?hetri

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?hetri_3

Computes the inverse of a complex Hermitian matrix using the factorization computed by ?hetrf_rk.

call chetri_3(uplo, n, A, lda, e, ipiv, work, lwork, info)

call zhetri_3(uplo, n, A, lda, e, ipiv, work, lwork, info)

Description

?hetri_3 computes the inverse 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 a 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 a Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

?hetri_3 sets the leading dimension of the workspace before calling ?hetri_3x, which actually computes the inverse.

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

Input Parameters

uplo

CHARACTER*1

Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.

= 'U': The upper triangle of A is stored.
= 'L': The lower triangle of A is stored.

n

INTEGER

The order of the matrix A. n ≥ 0.

A

COMPLEX for chetri_3

COMPLEX*16 for zhetri_3

Array, dimension (lda,n). On entry, 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 should be provided on entry in array e.
If uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L', factor L is the subdiagonal part of A.

lda

INTEGER

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

e

COMPLEX for chetri_3

COMPLEX*16 for zhetri_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.

lwork

INTEGER

The length of the array work.

If LDWORK = -1, a workspace query is assumed; the routine calculates only the optimal size of the work array and returns this value as the first entry of the work array, and no error message related to lwork is issued by XERBLA.

Output Parameters

A

COMPLEX for chetri_3

COMPLEX*16 for zhetri_3

On exit, if info = 0, the Hermitian inverse of the original matrix. If uplo = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced. If uplo = 'L', the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.

work

COMPLEX for chetri_3

COMPLEX*16 for zhetri_3

Array, dimension (n+NB+1)*(NB+3). On exit, if info = 0, work(1) returns the optimal lwork.

info

INTEGER

= 0: Successful exit.
< 0: If info = -i, the i^th argument had an illegal value.
> 0: If info = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.

Parent topic: Matrix Inversion: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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