CHPGVD - compute all the eigenvalues and, optionally, the
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 CHPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z,
LDZ, WORK, LWORK, RWORK, LRWORK, IWORK,
LIWORK, INFO )
CHARACTER JOBZ, UPLO
INTEGER INFO, ITYPE, LDZ, LIWORK, LRWORK,
LWORK, N
INTEGER IWORK( * )
REAL RWORK( * ), W( * )
COMPLEX AP( * ), BP( * ), WORK( * ), Z( LDZ, *
)
PURPOSE
CHPGVD computes all the eigenvalues and, optionally, the
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.
If eigenvectors are desired, it uses a divide and conquer
algorithm.
The divide and conquer algorithm makes very mild assump
tions 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 with
out guard digits, 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.
= '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 array, dimension
(N*(N+1)/2)
On entry, the upper or lower triangle of the Her
mitian 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 array, dimension
(N*(N+1)/2)
On entry, the upper or lower triangle of the Her
mitian 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.
W (output) REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) COMPLEX array, dimension (LDZ, N)
If JOBZ = 'V', then if INFO = 0, Z contains the
matrix Z of eigenvectors. 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 JOBZ =
'N', then Z is not referenced.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1,
and if JOBZ = 'V', LDZ >= max(1,N).
WORK (workspace) COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input) INTEGER
The dimension of array WORK. If N <= 1,
LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= N.
If JOBZ = 'V' and N > 1, LWORK >= 2*N.
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.
RWORK (workspace) REAL array, dimension (LRWORK)
On exit, if INFO = 0, RWORK(1) returns the optimal
LRWORK.
LRWORK (input) INTEGER
The dimension of array RWORK. If N <= 1,
LRWORK >= 1. If JOBZ = 'N' and N > 1, LRWORK >=
N. If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N +
2*N**2.
If LRWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of
the RWORK array, returns this value as the first
entry of the RWORK array, and no error message
related to LRWORK is issued by XERBLA.
IWORK (workspace/output) INTEGER array, dimension
(LIWORK)
On exit, if INFO = 0, IWORK(1) returns the optimal
LIWORK.
LIWORK (input) INTEGER
The dimension of 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 ille
gal value
> 0: CPPTRF or CHPEVD returned an error code:
<= N: if INFO = i, CHPEVD failed to converge; i
off-diagonal elements of an intermediate tridiago
nal form did not convergeto zero; > 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
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