?pbtrf

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?pbtrf

Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite band matrix.

Syntax

lapack_int LAPACKE_spbtrf (int matrix_layout , char uplo , lapack_int n , lapack_int kd , float * ab , lapack_int ldab );

lapack_int LAPACKE_dpbtrf (int matrix_layout , char uplo , lapack_int n , lapack_int kd , double * ab , lapack_int ldab );

lapack_int LAPACKE_cpbtrf (int matrix_layout , char uplo , lapack_int n , lapack_int kd , lapack_complex_float * ab , lapack_int ldab );

lapack_int LAPACKE_zpbtrf (int matrix_layout , char uplo , lapack_int n , lapack_int kd , lapack_complex_double * ab , lapack_int ldab );

Include Files

mkl.h

Description

The routine forms the Cholesky factorization of a symmetric positive-definite or, for complex data, Hermitian positive-definite band matrix A:

`A = U^TU` for real data, `A = U^HU` for complex data	if `uplo='U'`
`A = LL^T` for real data, `A = LL^H` for complex data	if `uplo='L'`

where L is a lower triangular matrix and U is upper triangular.

NOTE:

This routine supports the Progress Routine feature. See Progress Function for details.

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'. Indicates whether the upper or lower triangular part of `A` is stored in the array `ab`, and how `A` is factored: If `uplo = 'U'`, the upper triangle of `A` is stored. If `uplo = 'L'`, the lower triangle of `A` is stored.
n	The order of matrix `A`; `n`≥ 0.
kd	The number of superdiagonals or subdiagonals in the matrix `A`; `kd`≥ 0.
ab	Array, size max(1, ldab*n). The array `ab` contains either the upper or the lower triangular part of the matrix `A` (as specified by `uplo`) in band storage (see Matrix Storage Schemes).
ldab	The leading dimension of the array `ab`. `(ldab≥kd + 1)`

Output Parameters

ab	The upper or lower triangular part of `A` (in band storage) is overwritten by the Cholesky factor `U` or `L`, as specified by `uplo`.

Return Values

This function returns a value info.

If info=0, the execution is successful.

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

If info = i, the leading minor of order i (and therefore the matrix A itself) is not positive-definite, and the factorization could not be completed. This may indicate an error in forming the matrix A.

Application Notes

If uplo = 'U', the computed factor U is the exact factor of a perturbed matrix A + E, where

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

A similar estimate holds for uplo = 'L'.

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

After calling this routine, you can call the following routines:

?pbtrs	to solve `A*X = B`
?pbcon	to estimate the condition number of `A`.

Parent topic: Matrix Factorization: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?pbtrf

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