?sygst

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?sygst

Reduces a real symmetric-definite generalized eigenvalue problem to the standard form.

Syntax

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

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

Include Files

mkl.h

Description

The routine reduces real symmetric-definite generalized eigenproblems

A*z = λ*B*z, A*B*z = λ*z, or B*A*z = λ*z

to the standard form C*y = λ*y. Here A is a real symmetric matrix, and B is a real symmetric positive-definite matrix. Before calling this routine, call ?potrf to compute the Cholesky factorization: B = U^T*U or B = L*L^T.

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.

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

for uplo = 'U': C = inv(U^T)*A*inv(U), z = inv(U)*y;

for uplo = 'L': C = inv(L)*A*inv(L^T), z = inv(L^T)*y.

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

for uplo = 'U': C = U*A*U^T, z = inv(U)*y;

for uplo = 'L': C = L^T*A*L, z = inv(L^T)*y.

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

for uplo = 'U': C = U*A*U^T, z = U^T*y;

for uplo = 'L': C = L^T*A*L, z = L*y.

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 = U^T*U.

If uplo = 'L', the array a stores the lower triangle of A; you must supply B in the factored form B = L*L^T.

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 = U^T*U or B = L*L^T (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 inv(B) (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.

Parent topic: Generalized Symmetric-Definite Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?sygst