?hetri

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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Computes the inverse of a complex Hermitian matrix using U*D*U^H or L*D*L^H Bunch-Kaufman factorization.

Syntax

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

lapack_int LAPACKE_zhetri (int matrix_layout , char uplo , lapack_int n , lapack_complex_double * a , lapack_int lda , const lapack_int * ipiv );

Include Files

Description

The routine computes the inverse inv(A) of a complex Hermitian matrix A. Before calling this routine, call ?hetrf to factorize A.

Input Parameters

matrix_layout	Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
uplo	Must be 'U' or 'L'. Indicates how the input matrix `A` has been factored: If `uplo = 'U'`, the array `a` stores the Bunch-Kaufman factorization `A = UDU^H`. If `uplo = 'L'`, the array `a` stores the Bunch-Kaufman factorization `A = LDL^H`.
n	The order of the matrix `A`; `n≥ 0`.
a,	Array `a`(size max(1, lda*n)) contains the factorization of the matrix `A`, as returned by ?hetrf. The second dimension of `a` must be at least `max(1,n)`.
lda	The leading dimension of `a`; `lda≥ max(1, n)`.
ipiv	Array, size at least `max(1, n)`. The `ipiv` array, as returned by ?hetrf.

Output Parameters

a	Overwritten by the `n`-by-`n` matrix inv(`A`).

Return Values

This function returns a value info.

If info = 0, the execution is successful.

If info = -i, parameter i had an illegal value.

If info = i, the i-th diagonal element of D is zero, D is singular, and the inversion could not be completed.

Application Notes

The computed inverse X satisfies the following error bounds:

|D*U^H*P^T*X*P*U - I| ≤ c(n)ε(|D||U^H|P^T|X|P|U| + |D||D^-1|)

for uplo = 'U', and

|D*L^H*P^T*X*P*L - I| ≤ c(n)ε(|D||L^H|P^T|X|P|L| + |D||D^-1|)

for uplo = 'L'. Here c(n) is a modest linear function of n, and ε is the machine precision; I denotes the identity matrix.

The total number of floating-point operations is approximately (8/3)n³ for complex flavors.

The real counterpart of this routine is ?sytri.

Parent topic: Matrix Inversion: LAPACK Computational Routines