?hecon

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?hecon_3

Estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian matrix A.

lapack_int LAPACKE_checon_3 (int matrix_layout, char uplo, lapack_int n, const lapack_complex_float * A, lapack_int lda, const lapack_complex_float * e, const lapack_int * ipiv, float anorm, float * rcond);

lapack_int LAPACKE_zhecon_3 (int matrix_layout, char uplo, lapack_int n, const lapack_complex_double * A, lapack_int lda, const lapack_complex_double * e, const lapack_int * ipiv, double anorm, double * rcond);

Description

?hecon_3 estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian matrix A using the factorization computed by ?hetrf_rk: A = P*U*D*(U^H)*(P^T) or A = P*L*D*(L^H)*(P^T), where U (or L) is unit upper (or lower) triangular matrix, U^H (or L^H) is the conjugate of U (or L), P is a permutation matrix, P^T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as rcond = 1 / (anorm * norm(inv(A))).

This routine uses BLAS3 solver ?hetrs_3.

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 details of the factorization are stored as an upper or lower triangular matrix: = 'U': Upper triangular, form is A = P*U*D*(U^H)*(P^T); = 'L': Lower triangular, form is A = P*L*D*(L^H)*(P^T).

n

The order of the matrix A. n ≥ 0.

A

Array of size max(1, lda*n). Diagonal of the block diagonal matrix D and factor U or L as computed by ?hetrf_rk:

Only diagonal elements of the Hermitian block diagonal matrix D on the diagonal of A—that is, D(k,k) = A(k,k). Superdiagonal (or subdiagonal) elements of D must be provided on entry in array e.

—and—
If uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L', factor L in the subdiagonal part of A.

lda

The leading dimension of the array A.

e

Array of size n. On entry, contains the superdiagonal (or subdiagonal) elements of the Hermitian 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 not referenced. If uplo = 'L', e(i) = D(i+1,i), i=1:N-1, and e(n) is not referenced.

NOTE:

For 1-by-1 diagonal block D(k), where 1 ≤ k ≤ n, the element e[k-1] is not referenced in both the uplo = 'U' and uplo = 'L' cases.

ipiv

Array of size n. Details of the interchanges and the block structure of D as determined by ?hetrf_rk.

anorm

The 1-norm of the original matrix A.

Output Parameters

rcond: The reciprocal of the condition number of the matrix A, computed as rcond = 1/(anorm * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.

Return Values

This function returns a value info.

= 0: Successful exit.

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

Parent topic: Estimating the Condition Number: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?hecon_3