SSPGVD - compute all the eigenvalues, and optionally, the
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 SSPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z,
LDZ, WORK, LWORK, IWORK, LIWORK, INFO )
CHARACTER JOBZ, UPLO
INTEGER INFO, ITYPE, LDZ, LIWORK, LWORK, N
INTEGER IWORK( * )
REAL AP( * ), BP( * ), W( * ), WORK( * ), Z(
LDZ, * )
PURPOSE
SSPGVD computes all the eigenvalues, and optionally, the
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, 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.
UPLO (input) CHARACTER*1
= 'U': Upper triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored.
The order of the matrices A and B. N >= 0.
AP (input/output) REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the sym
metric 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) REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the sym
metric 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**T*U or B = L*L**T,
in the same storage format as B.
W (output) REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) REAL 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**T*B*Z
= I; if ITYPE = 3, Z**T*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/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. If N <= 1,
LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >=
2*N. If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N +
2*N**2.
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.
(LIWORK)
On exit, if INFO = 0, IWORK(1) returns the optimal
LIWORK.
LIWORK (input) INTEGER
The dimension of the 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: SPPTRF or SSPEVD returned an error code:
<= N: if INFO = i, SSPEVD failed to converge; i
off-diagonal elements of an intermediate tridiago
nal form did not converge to zero; > 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 Ken
tucky, USA
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