?potri

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

Syntax

FORTRAN 77:

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 )

FORTRAN 95:

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

lapack_int LAPACKE_<?>potri( int matrix_order, char uplo, lapack_int n, <datatype>* a, lapack_int lda );

Include Files

Fortran: mkl_lapack.fi and mkl_lapack.h
Fortran 95: lapack.f90
C: mkl_lapacke.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

The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.

uplo	CHARACTER*1. Must be 'U' or 'L'. Indicates whether `A` is upper or lower triangular: If `uplo = 'U'`, then `A` is upper triangular. If `uplo = 'L'`, then `A` is lower triangular.
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(lda,*)`. 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 n-by-n matrix inv(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.

Fortran 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 Fortran 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)εκ₂(A), ||AX - I||₂ ≤ c(n)εκ₂(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.