-
tor fails to converge, then that column of Z con-
tains the latest approximation to the eigenvector,
and the index of the eigenvector is returned in
IFAIL. If JOBZ = 'N', then Z is not referenced.
Note: the user must ensure that at least
max(1,NFOUND) columns are supplied in the array Z;
if RANGE = 'V', the exact value of NFOUND 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).
WORK (workspace)
dimension(N)
WORK2 (workspace)
dimension(7*N)
IWORK3 (workspace)
dimension(5*N)
IFAIL (output)
If JOBZ = 'V', then if INFO = 0, the first NFOUND
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 ille-
gal value
> 0: if INFO = i, then i eigenvectors failed to
converge. Their indices are stored in array
IFAIL.