.TH ZHPGVX 1 "November 2006" " LAPACK driver routine (version 3.1) " " LAPACK driver routine (version 3.1) "
.SH NAME
ZHPGVX - selected eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x
.SH SYNOPSIS
.TP 19
SUBROUTINE ZHPGVX(
ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
IWORK, IFAIL, INFO )
.TP 19
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CHARACTER
JOBZ, RANGE, UPLO
.TP 19
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INTEGER
IL, INFO, ITYPE, IU, LDZ, M, N
.TP 19
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DOUBLE
PRECISION ABSTOL, VL, VU
.TP 19
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INTEGER
IFAIL( * ), IWORK( * )
.TP 19
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DOUBLE
PRECISION RWORK( * ), W( * )
.TP 19
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COMPLEX*16
AP( * ), BP( * ), WORK( * ), Z( LDZ, * )
.SH PURPOSE
ZHPGVX computes selected eigenvalues and, optionally, eigenvectors
of a complex generalized Hermitian-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 Hermitian, stored in packed format, 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.
.br
.SH ARGUMENTS
.TP 8
ITYPE (input) INTEGER
Specifies the problem type to be solved:
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= 1: A*x = (lambda)*B*x
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= 2: A*B*x = (lambda)*x
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= 3: B*A*x = (lambda)*x
.TP 8
JOBZ (input) CHARACTER*1
.br
= \(aqN\(aq: Compute eigenvalues only;
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= \(aqV\(aq: Compute eigenvalues and eigenvectors.
.TP 8
RANGE (input) CHARACTER*1
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= \(aqA\(aq: all eigenvalues will be found;
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= \(aqV\(aq: all eigenvalues in the half-open interval (VL,VU]
will be found;
= \(aqI\(aq: the IL-th through IU-th eigenvalues will be found.
.TP 8
UPLO (input) CHARACTER*1
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= \(aqU\(aq: Upper triangles of A and B are stored;
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= \(aqL\(aq: Lower triangles of A and B are stored.
.TP 8
N (input) INTEGER
The order of the matrices A and B. N >= 0.
.TP 8
AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermitian matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = \(aqU\(aq, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = \(aqL\(aq, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
On exit, the contents of AP are destroyed.
.TP 8
BP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermitian matrix
B, packed columnwise in a linear array. The j-th column of B
is stored in the array BP as follows:
if UPLO = \(aqU\(aq, BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
if UPLO = \(aqL\(aq, BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
On exit, the triangular factor U or L from the Cholesky
factorization B = U**H*U or B = L*L**H, in the same storage
format as B.
.TP 8
VL (input) DOUBLE PRECISION
VU (input) DOUBLE PRECISION
If RANGE=\(aqV\(aq, the lower and upper bounds of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = \(aqA\(aq or \(aqI\(aq.
.TP 8
IL (input) INTEGER
IU (input) INTEGER
If RANGE=\(aqI\(aq, 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 = \(aqA\(aq or \(aqV\(aq.
.TP 8
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 AP to tridiagonal form.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*DLAMCH(\(aqS\(aq), not zero.
If this routine returns with INFO>0, indicating that some
eigenvectors did not converge, try setting ABSTOL to
2*DLAMCH(\(aqS\(aq).
.TP 8
M (output) INTEGER
The total number of eigenvalues found. 0 <= M <= N.
If RANGE = \(aqA\(aq, M = N, and if RANGE = \(aqI\(aq, M = IU-IL+1.
.TP 8
W (output) DOUBLE PRECISION array, dimension (N)
On normal exit, the first M elements contain the selected
eigenvalues in ascending order.
.TP 8
Z (output) COMPLEX*16 array, dimension (LDZ, N)
If JOBZ = \(aqN\(aq, then Z is not referenced.
If JOBZ = \(aqV\(aq, 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 associated with W(i).
The eigenvectors are normalized as follows:
if ITYPE = 1 or 2, Z**H*B*Z = I;
if ITYPE = 3, Z**H*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 = \(aqV\(aq, the exact value of M
is not known in advance and an upper bound must be used.
.TP 8
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = \(aqV\(aq, LDZ >= max(1,N).
.TP 8
WORK (workspace) COMPLEX*16 array, dimension (2*N)
.TP 8
RWORK (workspace) DOUBLE PRECISION array, dimension (7*N)
.TP 8
IWORK (workspace) INTEGER array, dimension (5*N)
.TP 8
IFAIL (output) INTEGER array, dimension (N)
If JOBZ = \(aqV\(aq, then if INFO = 0, the first M elements of
IFAIL are zero. If INFO > 0, then IFAIL contains the
indices of the eigenvectors that failed to converge.
If JOBZ = \(aqN\(aq, then IFAIL is not referenced.
.TP 8
INFO (output) INTEGER
= 0: successful exit
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< 0: if INFO = -i, the i-th argument had an illegal value
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> 0: ZPPTRF or ZHPEVX returned an error code:
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<= N: if INFO = i, ZHPEVX 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 factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.
.SH FURTHER DETAILS
Based on contributions by
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Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA