?hetrf_rook

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_rook

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

Syntax

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

lapack_int LAPACKE_zhetrf_rook (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 bounded 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 (or L ) is a product of permutation and unit upper ( or lower) triangular matrices, and D is a Hermitian block-diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS.

Input Parameters

matrix_layout	Specifies whether matrix storage layout for array b 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: If `uplo = 'U'`, the array `a` stores the upper triangular part of the matrix `A`. If `uplo = 'L'`, the array `a` stores the lower triangular part of the matrix `A`.
n	The order of matrix `A`; `n`≥ 0.
a	Array a, size `(lda*n)` The array `a` contains the upper or the lower triangular part of the matrix `A` (see `uplo`). If uplo = 'U', the leading n-by-n upper triangular part of a contains the upper triangular part of the matrix `A`, and the strictly lower triangular part of a is not referenced. If uplo = 'L', the leading n-by-n lower triangular part of a contains the lower triangular part of the matrix `A`, and the strictly upper triangular part of a is not referenced.
lda	The leading dimension of `a`; at least `max(1, n)`.

Output Parameters

The block diagonal matrix D and the multipliers used to obtain the factor U or L (see Application Notes for further details).

ipiv

If uplo = 'U':

If ipiv(k) > 0, then rows and columns k and ipiv(k) were interchanged and D_{k, k} is a 1-by-1 diagonal block.

If ipiv(k) < 0 and ipiv(k - 1) < 0, then rows and columns k and -ipiv(k) were interchanged and rows and columns k - 1 and -ipiv(k - 1) were interchanged, D_{k - 1:k,k - 1:k} is a 2-by-2 diagonal block.
If uplo = 'L':

If ipiv(k) > 0, then rows and columns k and ipiv(k) were interchanged and D_k,k is a 1-by-1 diagonal block.

If ipiv(k) < 0 and ipiv(k + 1) < 0, then rows and columns k and -ipiv(k) were interchanged and rows and columns k + 1 and -ipiv(k + 1) were interchanged, D_{k:k + 1,k:k + 1} is a 2-by-2 diagonal block.

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.

If info = i, D_ii is exactly 0. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by 0 will occur if you use D for solving a system of linear equations.

Application Notes

If uplo = 'U', thenA = U*D*U^H, where

U = P(n)*U(n)* ... *P(k)U(k)* ...,

i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by ipiv(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).

If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k).

If uplo = 'L', then A = L*D*L^H, where

L = P(1)*L(1)* ... *P(k)*L(k)* ...,

i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by ipiv(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).

If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Parent topic: Matrix Factorization: LAPACK Computational Routines

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?hetrf_rook

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