CHPGVD(1) LAPACK driver routine (version 3.2) CHPGVD(1)NAME
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
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 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.
AP (input/output) COMPLEX 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 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.
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 (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the required 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. If LWORK = -1, then a workspace query is
assumed; the routine only calculates the required sizes of the
WORK, RWORK and IWORK arrays, returns these values as the first
entries of the WORK, RWORK and IWORK arrays, and no error mes‐
sage related to LWORK or LRWORK or LIWORK is issued by XERBLA.
RWORK (workspace) REAL array, dimension (MAX(1,LRWORK))
On exit, if INFO = 0, RWORK(1) returns the required 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
required sizes of the WORK, RWORK and IWORK arrays, returns
these values as the first entries of the WORK, RWORK and IWORK
arrays, and no error message related to LWORK or LRWORK or
LIWORK is issued by XERBLA.
IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the required 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 cal‐
culates the required sizes of the WORK, RWORK and IWORK arrays,
returns these values as the first entries of the WORK, RWORK
and IWORK arrays, and no error message related to LWORK or
LRWORK 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: CPPTRF or CHPEVD returned an error code:
<= N: if INFO = i, CHPEVD failed to converge; i off-diagonal
elements of an intermediate tridiagonal 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 factoriza‐
tion of B could not be completed and no eigenvalues or eigen‐
vectors were computed.
FURTHER DETAILS
Based on contributions by
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
LAPACK driver routine (version 3November 2008 CHPGVD(1)
