# ?sygvd

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

## Syntax

FORTRAN 77:

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

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

Fortran 95:

call sygvd(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.

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 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 ssygvd

  DOUBLE PRECISION for dsygvd.

  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.

  Constraints:

  If `n ≤ 1`, `lwork ≥ 1`;

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

  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

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 ssygvd

  DOUBLE PRECISION for dsygvd.

  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.

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`= i ≤ n, and `jobz = 'N'`, then the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; if `jobz = 'V'`, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns `info/(n+1)` through `mod(info,n+1)`;

  If `info > 0= 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 sygvd 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

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