?potri

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?potri

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

Syntax

call spotri( uplo, n, a, lda, info )

call dpotri( uplo, n, a, lda, info )

call cpotri( uplo, n, a, lda, info )

call zpotri( uplo, n, a, lda, info )

call potri( a [,uplo] [,info] )

Include Files

mkl.fi, lapack.f90

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

uplo	CHARACTER*1. 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	INTEGER. The order of the matrix `A`; `n≥ 0`.
a	REAL for spotri DOUBLE PRECISION for dpotri COMPLEX for cpotri DOUBLE COMPLEX for zpotri. Array `a`(size `lda` by *) Contains the factorization of the matrix `A`, as returned by ?potrf. The second dimension of `a` must be at least `max(1, n)`.
lda	INTEGER. The leading dimension of `a`. `lda≥ max(1, n).`

Output Parameters

Overwritten by the upper or lower triangle of the inverse of A.

info

INTEGER.

If info = 0, the execution is successful.

If info = -i, the i-th parameter 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.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine potri interface are as follows:

a	Holds the matrix `A` of size (`n,n`).
uplo	Must be 'U' or 'L'. The default value is 'U'.

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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