.TH DSPGVX l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
DSPGVX - 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
.SH SYNOPSIS
.TP 19
SUBROUTINE DSPGVX(
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
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INTEGER
IL, INFO, ITYPE, IU, LDZ, M, N
.TP 19
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DOUBLE
PRECISION ABSTOL, VL, VU
.TP 19
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INTEGER
IFAIL( * ), IWORK( * )
.TP 19
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DOUBLE
PRECISION AP( * ), BP( * ), W( * ), WORK( * ),
Z( LDZ, * )
.SH PURPOSE
DSPGVX 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
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= 2:  A*B*x = (lambda)*x
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= 3:  B*A*x = (lambda)*x
.TP 8
JOBZ    (input) CHARACTER*1
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= 'N':  Compute eigenvalues only;
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= 'V':  Compute eigenvalues and eigenvectors.
.TP 8
RANGE   (input) CHARACTER*1
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= 'A': all eigenvalues will be found.
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= 'V': all eigenvalues in the half-open interval (VL,VU]
will be found.
= 'I': the IL-th through IU-th eigenvalues will be found.
.TP 8
UPLO    (input) CHARACTER*1
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= 'U':  Upper triangle of A and B are stored;
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= 'L':  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) DOUBLE PRECISION 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 = '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.
.TP 8
BP      (input/output) DOUBLE PRECISION 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 = '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.
.TP 8
VL      (input) DOUBLE PRECISION
VU      (input) DOUBLE PRECISION
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'.
.TP 8
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 referenced if RANGE = 'A' or 'V'.
.TP 8
ABSTOL  (input) DOUBLE PRECISION
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*DLAMCH('S'), not zero.
If this routine returns with INFO>0, indicating that some
eigenvectors did not converge, try setting ABSTOL to
2*DLAMCH('S').
.TP 8
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.
.TP 8
W       (output) DOUBLE PRECISION array, dimension (N)
On normal exit, the first M elements contain the selected
eigenvalues in ascending order.
.TP 8
Z       (output) DOUBLE PRECISION 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 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 = 'V', 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 = 'V', LDZ >= max(1,N).
.TP 8
WORK    (workspace) DOUBLE PRECISION array, dimension (8*N)
.TP 8
IWORK   (workspace) INTEGER array, dimension (5*N)
.TP 8
IFAIL   (output) INTEGER array, dimension (N)
If JOBZ = 'V', 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 = 'N', then IFAIL is not referenced.
.TP 8
INFO    (output) INTEGER
= 0:  successful exit
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< 0:  if INFO = -i, the i-th argument had an illegal value
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> 0:  DPPTRF or DSPEVX returned an error code:
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<= N:  if INFO = i, DSPEVX 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
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   Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA