# ?sytri

Computes the inverse of a symmetric matrix.

## Syntax

FORTRAN 77:

call ssytri( uplo, n, a, lda, ipiv, work, info )

call dsytri( uplo, n, a, lda, ipiv, work, info )

call csytri( uplo, n, a, lda, ipiv, work, info )

call zsytri( uplo, n, a, lda, ipiv, work, info )

Fortran 95:

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

## Description

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

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

n

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

a, work

  REAL for ssytri

  DOUBLE PRECISION for dsytri

  COMPLEX for csytri

  DOUBLE COMPLEX for zsytri.

  Arrays:

  `a(lda,*)` contains the factorization of the matrix A, as returned by ?sytrf.

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

  `work(*)` is a workspace array.

  The dimension of work must be at least `max(1,2*n)`.

lda

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

ipiv

  INTEGER.

  Array, DIMENSION at least `max(1, n)`.

  The ipiv array, as returned by ?sytrf.

## 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 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 sytri interface are as follows:

a

  Holds the matrix A of size (`n,n`).

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