# ?spgvd

Computes all eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem with matrices in packed storage. If eigenvectors are desired, it uses a divide and conquer method.

## Syntax

FORTRAN 77:

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)

Fortran 95:

call spgvd(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.

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, its 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(*), 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.

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 = i ≤ n`, 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 ≤ 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 spgvd 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.

## 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.

Parent topic: Generalized Symmetric Definite Eigenproblems