?heevd

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?heevd

Computes all eigenvalues and, optionally, all eigenvectors of a complex Hermitian matrix using divide and conquer algorithm.

Syntax

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

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

Include Files

mkl.h

Description

The routine computes all the eigenvalues, and optionally all the eigenvectors, of a complex Hermitian matrix A. In other words, it can compute the spectral factorization of A as: A = Z*Λ*Z^H.

Here Λ is a real diagonal matrix whose diagonal elements are the eigenvalues λ_i, and Z is the (complex) unitary matrix whose columns are the eigenvectors z_i. Thus,

A*z_i = λ_i*z_i 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 complex Hermetian eigenvalue problems the default choice should be heevr function as its underlying algorithm is faster and uses less workspace. ?heevd 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

a (size max(1, lda*n)) is an array containing either upper or lower triangular part of the Hermitian 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 unitary 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 off-diagonal elements of an intermediate tridiagonal form 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 real analogue of this routine is syevd. See also hpevd for matrices held in packed storage, and hbevd for banded matrices.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Driver Routines

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?heevd