?hetrf

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?hetrf

Computes the Bunch-Kaufman factorization of a complex Hermitian matrix.

Syntax

lapack_int LAPACKE_chetrf (int matrix_layout , char uplo , lapack_int n , lapack_complex_float * a , lapack_int lda , lapack_int * ipiv );

lapack_int LAPACKE_zhetrf (int matrix_layout , char uplo , lapack_int n , lapack_complex_double * a , lapack_int lda , lapack_int * ipiv );

Include Files

mkl.h

Description

The routine computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method:

if uplo='U', A = U*D*U^H
if uplo='L', A = L*D*L^H,

where A is the input matrix, U and L are products of permutation and triangular matrices with unit diagonal (upper triangular for U and lower triangular for L), and D is a Hermitian block-diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks. U and L have 2-by-2 unit diagonal blocks corresponding to the 2-by-2 blocks of D.

NOTE:

This routine supports the Progress Routine feature. See Progress Routine 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 and how `A` is factored: If `uplo = 'U'`, the array `a` stores the upper triangular part of the matrix `A`, and `A` is factored as `UDU^H`. If `uplo = 'L'`, the array `a` stores the lower triangular part of the matrix `A`, and `A` is factored as `LDL^H`.
n	The order of matrix `A`; `n`≥ 0.
a	Array, size max(1, lda*n). The array `a` contains the upper or the lower triangular part of the matrix `A` (see `uplo`).
lda	The leading dimension of `a`; at least `max(1, n)`.

Output Parameters

The upper or lower triangular part of a is overwritten by details of the block-diagonal matrix D and the multipliers used to obtain the factor U (or L).

ipiv

Array, size at least max(1, n). Contains details of the interchanges and the block structure of D. If ipiv[i-1] = k >0, then d_ii is a 1-by-1 block, and the i-th row and column of A was interchanged with the k-th row and column.

If uplo = 'U' and ipiv[i] =ipiv[i-1] = -m < 0, then D has a 2-by-2 block in rows/columns i and i+1, and i-th row and column of A was interchanged with the m-th row and column.

If uplo = 'L' and ipiv[i] =ipiv[i-1] = -m < 0, then D has a 2-by-2 block in rows/columns i and i+1, and (i+1)-th row and column of A was interchanged with the m-th row and column.

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, d_ii is 0. The factorization has been completed, but D is exactly singular. Division by 0 will occur if you use D for solving a system of linear equations.

Application Notes

This routine is suitable for Hermitian matrices that are not known to be positive-definite. If A is in fact positive-definite, the routine does not perform interchanges, and no 2-by-2 diagonal blocks occur in D.

The 2-by-2 unit diagonal blocks and the unit diagonal elements of U and L are not stored. The remaining elements of U and L are stored in the corresponding columns of the array a, but additional row interchanges are required to recover U or L explicitly (which is seldom necessary).

Ifipiv[i-1] = i for all i =1...n, then all off-diagonal elements of U (L) are stored explicitly in the corresponding elements of the array a.

If uplo = 'U', the computed factors U and D are the exact factors of a perturbed matrix A + E, where

|E| ≤ c(n)ε P|U||D||U^T|P^T

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

A similar estimate holds for the computed L and D when uplo = 'L'.

The total number of floating-point operations is approximately (4/3)n³.

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

?hetrs	to solve `A*X = B`
?hecon	to estimate the condition number of `A`
?hetri	to compute the inverse of `A`.

Parent topic: Matrix Factorization: LAPACK Computational Routines

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