?potri

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?potri

Computes the inverse of a symmetric (Hermitian) positive-definite matrix using the Cholesky factorization.

Syntax

lapack_int LAPACKE_spotri (int matrix_layout , char uplo , lapack_int n , float * a , lapack_int lda );

lapack_int LAPACKE_dpotri (int matrix_layout , char uplo , lapack_int n , double * a , lapack_int lda );

lapack_int LAPACKE_cpotri (int matrix_layout , char uplo , lapack_int n , lapack_complex_float * a , lapack_int lda );

lapack_int LAPACKE_zpotri (int matrix_layout , char uplo , lapack_int n , lapack_complex_double * a , lapack_int lda );

Include Files

mkl.h

Description

The routine computes the inverse inv(A) of a symmetric positive definite or, for complex flavors, Hermitian positive-definite matrix A. Before calling this routine, call ?potrf 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 upper triangle of `A` is stored. If `uplo = 'L'`, the lower triangle of `A` is stored.
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 ?potrf.
lda	The leading dimension of `a`. `lda≥ max(1, n).`

Output Parameters

a	Overwritten by the upper or lower triangle of the inverse of `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 the Cholesky factor (and therefore the factor itself) is zero, and the inversion could not be completed.

Application Notes

The computed inverse X satisfies the following error bounds:

||XA - I||₂
					≤
					c(n)ε
					κ
					₂(A), ||AX -  I||₂
					≤
					c(n)ε
					κ
					₂(A),

where c(n) is a modest linear function of n, and ε is the machine precision; I denotes the identity matrix.

The 2-norm ||A||₂ of a matrix A is defined by ||A||₂ = max_x·x=1(Ax·Ax)^1/2, and the condition number κ₂(A) is defined by κ₂(A) = ||A||₂ ||A^-1||₂.

The total number of floating-point operations is approximately (2/3)n³ for real flavors and (8/3)n³ for complex flavors.

Parent topic: Matrix Inversion: LAPACK Computational Routines

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