sspgvd(3)

NAME

SSPGVD - compute all the eigenvalues, and optionally, the

eigenvectors of a real generalized symmetric-definite eigenprob

lem, 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 eigenprob

lem, 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 hexadeci

mal or decimal machines without 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.

N (input) INTEGER

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.

IWORK (workspace/output) INTEGER array, dimension (LI

WORK)

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 tridiagonal 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

LAPACK version 3.0 15 June 2000

SSPGVD(3)