?sytrf

Developer Reference for Intel® oneAPI Math Kernel Library for C

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Date 3/22/2024

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?sytrf_rk

Computes the factorization of a real or complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

lapack_int LAPACKE_ssytrf_rk (int matrix_layout, char uplo, lapack_int n, float * A, lapack_int lda, float * e, lapack_int * ipiv);

lapack_int LAPACKE_dsytrf_rk (int matrix_layout, char uplo, lapack_int n, double * A, lapack_int lda, double * e, lapack_int * ipiv);

lapack_int LAPACKE_csytrf_rk (int matrix_layout, char uplo, lapack_int n, lapack_complex_float * A, lapack_int lda, lapack_complex_float * e, lapack_int * ipiv);

lapack_int LAPACKE_zsytrf_rk (int matrix_layout, char uplo, lapack_int n, lapack_complex_double * A, lapack_int lda, lapack_complex_double * e, lapack_int * ipiv);

Description

?sytrf_rk computes the factorization of a real or complex symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method: A= P*U*D*(U^T)*(P^T) or A = P*L*D*(L^T)*(P^T), where U (or L) is unit upper (or lower) triangular matrix, U^T (or L^T) is the transpose of U (or L), P is a permutation matrix, P^T is the transpose of P, and D is symmetric and block diagonal 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 is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

uplo

Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:

= 'U': Upper triangular
= 'L': Lower triangular

n

The order of the matrix A. n ≥ 0.

A

Array of size max(1, lda*n). On entry, the symmetric matrix A. 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 the array A.

Output Parameters

A

On exit, contains:

Only diagonal elements of the symmetric block diagonal matrix D on the diagonal of A; that is, D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array e).
If uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L', factor L in the subdiagonal part of A.

e

Array of size n. On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks. If uplo = 'U', e(i) = D(i-1,i), i=2:N, and e(1) is set to 0. If uplo = 'L', e(i) = D(i+1,i), i=1:N-1, and e(n) is set to 0.

NOTE:

For 1-by-1 diagonal block D(k), where 1 ≤ k ≤ n, the element e[k-1] is set to 0 in both the uplo = 'U' and uplo = 'L' cases.

ipiv

Array of size n.ipiv describes the permutation matrix P in the factorization of matrix A as follows: The absolute value of ipiv(k) represents the index of the row and column that were interchanged with the k^th row and column. The value of uplo describes the order in which the interchanges were applied. Also, the sign of ipiv represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, which correspond to 1 or 2 interchanges at each factorization step. If uplo = 'U' (in factorization order, k decreases from n to 1):

A single positive entry ipiv(k) > 0 means that D(k,k) is a 1-by-1 diagonal block. If ipiv(k) != k, rows and columns k and ipiv(k) were interchanged in the matrix A(1:N,1:N). If ipiv(k) = k, no interchange occurred.
A pair of consecutive negative entries ipiv(k) < 0 and ipiv(k-1). < 0 means that D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (Note that negative entries in ipiv appear only in pairs.)
- If -ipiv(k) != k, rows and columns k and -ipiv(k) were interchanged in the matrix A(1:N,1:N). If -ipiv(k) = k, no interchange occurred.
- If -ipiv(k-1) != k-1, rows and columns k-1 and -ipiv(k-1) were interchanged in the matrix A(1:N,1:N). If -ipiv(k-1) = k-1, no interchange occurred.
In both cases 1 and 2, always ABS( ipiv(k) ) ≤ k.

NOTE:

Any entry ipiv(k) is always nonzero on output.

If uplo = 'L' (in factorization order, k increases from 1 to n):

A single positive entry ipiv(k) > 0 means that D(k,k) is a 1-by-1 diagonal block. If ipiv(k) != k, rows and columns k and ipiv(k) were interchanged in the matrix A(1:N,1:N). If ipiv(k) = k, no interchange occurred.
A pair of consecutive negative entries ipiv(k) < 0 and ipiv(k+1) < 0 means that D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (Note that negative entries in ipiv appear only in pairs.)
- If -ipiv(k) != k, rows and columns k and -ipiv(k) were interchanged in the matrix A(1:N,1:N). If -ipiv(k) = k, no interchange occurred.
- If -ipiv(k+1) != k+1, rows and columns k-1 and -ipiv(k-1) were interchanged in the matrix A(1:N,1:N). If -ipiv(k+1) = k+1, no interchange occurred.
In both cases 1 and 2, always ABS( ipiv(k) ) ≥ k.

NOTE:

Any entry ipiv(k) is always nonzero on output.

Return Values

This function returns a value info.

= 0: Successful exit.

< 0: If info = -k, the k^th argument had an illegal value.

> 0: If info = k, the matrix A is singular. If uplo = 'U', column k in the upper triangular part of A contains all zeros. If uplo = 'L', column k in the lower triangular part of A contains all zeros. Therefore, D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.

Parent topic: Matrix Factorization: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?sytrf_rk

Description

Input Parameters

Output Parameters

Return Values