Computes all eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem using a divide and conquer method.

## Syntax

lapack_int LAPACKE_ssygvd (int matrix_layout, lapack_int itype, char jobz, char uplo, lapack_int n, float* a, lapack_int lda, float* b, lapack_int ldb, float* w);

lapack_int LAPACKE_dsygvd (int matrix_layout, lapack_int itype, char jobz, char uplo, lapack_int n, double* a, lapack_int lda, double* b, lapack_int ldb, double* w);

• mkl.h

## 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 and B is also positive definite.

It uses a divide and conquer algorithm.

## Input Parameters

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

itype

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

Must be 'N' or 'V'.

If jobz = 'N', then compute eigenvalues only.

If jobz = 'V', then compute eigenvalues and eigenvectors.

uplo

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

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

a, b

Arrays:

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

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

lda

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

ldb

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

## 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 infon, 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

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

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

## Return Values

This function returns a value info.

If info=0, the execution is successful.

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

If info > 0, an error code is returned as specified below.

• For infon:

• If info = i 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).

• For info > n:

• 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.

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