# ?hpgvd

Computes all eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian positive-definite eigenproblem with matrices in packed storage using a divide and conquer method.

## Syntax

lapack_int LAPACKE_chpgvd( int matrix_layout, lapack_int itype, char jobz, char uplo, lapack_int n, lapack_complex_float* ap, lapack_complex_float* bp, float* w, lapack_complex_float* z, lapack_int ldz );

lapack_int LAPACKE_zhpgvd( int matrix_layout, lapack_int itype, char jobz, char uplo, lapack_int n, lapack_complex_double* ap, lapack_complex_double* bp, double* w, lapack_complex_double* z, lapack_int ldz );

• mkl.h

## Description

The routine computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian positive-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 Hermitian, stored in packed format, and B is also positive definite.

If eigenvectors are desired, 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 ap and bp store the upper triangles of A and B;

If uplo = 'L', arrays ap and bp store the lower triangles of A and B.

n

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

ap, bp

Arrays:

ap contains the packed upper or lower triangle of the Hermitian matrix A, as specified by uplo.

The dimension of ap must be at least max(1, n*(n+1)/2).

bp contains the packed upper or lower triangle of the Hermitian matrix B, as specified by uplo.

The dimension of bp must be at least max(1, n*(n+1)/2).

ldz

The leading dimension of the output array z; ldz 1. If jobz = 'V', ldz max(1, n).

## Output Parameters

ap

On exit, the contents of ap are overwritten.

bp

On exit, contains the triangular factor U or L from the Cholesky factorization B = UH*U or B = L*LH, in the same storage format as B.

w

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

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

z

Array z (size at least max(1, ldz*n)).

If jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:

if itype = 1 or 2, ZH*B*Z = I;

if itype = 3, ZH*inv(B)*Z = I;

If jobz = 'N', then z is not referenced.

## 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, cpptrf/zpptrf and chpevd/zhpevd returned an error code:

If info = in, chpevd/zhpevd failed to converge, and i off-diagonal elements of an intermediate tridiagonal did not converge to zero;

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.

For more complete information about compiler optimizations, see our Optimization Notice.