Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 6/24/2024
Public

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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

a

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 = maxx·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.