Contents
ssbgvx - compute selected eigenvalues, and optionally,
eigenvectors of a real generalized symmetric-definite banded
eigenproblem, of the form A*x=(lambda)*B*x
SUBROUTINE SSBGVX(JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB,
Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
INFO)
CHARACTER * 1 JOBZ, RANGE, UPLO
INTEGER N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
INTEGER IWORK(*), IFAIL(*)
REAL VL, VU, ABSTOL
REAL AB(LDAB,*), BB(LDBB,*), Q(LDQ,*), W(*), Z(LDZ,*),
WORK(*)
SUBROUTINE SSBGVX_64(JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
IFAIL, INFO)
CHARACTER * 1 JOBZ, RANGE, UPLO
INTEGER*8 N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
INTEGER*8 IWORK(*), IFAIL(*)
REAL VL, VU, ABSTOL
REAL AB(LDAB,*), BB(LDBB,*), Q(LDQ,*), W(*), Z(LDZ,*),
WORK(*)
F95 INTERFACE
SUBROUTINE SBGVX(JOBZ, RANGE, UPLO, [N], KA, KB, AB, [LDAB], BB,
[LDBB], Q, [LDQ], VL, VU, IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK],
[IWORK], IFAIL, [INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
INTEGER :: N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
INTEGER, DIMENSION(:) :: IWORK, IFAIL
REAL :: VL, VU, ABSTOL
REAL, DIMENSION(:) :: W, WORK
REAL, DIMENSION(:,:) :: AB, BB, Q, Z
SUBROUTINE SBGVX_64(JOBZ, RANGE, UPLO, [N], KA, KB, AB, [LDAB], BB,
[LDBB], Q, [LDQ], VL, VU, IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK],
[IWORK], IFAIL, [INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
INTEGER(8) :: N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ,
INFO
INTEGER(8), DIMENSION(:) :: IWORK, IFAIL
REAL :: VL, VU, ABSTOL
REAL, DIMENSION(:) :: W, WORK
REAL, DIMENSION(:,:) :: AB, BB, Q, Z
C INTERFACE
#include <sunperf.h>
void ssbgvx(char jobz, char range, char uplo, int n, int ka,
int kb, float *ab, int ldab, float *bb, int ldbb,
float *q, int ldq, float vl, float vu, int il, int
iu, float abstol, int *m, float *w, float *z, int
ldz, int *ifail, int *info);
void ssbgvx_64(char jobz, char range, char uplo, long n,
long ka, long kb, float *ab, long ldab, float *bb,
long ldbb, float *q, long ldq, float vl, float vu,
long il, long iu, float abstol, long *m, float *w,
float *z, long ldz, long *ifail, long *info);
ssbgvx computes selected eigenvalues, and optionally, eigen-
vectors of a real generalized symmetric-definite banded
eigenproblem, of the form A*x=(lambda)*B*x. Here A and B
are assumed to be symmetric and banded, and B is also posi-
tive 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.
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 sym-
metric 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 sym-
metric 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(ka+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 fac-
torization B = S**T*S, as returned by SPBSTF.
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 tri-
diagonal 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)
See the description of VL.
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)
See the description of IL.
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.
Eigenvalues will be computed most accurately when
ABSTOL 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 ABSTOL to 2*SLAMCH('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 (input) If JOBZ = 'V', then if INFO = 0, Z contains the
matrix Z of eigenvectors, with the i-th column of
Z holding the eigenvector associated with W(i).
The eigenvectors are normalized so Z**T*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 >= max(1,N).
WORK (workspace)
dimension(7*N)
IWORK (workspace/output)
dimension(5*N)
IFAIL (output)
If JOBZ = 'V', then if INFO = 0, the first M ele-
ments of IFAIL are zero. If INFO > 0, then IFAIL
contains the indices of the eigenvalues 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
<= N: if INFO = i, then i eigenvectors failed to
converge. Their indices are stored in IFAIL. > N
: SPBSTF returned an error code; i.e., if INFO = N
+ i, for 1 <= i <= N, then the leading minor of
order i of B is not positive definite. The fac-
torization of B could not be completed and no
eigenvalues or eigenvectors were computed.
Based on contributions by
Mark Fahey, Department of Mathematics, Univ. of Kentucky,
USA