DSBGVX(3S)DSBGVX(3S)NAMEDSBGVX - compute selected eigenvalues, and optionally, eigenvectors of a
real generalized symmetric-definite banded eigenproblem, of the form
A*x=(lambda)*B*x
SYNOPSIS
SUBROUTINE DSBGVX( 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 JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M, N
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
W( * ), WORK( * ), Z( LDZ, * )
IMPLEMENTATION
These routines are part of the SCSL Scientific Library and can be loaded
using either the -lscs or the -lscs_mp option. The -lscs_mp option
directs the linker to use the multi-processor version of the library.
When linking to SCSL with -lscs or -lscs_mp, the default integer size is
4 bytes (32 bits). Another version of SCSL is available in which integers
are 8 bytes (64 bits). This version allows the user access to larger
memory sizes and helps when porting legacy Cray codes. It can be loaded
by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
only one of the two versions; 4-byte integer and 8-byte integer library
calls cannot be mixed.
PURPOSEDSBGVX computes selected eigenvalues, and optionally, eigenvectors 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 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) CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
RANGE (input) CHARACTER*1
= '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.
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DSBGVX(3S)DSBGVX(3S)
UPLO (input) CHARACTER*1
= 'U': Upper triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored.
N (input) INTEGER
The order of the matrices A and B. N >= 0.
KA (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U', or
the number of subdiagonals if UPLO = 'L'. KA >= 0.
KB (input) INTEGER
The number of superdiagonals of the matrix B if UPLO = 'U', or
the number of subdiagonals if UPLO = 'L'. KB >= 0.
AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
On entry, the upper or lower triangle of the symmetric 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) INTEGER
The leading dimension of the array AB. LDAB >= KA+1.
BB (input/output) DOUBLE PRECISION array, dimension (LDBB, N)
On entry, the upper or lower triangle of the symmetric 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 factorization B =
S**T*S, as returned by DPBSTF.
LDBB (input) INTEGER
The leading dimension of the array BB. LDBB >= KB+1.
Q (output) DOUBLE PRECISION array, dimension (LDQ, N)
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) INTEGER
The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1.
If JOBZ = 'V', LDQ >= max(1,N).
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DSBGVX(3S)DSBGVX(3S)
VL (input) DOUBLE PRECISION
VU (input) DOUBLE PRECISION 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) INTEGER
IU (input) INTEGER 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) DOUBLE PRECISION
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 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) INTEGER
The total number of eigenvalues found. 0 <= M <= N. If RANGE =
'A', M = N, and if RANGE = 'I', M = IU-IL+1.
W (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) DOUBLE PRECISION array, dimension (LDZ, N)
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) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
'V', LDZ >= max(1,N).
WORK (workspace/output) DOUBLE PRECISION array, dimension (7N)
IWORK (workspace/output) INTEGER array, dimension (5N)
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DSBGVX(3S)DSBGVX(3S)
IFAIL (input) INTEGER array, dimension (M)
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
eigenvalues that failed to converge. If JOBZ = 'N', then IFAIL
is not referenced.
INFO (output) INTEGER
= 0 : successful exit
< 0 : if INFO = -i, the i-th argument had an illegal value
<= N: if INFO = i, then i eigenvectors failed to converge. Their
indices are stored in IFAIL. > N : DPBSTF 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
factorization 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
SEE ALSOINTRO_LAPACK(3S), INTRO_SCSL(3S)
This man page is available only online.
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