# ?pptri

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

## Syntax

FORTRAN 77:

call spptri( uplo, n, ap, info )

call dpptri( uplo, n, ap, info )

call cpptri( uplo, n, ap, info )

call zpptri( uplo, n, ap, info )

Fortran 95:

call pptri( 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 in packed form. Before calling this routine, call ?pptrf to factorize A.

## Input Parameters

uplo

  CHARACTER*1. Must be 'U' or 'L'.

  Indicates whether the upper or lower triangular factor is stored in ap:

  If `uplo = 'U'`, then the upper triangular factor is stored.

  If `uplo = 'L'`, then the lower triangular factor is stored.

n

  INTEGER. The order of the matrix A; `n ≥ 0`.

ap

  REAL for spptri

  DOUBLE PRECISION for dpptri

  COMPLEX for cpptri

  DOUBLE COMPLEX for zpptri.

  Array, DIMENSION at least max(1, n(n+1)/2).

  Contains the factorization of the packed matrix A, as returned by ?pptrf.

  The dimension ap must be at least max(1,n(n+1)/2).

## Output Parameters

ap

  Overwritten by the packed 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 pptri interface are as follows:

a

  Stands for argument ap in Fortan 77 interface. Holds the array A of size `(n*(n+1)/2)`.

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