# ?getri

Computes the inverse of an LU-factored general matrix.

## Syntax

FORTRAN 77:

call sgetri( n, a, lda, ipiv, work, lwork, info )

call dgetri( n, a, lda, ipiv, work, lwork, info )

call cgetri( n, a, lda, ipiv, work, lwork, info )

call zgetri( n, a, lda, ipiv, work, lwork, info )

Fortran 95:

call getri( a, ipiv [,info] )

## Description

This routine computes the inverse `inv(A)` of a general matrix A. Before calling this routine, call ?getrf to factorize A.

## Input Parameters

n

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

a, work

  REAL for sgetri

  DOUBLE PRECISION for dgetri

  COMPLEX for cgetri

  DOUBLE COMPLEX for zgetri.

  Arrays: `a(lda,*)`, `work(*)`.

  `a(lda,*)` contains the factorization of the matrix A, as returned by ?getrf: `A = P*L*U`.

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

  `work(*)` is a workspace array of dimension at least `max(1,lwork)`.

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 ?getrf.

lwork

  INTEGER. The size of the work array; lwork ≥ n.

  If `lwork = -1`, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla.

See Application Notes below for the suggested value of lwork.

## Output Parameters

a

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

`work(1)`

  If `info = 0`, on exit `work(1)` contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.

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 factor U is zero, U 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 getri interface are as follows:

a

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

ipiv

  Holds the vector of length (n).

## Application Notes

For better performance, try using `lwork = n*blocksize`, where blocksize is a machine-dependent value (typically, 16 to 64) required for optimum performance of the blocked algorithm.

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set `lwork = -1`.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (`work(1)`) for subsequent runs.

If you set `lwork = -1`, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

The computed inverse X satisfies the following error bound:
`|XA - I| ≤ c(n)ε|X|P|L||U|,`

where `c(n)` is a modest linear function of n; ε is the machine precision; I denotes the identity matrix; P, L, and U are the factors of the matrix factorization `A = P*L*U`.

The total number of floating-point operations is approximately `(4/3)n3` for real flavors and `(16/3)n3` for complex flavors.

Parent topic: Routines for Matrix Inversion