.TH  SSPGVX 1 "November 2006" " LAPACK driver routine (version 3.1) " " LAPACK driver routine (version 3.1) " 
.SH NAME
SSPGVX - 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
.SH SYNOPSIS
.TP 19
SUBROUTINE SSPGVX(
ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
IFAIL, INFO )
.TP 19
.ti +4
CHARACTER
JOBZ, RANGE, UPLO
.TP 19
.ti +4
INTEGER
IL, INFO, ITYPE, IU, LDZ, M, N
.TP 19
.ti +4
REAL
ABSTOL, VL, VU
.TP 19
.ti +4
INTEGER
IFAIL( * ), IWORK( * )
.TP 19
.ti +4
REAL
AP( * ), BP( * ), W( * ), WORK( * ),
Z( LDZ, * )
.SH 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.
.br

.SH ARGUMENTS
.TP 8
ITYPE   (input) INTEGER
Specifies the problem type to be solved:
.br
= 1:  A*x = (lambda)*B*x
.br
= 2:  A*B*x = (lambda)*x
.br
= 3:  B*A*x = (lambda)*x
.TP 8
JOBZ    (input) CHARACTER*1
.br
= \(aqN\(aq:  Compute eigenvalues only;
.br
= \(aqV\(aq:  Compute eigenvalues and eigenvectors.
.TP 8
RANGE   (input) CHARACTER*1
.br
= \(aqA\(aq: all eigenvalues will be found.
.br
= \(aqV\(aq: all eigenvalues in the half-open interval (VL,VU]
will be found.
= \(aqI\(aq: the IL-th through IU-th eigenvalues will be found.
.TP 8
UPLO    (input) CHARACTER*1
.br
= \(aqU\(aq:  Upper triangle of A and B are stored;
.br
= \(aqL\(aq:  Lower triangle of A and B are stored.
.TP 8
N       (input) INTEGER
The order of the matrix pencil (A,B).  N >= 0.
.TP 8
AP      (input/output) REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array.  The j-th column of A
is stored in the array AP as follows:
if UPLO = \(aqU\(aq, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = \(aqL\(aq, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed.
.TP 8
BP      (input/output) REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
B, packed columnwise in a linear array.  The j-th column of B
is stored in the array BP as follows:
if UPLO = \(aqU\(aq, BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
if UPLO = \(aqL\(aq, 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.
.TP 8
VL      (input) REAL
VU      (input) REAL
If RANGE=\(aqV\(aq, the lower and upper bounds of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = \(aqA\(aq or \(aqI\(aq.
.TP 8
IL      (input) INTEGER
IU      (input) INTEGER
If RANGE=\(aqI\(aq, 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 = \(aqA\(aq or \(aqV\(aq.
.TP 8
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 2*SLAMCH(\(aqS\(aq), not zero.
If this routine returns with INFO>0, indicating that some
eigenvectors did not converge, try setting ABSTOL to
2*SLAMCH(\(aqS\(aq).
.TP 8
M       (output) INTEGER
The total number of eigenvalues found.  0 <= M <= N.
If RANGE = \(aqA\(aq, M = N, and if RANGE = \(aqI\(aq, M = IU-IL+1.
.TP 8
W       (output) REAL array, dimension (N)
On normal exit, the first M elements contain the selected
eigenvalues in ascending order.
.TP 8
Z       (output) REAL array, dimension (LDZ, max(1,M))
If JOBZ = \(aqN\(aq, then Z is not referenced.
If JOBZ = \(aqV\(aq, 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 associated 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 = \(aqV\(aq, the exact value of M
is not known in advance and an upper bound must be used.
.TP 8
LDZ     (input) INTEGER
The leading dimension of the array Z.  LDZ >= 1, and if
JOBZ = \(aqV\(aq, LDZ >= max(1,N).
.TP 8
WORK    (workspace) REAL array, dimension (8*N)
.TP 8
IWORK   (workspace) INTEGER array, dimension (5*N)
.TP 8
IFAIL   (output) INTEGER array, dimension (N)
If JOBZ = \(aqV\(aq, 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 = \(aqN\(aq, then IFAIL is not referenced.
.TP 8
INFO    (output) INTEGER
= 0:  successful exit
.br
< 0:  if INFO = -i, the i-th argument had an illegal value
.br
> 0:  SPPTRF or SSPEVX returned an error code:
.br
<= N:  if INFO = i, SSPEVX 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.
.SH FURTHER DETAILS
Based on contributions by
.br
   Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

