# ?spgv

Computes all eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem with matrices in packed storage.

## Syntax

call sspgv(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, info)

call dspgv(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, info)

call spgv(ap, bp, w [,itype] [,uplo] [,z] [,info])

## Include Files

• mkl.fi, lapack.f90

## Description

The routine computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form

A*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x.

Here A and B are assumed to be symmetric, stored in packed format, and B is also positive definite.

## Input Parameters

itype

INTEGER. Must be 1 or 2 or 3. Specifies the problem type to be solved:

if itype = 1, the problem type is A*x = lambda*B*x;

if itype = 2, the problem type is A*B*x = lambda*x;

if itype = 3, the problem type is B*A*x = lambda*x.

jobz

CHARACTER*1. Must be 'N' or 'V'.

If jobz = 'N', then compute eigenvalues only.

If jobz = 'V', then compute eigenvalues and eigenvectors.

uplo

CHARACTER*1. Must be 'U' or 'L'.

If uplo = 'U', arrays ap and bp store the upper triangles of A and B;

If uplo = 'L', arrays ap and bp store the lower triangles of A and B.

n

INTEGER. The order of the matrices A and B (n 0).

ap, bp, work

REAL for sspgv

DOUBLE PRECISION for dspgv.

Arrays:

ap(*) contains the packed upper or lower triangle of the symmetric matrix A, as specified by uplo.

The dimension of ap must be at least max(1, n*(n+1)/2).

bp(*) contains the packed upper or lower triangle of the symmetric matrix B, as specified by uplo.

The dimension of bp must be at least max(1, n*(n+1)/2).

work(*) is a workspace array, size at least max(1, 3n).

ldz

INTEGER. The leading dimension of the output array z; ldz 1. If jobz = 'V', ldz max(1, n).

## Output Parameters

ap

On exit, the contents of ap are overwritten.

bp

On exit, contains the triangular factor U or L from the Cholesky factorization B = UT*U or B = L*LT, in the same storage format as B.

w, z

REAL for sspgv

DOUBLE PRECISION for dspgv.

Arrays:

w(*), size at least max(1, n).

If info = 0, contains the eigenvalues in ascending order.

z(ldz,*) .

The second dimension of z must be at least max(1, 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, ZT*B*Z = I;

if itype = 3, ZT*inv(B)*Z = I;

If jobz = 'N', then z is not referenced.

info

INTEGER.

If info = 0, the execution is successful.

If info = -i, the i-th argument had an illegal value.

If info > 0, spptrf/dpptrf and sspev/dspev returned an error code:

If info = in, sspev/dspev failed to converge, and i off-diagonal elements of an intermediate tridiagonal did not converge to zero;

If info = n + i, for 1 in, 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.

## LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine spgv interface are the following:

ap

Holds the array A of size (n*(n+1)/2).

bp

Holds the array B of size (n*(n+1)/2).

w

Holds the vector with the number of elements n.

z

Holds the matrix Z of size (n, n).

itype

Must be 1, 2, or 3. The default value is 1.

uplo

Must be 'U' or 'L'. The default value is 'U'.

jobz

Restored based on the presence of the argument z as follows:

jobz = 'V', if z is present,

jobz = 'N', if z is omitted.

Para obtener información más completa sobre las optimizaciones del compilador, consulte nuestro Aviso de optimización.