# ?spgvd

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

## Syntax

call sspgvd(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, iwork, liwork, info)

call dspgvd(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, iwork, liwork, info)

call spgvd(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.

If eigenvectors are desired, it uses a divide and conquer algorithm.

## 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 sspgvd

DOUBLE PRECISION for dspgvd.

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, its dimension max(1, lwork).

ldz

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

lwork

INTEGER.

The dimension of the array work.

Constraints:

If n 1, lwork 1;

If jobz = 'N' and n>1, lwork 2n;

If jobz = 'V' and n>1, lwork 2n2+6n+1.

If lwork = -1, then a workspace query is assumed; the routine only calculates the required sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued by xerbla. See Application Notes for details.

iwork

INTEGER.

Workspace array, dimension max(1, lwork).

liwork

INTEGER.

The dimension of the array iwork.

Constraints:

If n 1, liwork 1;

If jobz = 'N' and n>1, liwork 1;

If jobz = 'V' and n>1, liwork 5n+3.

If liwork = -1, then a workspace query is assumed; the routine only calculates the required sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued by xerbla. See Application Notes for details.

## 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.

work(1)

On exit, if info = 0, then work(1) returns the required minimal size of lwork.

iwork(1)

On exit, if info = 0, then iwork(1) returns the required minimal size of liwork.

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 sspevd/dspevd returned an error code:

If info = in, sspevd/dspevd 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 spgvd 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.

## Application Notes

If it is not clear how much workspace to supply, use a generous value of lwork (or liwork) for the first run, or set lwork = -1 (liwork = -1).

If lwork (or liwork) has any of admissible sizes, which is no less than the minimal value described, then the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array (work, iwork) on exit. Use this value (work(1), iwork(1)) for subsequent runs.

If lwork = -1 (liwork = -1), then the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work, iwork). This operation is called a workspace query.

Note that if lwork (liwork) is less than the minimal required value and is not equal to -1, then the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

For more complete information about compiler optimizations, see our Optimization Notice.