DSYGVX - compute selected eigenvalues, and optionally,
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 DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B,
LDB, VL, VU, IL, IU, ABSTOL, M, W, Z,
LDZ, WORK, LWORK, IWORK, IFAIL, INFO )
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ,
LWORK, M, N
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W(
* ), WORK( * ), Z( LDZ, * )
PURPOSE
DSYGVX computes selected eigenvalues, and optionally,
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.
Eigenvalues and eigenvectors can be selected by specifying
either a range of values or a range of indices for the
desired eigenvalues.
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.
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.
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A and B are stored;
= 'L': Lower triangle of A and B are stored.
The order of the matrix pencil (A,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 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) INTEGER
The leading dimension of the array A. LDA >=
max(1,N).
B (input/output) DOUBLE PRECISION array, dimension
(LDA, N)
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) INTEGER
The leading dimension of the array B. LDB >=
max(1,N).
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 refer
enced if RANGE = 'A' or 'V'.
ABSTOL (input) DOUBLE PRECISION
The absolute error tolerance for the eigenvalues.
An approximate eigenvalue is accepted as converged
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)
On normal exit, the first M elements contain the
selected eigenvalues in ascending order.
Z (output) DOUBLE PRECISION array, dimension
(LDZ, max(1,M))
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) 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.
The length of the array WORK. LWORK >=
max(1,8*N). For optimal efficiency, LWORK >=
(NB+3)*N, where NB is the blocksize for DSYTRD
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) INTEGER array, dimension (5*N)
IFAIL (output) INTEGER array, dimension (N)
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) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille
gal value
> 0: DPOTRF or DSYEVX returned an error code:
<= N: if INFO = i, DSYEVX 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.
FURTHER DETAILS
Based on contributions by
Mark Fahey, Department of Mathematics, Univ. of Ken
tucky, USA
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