Contents
ssygvx - compute selected eigenvalues, and optionally,
eigenvectors of a real generalized symmetric-definite eigen-
problem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or
B*A*x=(lambda)*x
SUBROUTINE SSYGVX(ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL,
VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL,
INFO)
CHARACTER * 1 JOBZ, RANGE, UPLO
INTEGER ITYPE, N, LDA, LDB, IL, IU, M, LDZ, LWORK, INFO
INTEGER IWORK(*), IFAIL(*)
REAL VL, VU, ABSTOL
REAL A(LDA,*), B(LDB,*), W(*), Z(LDZ,*), WORK(*)
SUBROUTINE SSYGVX_64(ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL,
VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL,
INFO)
CHARACTER * 1 JOBZ, RANGE, UPLO
INTEGER*8 ITYPE, N, LDA, LDB, IL, IU, M, LDZ, LWORK, INFO
INTEGER*8 IWORK(*), IFAIL(*)
REAL VL, VU, ABSTOL
REAL A(LDA,*), B(LDB,*), W(*), Z(LDZ,*), WORK(*)
F95 INTERFACE
SUBROUTINE SYGVX(ITYPE, JOBZ, RANGE, UPLO, [N], A, [LDA], B, [LDB],
VL, VU, IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK], [LWORK], [IWORK],
IFAIL, [INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
INTEGER :: ITYPE, N, LDA, LDB, IL, IU, M, LDZ, LWORK, INFO
INTEGER, DIMENSION(:) :: IWORK, IFAIL
REAL :: VL, VU, ABSTOL
REAL, DIMENSION(:) :: W, WORK
REAL, DIMENSION(:,:) :: A, B, Z
SUBROUTINE SYGVX_64(ITYPE, JOBZ, RANGE, UPLO, [N], A, [LDA], B, [LDB],
VL, VU, IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK], [LWORK], [IWORK],
IFAIL, [INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
INTEGER(8) :: ITYPE, N, LDA, LDB, IL, IU, M, LDZ, LWORK,
INFO
INTEGER(8), DIMENSION(:) :: IWORK, IFAIL
REAL :: VL, VU, ABSTOL
REAL, DIMENSION(:) :: W, WORK
REAL, DIMENSION(:,:) :: A, B, Z
C INTERFACE
#include <sunperf.h>
void ssygvx(int itype, char jobz, char range, char uplo, int
n, float *a, int lda, float *b, int ldb, float vl,
float vu, int il, int iu, float abstol, int *m,
float *w, float *z, int ldz, int *ifail, int
*info);
void ssygvx_64(long itype, char jobz, char range, char uplo,
long n, float *a, long lda, float *b, long ldb,
float vl, float vu, long il, long iu, float
abstol, long *m, float *w, float *z, long ldz,
long *ifail, long *info);
ssygvx computes selected eigenvalues, and optionally, eigen-
vectors of a real generalized symmetric-definite eigenprob-
lem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or
B*A*x=(lambda)*x. Here A and B are assumed to be symmetric
and B is also positive definite. Eigenvalues and eigenvec-
tors can be selected by specifying either a range of values
or a range of indices for the desired eigenvalues.
ITYPE (input)
Specifies the problem type to be solved:
= 1: A*x = (lambda)*B*x
= 2: A*B*x = (lambda)*x
= 3: B*A*x = (lambda)*x
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 and B are stored;
= 'L': Lower triangle of A and B are stored.
N (input) The order of the matrix pencil (A,B). N >= 0.
A (input/output)
On entry, the symmetric 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).
B (input/output)
On entry, the symmetric matrix B. If UPLO = 'U',
the leading N-by-N upper triangular part of B con-
tains the upper triangular part of the matrix B.
If UPLO = 'L', the leading N-by-N lower triangular
part of B contains the lower triangular part of
the matrix B.
On exit, if INFO <= N, the part of B containing
the matrix is overwritten by the triangular factor
U or L from the Cholesky factorization B = U**T*U
or B = L*L**T.
LDB (input)
The leading dimension of the array B. LDB >=
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*DLAMCH('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)
On normal exit, the first M elements contain the
selected eigenvalues in ascending order.
Z (input) If JOBZ = 'N', then Z is not referenced. 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 asso-
ciated with W(i). The eigenvectors are normalized
as follows: if ITYPE = 1 or 2, Z**T*B*Z = I; if
ITYPE = 3, Z**T*inv(B)*Z = I.
If an eigenvector fails to converge, then that
column of Z contains the latest approximation to
the eigenvector, and the index of the eigenvector
is returned in IFAIL. 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).
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input)
The length of the array WORK. LWORK >=
max(1,8*N). For optimal efficiency, LWORK >=
(NB+3)*N, where NB is the blocksize for SSYTRD
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.
IWORK (workspace)
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 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: SPOTRF or SSYEVX returned an error code:
<= N: if INFO = i, SSYEVX failed to converge; i
eigenvectors failed to converge. Their indices
are stored in array IFAIL. > N: 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