DSBGVD(3S)DSBGVD(3S)NAME
DSBGVD - compute all the eigenvalues, and optionally, the eigenvectors of
a real generalized symmetric-definite banded eigenproblem, of the form
A*x=(lambda)*B*x
SYNOPSIS
SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ,
WORK, LWORK, IWORK, LIWORK, INFO )
CHARACTER JOBZ, UPLO
INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N
INTEGER IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), 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.
PURPOSE
DSBGVD computes all the eigenvalues, and optionally, the 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. If eigenvectors are desired, it uses a
divide and conquer algorithm.
The divide and conquer algorithm makes very mild assumptions about
floating point arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits which
subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
conceivably fail on hexadecimal or decimal machines without guard digits,
but we know of none.
ARGUMENTS
JOBZ (input) CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
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DSBGVD(3S)DSBGVD(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.
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.
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DSBGVD(3S)DSBGVD(3S)
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 (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. If N <= 1, LWORK
>= 1. If JOBZ = 'N' and N > 1, LWORK >= 3*N. If JOBZ = 'V' and
N > 1, LWORK >= 1 + 5*N + 2*N**2.
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/output) INTEGER array, dimension (LIWORK)
On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.
LIWORK (input) INTEGER
The dimension of the array IWORK. If JOBZ = 'N' or N <= 1,
LIWORK >= 1. If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
If LIWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the IWORK array, returns this
value as the first entry of the IWORK array, and no error message
related to LIWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is:
<= N: the algorithm failed to converge: i off-diagonal elements
of an intermediate tridiagonal form did not converge to zero; >
N: if INFO = N + i, for 1 <= i <= N, then DPBSTF
returned INFO = i: 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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