Developer Reference for Intel® oneAPI Math Kernel Library for C

ID 766684
Date 3/22/2024
Public

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?pbstf

Computes a split Cholesky factorization of a real symmetric or complex Hermitian positive-definite banded matrix used in ?sbgst/?hbgst .

Syntax

lapack_int LAPACKE_spbstf (int matrix_layout, char uplo, lapack_int n, lapack_int kb, float* bb, lapack_int ldbb);

lapack_int LAPACKE_dpbstf (int matrix_layout, char uplo, lapack_int n, lapack_int kb, double* bb, lapack_int ldbb);

lapack_int LAPACKE_cpbstf (int matrix_layout, char uplo, lapack_int n, lapack_int kb, lapack_complex_float* bb, lapack_int ldbb);

lapack_int LAPACKE_zpbstf (int matrix_layout, char uplo, lapack_int n, lapack_int kb, lapack_complex_double* bb, lapack_int ldbb);

Include Files

  • mkl.h

Description

The routine computes a split Cholesky factorization of a real symmetric or complex Hermitian positive-definite band matrix B. It is to be used in conjunction with sbgst/hbgst.

The factorization has the form B = ST*S (or B = SH*S for complex flavors), where S is a band matrix of the same bandwidth as B and the following structure: S is upper triangular in the first (n+kb)/2 rows and lower triangular in the remaining rows.

Input Parameters

matrix_layout

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

uplo

Must be 'U' or 'L'.

If uplo = 'U', bb stores the upper triangular part of B.

If uplo = 'L', bb stores the lower triangular part of B.

n

The order of the matrix B (n 0).

kb

The number of super- or sub-diagonals in B

(kb 0).

bb

bb(size at least max(1, ldbb*n) for column major layout and at least max(1, ldbb*(kb + 1)) for row major layout) is an array containing either upper or lower triangular part of the matrix B (as specified by uplo) in band storage format.

ldbb

The leading dimension of bb; must be at least kb+1for column major and at least max(1, n) for row major.

Output Parameters

bb

On exit, this array is overwritten by the elements of the split Cholesky factor S.

Return Values

This function returns a value info.

If info=0, the execution is successful.

If info = i, then the factorization could not be completed, because the updated element bii would be the square root of a negative number; hence the matrix B is not positive-definite.

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

Application Notes

The computed factor S is the exact factor of a perturbed matrix B + E, where


Equation

c(n) is a modest linear function of n, and ε is the machine precision.

The total number of floating-point operations for real flavors is approximately n(kb+1)2. The number of operations for complex flavors is 4 times greater. All these estimates assume that kb is much less than n.

After calling this routine, you can call sbgst/hbgst to solve the generalized eigenproblem Az = λBz, where A and B are banded and B is positive-definite.