Computes all eigenvalues and, optionally, all eigenvectors of a real symmetric matrix using divide and conquer algorithm.

## Syntax

lapack_int LAPACKE_ssyevd (int matrix_layout, char jobz, char uplo, lapack_int n, float* a, lapack_int lda, float* w);

lapack_int LAPACKE_dsyevd (int matrix_layout, char jobz, char uplo, lapack_int n, double* a, lapack_int lda, double* w);

• mkl.h

## Description

The routine computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric matrix A. In other words, it can compute the spectral factorization of A as: A = Z*λ*ZT.

Here Λ is a diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the orthogonal matrix whose columns are the eigenvectors zi. Thus,

A*zi = λi*zi for i = 1, 2, ..., n.

If the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan variant of the QL or QR algorithm.

Note that for most cases of real symmetric eigenvalue problems the default choice should be syevr function as its underlying algorithm is faster and uses less workspace. ?syevd requires more workspace but is faster in some cases, especially for large matrices.

## Input Parameters

matrix_layout

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

jobz

Must be 'N' or 'V'.

If jobz = 'N', then only eigenvalues are computed.

If jobz = 'V', then eigenvalues and eigenvectors are computed.

uplo

Must be 'U' or 'L'.

If uplo = 'U', a stores the upper triangular part of A.

If uplo = 'L', a stores the lower triangular part of A.

n

The order of the matrix A (n 0).

a

Array, size (lda, *).

a (size max(1, lda*n)) is an array containing either upper or lower triangular part of the symmetric matrix A, as specified by uplo.

lda

The leading dimension of the array a.

Must be at least max(1, n).

## Output Parameters

w

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

If info = 0, contains the eigenvalues of the matrix A in ascending order. See also info.

a

If jobz = 'V', then on exit this array is overwritten by the orthogonal matrix Z which contains the eigenvectors of A.

## Return Values

This function returns a value info.

If info=0, the execution is successful.

If info = i, and jobz = 'N', then the algorithm failed to converge; i indicates the number of off-diagonal elements of an intermediate tridiagonal form which did not converge to zero.

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

## Application Notes

The computed eigenvalues and eigenvectors are exact for a matrix A+E such that ||E||2 = O(ε)*||A||2, where ε is the machine precision.

The complex analogue of this routine is heevd

Para obter informações mais completas sobre otimizações do compilador, consulte nosso aviso de otimização.