ZHPGVX(1) LAPACK driver routine (version 3.2) ZHPGVX(1)NAME
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
SYNOPSIS
SUBROUTINE ZHPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU,
ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL,
INFO )
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, ITYPE, IU, LDZ, M, N
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION RWORK( * ), W( * )
COMPLEX*16 AP( * ), BP( * ), WORK( * ), Z( LDZ, * )
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.
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 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.
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 = 'U', AP(i +
(j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
(j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, the contents
of AP are destroyed.
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 = 'U', BP(i +
(j-1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', 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.
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 approxi‐
mate 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 pre‐
cision. 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('S'), not zero. If
this routine returns with INFO>0, indicating that some eigen‐
vectors 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 ei‐
genvalues in ascending order.
Z (output) COMPLEX*16 array, dimension (LDZ, N)
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 ei‐
genvalues, with the i-th column of Z holding the eigenvector
associated with W(i). The eigenvectors are normalized as fol‐
lows: 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) col‐
umns 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) COMPLEX*16 array, dimension (2*N)
RWORK (workspace) DOUBLE PRECISION array, dimension (7*N)
IWORK (workspace) INTEGER array, dimension (5*N)
IFAIL (output) INTEGER array, dimension (N)
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
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 illegal value
> 0: ZPPTRF or ZHPEVX returned an error code:
<= 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.
FURTHER DETAILS
Based on contributions by
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
LAPACK driver routine (version 3November 2008 ZHPGVX(1)