# ?sptri

Computes the inverse of a symmetric matrix using packed storage.

## Syntax

FORTRAN 77:

call ssptri( uplo, n, ap, ipiv, work, info )

call dsptri( uplo, n, ap, ipiv, work, info )

call csptri( uplo, n, ap, ipiv, work, info )

call zsptri( uplo, n, ap, ipiv, work, info )

Fortran 95:

call sptri( a, ipiv [,uplo] [,info] )

## Description

This routine computes the inverse `inv(A)` of a packed symmetric matrix A. Before calling this routine, call ?sptrf 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 array ap stores the Bunch-Kaufman factorization `A = P*U*D*UT*PT`.

  If `uplo = 'L'`, the array ap stores the Bunch-Kaufman factorization `A = P*L*D*LT*PT`.

n

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

ap, work

  REAL for ssptri

  DOUBLE PRECISION for dsptri

  COMPLEX for csptri

  DOUBLE COMPLEX for zsptri.

  Arrays:

  `ap(*)` contains the factorization of the matrix A, as returned by ?sptrf.

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

  `work(*)` is a workspace array.

  The dimension of work must be at least `max(1,n)`.

ipiv

  INTEGER.

  Array, DIMENSION at least `max(1, n)`. The ipiv array, as returned by ?sptrf.

## Output Parameters

ap

  Overwritten by the n-by-n matrix `inv(A)` in packed form.

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 D is zero, D is singular, 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 sptri interface are as follows:

a

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

ipiv

  Holds the vector of length (n).

uplo

  Must be 'U' or 'L'. The default value is 'U'.

## Application Notes

The computed inverse X satisfies the following error bounds:
`|D*UT*PT*X*P*U - I| ≤ c(n)ε(|D||UT|PT|X|P|U| + |D||D-1|)`

for uplo = 'U', and
`|D*LT*PT*X*P*L - I| ≤ c(n)ε(|D||LT|PT|X|P|L| + |D||D-1|)`

for `uplo = 'L'`. Here `c(n)` is a modest linear function of n, and ε is the machine precision; I denotes the identity matrix.

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