Developer Reference for Intel® oneAPI Math Kernel Library for C

ID 766684
Date 7/13/2023
Public

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?hegst

Reduces a complex Hermitian positive-definite generalized eigenvalue problem to the standard form.

Syntax

lapack_int LAPACKE_chegst (int matrix_layout, lapack_int itype, char uplo, lapack_int n, lapack_complex_float* a, lapack_int lda, const lapack_complex_float* b, lapack_int ldb);

lapack_int LAPACKE_zhegst (int matrix_layout, lapack_int itype, char uplo, lapack_int n, lapack_complex_double* a, lapack_int lda, const lapack_complex_double* b, lapack_int ldb);

Include Files

  • mkl.h

Description

The routine reduces a complex Hermitian positive-definite generalized eigenvalue problem to standard form.

itype

Problem

Result

1

A*x = λ*B*x

A overwritten by inv(UH)*A*inv(U) or inv(L)*A*inv(LH)

2

A*B*x = λ*x

A overwritten by U*A*UH or LH*A*L

3

B*A*x = λ*x
     

Before calling this routine, you must call ?potrf to compute the Cholesky factorization: B = UH*U or B = L*LH.

Input Parameters

itype

Must be 1 or 2 or 3.

If itype = 1, the generalized eigenproblem is A*z = lambda*B*z

for uplo = 'U': C = (UH)-1*A*U-1;

for uplo = 'L': C = L-1*A*(LH)-1.

If itype = 2, the generalized eigenproblem is A*B*z = lambda*z

for uplo = 'U': C = U*A*UH;

for uplo = 'L': C = LH*A*L.

If itype = 3, the generalized eigenproblem is B*A*z = lambda*z

for uplo = 'U': C = U*A*UH;

for uplo = 'L': C = LH*A*L.

uplo

Must be 'U' or 'L'.

If uplo = 'U', the array a stores the upper triangle of A; you must supply B in the factored form B = UH*U.

If uplo = 'L', the array a stores the lower triangle of A; you must supply B in the factored form B = L*LH.

n

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

a, b

Arrays:

a (size max(1, lda*n)) contains the upper or lower triangle of A.

b (size max(1, ldb*n)) contains the Cholesky-factored matrix B:

B = UH*U or B = L*LH (as returned by ?potrf).

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

The upper or lower triangle of A is overwritten by the upper or lower triangle of C, as specified by the arguments itype and uplo.

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.

Application Notes

Forming the reduced matrix C is a stable procedure. However, it involves implicit multiplication by B-1 (if itype = 1) or B (if itype = 2 or 3). When the routine is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if B is ill-conditioned with respect to inversion.

The approximate number of floating-point operations is n3.