
SSPGVX(l) ) SSPGVX(l)
NAME
SSPGVX  compute selected eigenvalues, and optionally, eigenvectors of a real generalized
symmetricdefinite 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
symmetricdefinite 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 halfopen interval (VL,VU] will be found. = 'I':
the ILth through IUth 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.
N (input) INTEGER
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 symmetric matrix A, packed columnwise
in a linear array. The jth column of A is stored in the array AP as follows: if
UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
(j1)*(2*nj)/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 symmetric matrix B, packed columnwise
in a linear array. The jth column of B is stored in the array BP as follows: if
UPLO = 'U', BP(i + (j1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i +
(j1)*(2*nj)/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 referenced 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 1norm 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('S'), not zero. If this routine returns 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 = IUIL+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 corre
sponding to the selected eigenvalues, with the ith column of Z holding the eigen
vector 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.
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 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.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
> 0: SPPTRF or SSPEVX returned an error code:
<= 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 factoriza
tion 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 Kentucky, USA
LAPACK version 3.0 15 June 2000 SSPGVX(l) 
