# ?spgv

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

## Syntax

FORTRAN 77:

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)

Fortran 95:

call spgv(a, b, w [,itype] [,uplo] [,z] [,info])

## Description

This 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, DIMENSION 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(*), DIMENSION 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 = i ≤ n`, 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 ≤ 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.

## Fortran 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 Fortran 95 Interface Conventions.

Specific details for the routine spgv interface are the following:

a

  Stands for argument ap in Fortan 77 interface. Holds the array A of size (n*(n+1)/2).

b

  Stands for argument bp in Fortan 77 interface. Holds the array B of size (n*(n+1)/2).

w

  Holds the vector of length (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.

Parent topic: Generalized Symmetric Definite Eigenproblems