?hecon

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?hecon_3

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

call checon_3 (uplo, n, A, lda, e, ipiv, anorm, rcond, work, info)

call zhecon_3(uplo, n, A, lda, e, ipiv, anorm, rcond, work, info)

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

uplo

CHARACTER*1. 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

INTEGER. The order of the matrix A. n ≥ 0.

A

COMPLEX for checon_3

COMPLEX*16 for zhecon_3

Array, dimension (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

INTEGER

The leading dimension of the array A.lda ≥ max(1, n).

e

COMPLEX for checon_3

COMPLEX*16 for zhecon_3

Array, dimension (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) is not referenced in both the uplo = 'U' and uplo = 'L' cases.

ipiv

INTEGER

Array, dimension (n). Details of the interchanges and the block structure of D as determined by ?hetrf_rk.

anorm

COMPLEX for checon_3

COMPLEX*16 for zhecon_3

The 1-norm of the original matrix A.

Output Parameters

rcond

COMPLEX for checon_3

COMPLEX*16 for zhecon_3

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.

work

COMPLEX for checon_3

COMPLEX*16 for zhecon_3

Array, dimension (2*n).

info

INTEGER.

= 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 Fortran

?hecon_3