# ?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] )

## Description

This 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 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 first dimension of a; `lda ≥ max(1, n)`.

## Output Parameters

a

  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 = 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.

Parent topic: Routines for Matrix Inversion