SSPGVX - compute selected eigenvalues, and optionally,
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 SSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP,
VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ,
WORK, IWORK, IFAIL, INFO )
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, ITYPE, IU, LDZ, M, N
REAL ABSTOL, VL, VU
INTEGER IFAIL( * ), IWORK( * )
REAL AP( * ), BP( * ), W( * ), WORK( * ), Z(
LDZ, * )
PURPOSE
SSPGVX computes selected eigenvalues, and optionally,
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 storage, 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 triangle of A and B are stored;
= 'L': Lower triangle of A and B are stored.
The order of the matrix pencil (A,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.
VL (input) REAL
VU (input) REAL 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 refer
enced if RANGE = 'A' or 'V'.
ABSTOL (input) REAL
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 A to
tridiagonal form.
Eigenvalues will be computed most accurately when
ABSTOL is set to twice the underflow threshold
with INFO>0, indicating that some eigenvectors did
not converge, try setting ABSTOL to 2*SLAMCH('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) REAL array, dimension (N)
On normal exit, the first M elements contain the
selected eigenvalues in ascending order.
Z (output) REAL array, dimension (LDZ, max(1,M))
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 eigenvalues, with
the i-th column of Z holding the eigenvector asso
ciated with W(i). 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 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 = '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) REAL array, dimension (8*N)
IWORK (workspace) INTEGER array, dimension (5*N)
IFAIL (output) INTEGER array, dimension (N)
If JOBZ = 'V', then if INFO = 0, the first M ele
ments 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 ille
gal value
> 0: SPPTRF or SSPEVX returned an error code:
<= N: if INFO = i, SSPEVX failed to converge; i
eigenvectors failed to converge. Their indices
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 Ken
tucky, USA
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