# ?sygv

Computes all eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem.

## Syntax

FORTRAN 77:

call ssygv(itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, info)

call dsygv(itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, info)

Fortran 95:

call sygv(a, b, w [,itype] [,jobz] [,uplo] [,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 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 a and b store the upper triangles of A and B;

  If `uplo = 'L'`, arrays a and b store the lower triangles of A and B.

n

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

a, b, work

  REAL for ssygv

  DOUBLE PRECISION for dsygv.

  Arrays:

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

  The second dimension of a must be at least max(1, n).

  b(ldb,*) contains the upper or lower triangle of the symmetric positive definite matrix B, as specified by uplo.

  The second dimension of b must be at least max(1, n).

  work is a workspace array, its dimension `max(1, lwork)`.
lda

  INTEGER. The first dimension of a; at least max(1, n).

ldb

  INTEGER. The first dimension of b; at least max(1, n).

lwork

  INTEGER.

  The dimension of the array work;

  `lwork ≥ max`(1, 3n-1).

  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.

  See Application Notes for the suggested value of lwork.

## Output Parameters

a

  On exit, if `jobz = 'V'`, then if `info = 0`, a 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 on exit the upper triangle (if `uplo = 'U'`) or the lower triangle (if `uplo = 'L'`) of A, including the diagonal, is destroyed.

b

  On exit, if `info ≤ n`, the part of b containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization `B = UT*U` or `B = L*LT`.

w

  REAL for ssygv

  DOUBLE PRECISION for dsygv.

  Array, DIMENSION at least max(1, n).

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

work(1)

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

info

  INTEGER.

  If `info = 0`, the execution is successful.

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

  If `info > 0`, spotrf/dpotrf and ssyev/dsyev returned an error code:

  If `info = i ≤ n`, ssyev/dsyev 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 sygv interface are the following:

a

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

b

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

w

  Holds the vector of length (n).

itype

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

jobz

  Must be 'N' or 'V'. The default value is 'N'.

uplo

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

## Application Notes

For optimum performance use `lwork ≥ (nb+2)*n`, where nb is the blocksize for ssytrd/dsytrd returned by ilaenv.

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

Parent topic: Generalized Symmetric Definite Eigenproblems