DSYGVD(1) LAPACK driver routine (version 3.2) DSYGVD(1)NAME
DSYGVD - computes all the eigenvalues, and optionally, the eigenvectors
of a real generalized symmetric-definite eigenproblem, of the form
A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x
SYNOPSIS
SUBROUTINE DSYGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
LWORK, IWORK, LIWORK, INFO )
CHARACTER JOBZ, UPLO
INTEGER INFO, ITYPE, LDA, LDB, LIWORK, LWORK, N
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( *
)
PURPOSE
DSYGVD computes all the eigenvalues, and optionally, the eigenvectors
of a real generalized symmetric-definite eigenproblem, 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. If eigen‐
vectors 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 dig‐
its, but we know of none.
ARGUMENTS
ITYPE (input) INTEGER
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) CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
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.
A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains 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, if JOBZ = 'V', then if INFO = 0, A contains
the matrix Z of eigenvectors. 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 JOBZ = 'N', then on exit the upper tri‐
angle (if UPLO='U') or the lower triangle (if UPLO='L') of A,
including the diagonal, is destroyed.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
On entry, the symmetric matrix B. If UPLO = 'U', the leading
N-by-N upper triangular part of B contains 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) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
W (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
WORK (workspace/output) DOUBLE PRECISION array, dimension
(MAX(1,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 >= 2*N+1. If JOBZ
= 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2. If LWORK = -1,
then a workspace query is assumed; the routine only calculates
the optimal sizes of the WORK and IWORK arrays, returns these
values as the first entries of the WORK and IWORK arrays, and
no error message related to LWORK or LIWORK is issued by
XERBLA.
IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK (input) INTEGER
The dimension of the array IWORK. If N <= 1,
LIWORK >= 1. If JOBZ = 'N' and 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 sizes of the WORK and IWORK arrays, returns these val‐
ues as the first entries of the WORK and IWORK arrays, and no
error message related to LWORK or LIWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: DPOTRF or DSYEVD returned an error code:
<= N: if INFO = i and JOBZ = 'N', then the algorithm failed to
converge; i off-diagonal elements of an intermediate tridiago‐
nal form did not converge to zero; if INFO = i and JOBZ = 'V',
then the algorithm failed to compute an eigenvalue while work‐
ing on the submatrix lying in rows and columns INFO/(N+1)
through mod(INFO,N+1); > N: if INFO = N + i, for 1 <= i <= N,
then the leading minor of order i of B is not positive defi‐
nite. The factorization of B could not be completed and no ei‐
genvalues or eigenvectors were computed.
FURTHER DETAILS
Based on contributions by
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA Modi‐
fied so that no backsubstitution is performed if DSYEVD fails to con‐
verge (NEIG in old code could be greater than N causing out of bounds
reference to A - reported by Ralf Meyer). Also corrected the descrip‐
tion of INFO and the test on ITYPE. Sven, 16 Feb 05.
LAPACK driver routine (version 3November 2008 DSYGVD(1)
