NAG Library Routine Document

Integer, Intent (In)	::	n, lda, ipiv(*), lwork
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,*)
Real (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))

C Header Interface

#include <nagmk26.h>

void	f07ajf_ (const Integer n, double a[], const Integer lda, const Integer ipiv[], double work[], const Integer lwork, Integer info)

The routine may be called by its LAPACK name dgetri.

3

Description

f07ajf (dgetri) is used to compute the inverse of a real matrix

A

, the routine must be preceded by a call to f07adf (dgetrf), which computes the

L U

factorization of

A

A = P L U

. The inverse of

A

is computed by forming

U^{- 1}

and then solving the equation

X P L = U^{- 1}

for

X

4

References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

5

Arguments

1: $n$ – IntegerInput: On entry: $n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $a (lda *)$ – Real (Kind=nag_wp) arrayInput/Output: Note: the second dimension of the array a must be at least $\max (1, n)$ .

On entry: the $L U$ factorization of $A$ , as returned by f07adf (dgetrf).

On exit: the factorization is overwritten by the $n$ by $n$ matrix $A^{- 1}$ .
3: $lda$ – IntegerInput: On entry: the first dimension of the array a as declared in the (sub)program from which f07ajf (dgetri) is called.

Constraint: $lda \geq \max (1, n)$ .
4: $ipiv (*)$ – Integer arrayInput: Note: the dimension of the array ipiv must be at least $\max (1, n)$ .

On entry: the pivot indices, as returned by f07adf (dgetrf).
5: $work (\max (1, lwork))$ – Real (Kind=nag_wp) arrayWorkspace: On exit: if $info = 0$ , $work (1)$ contains the minimum value of lwork required for optimum performance.
6: $lwork$ – IntegerInput: On entry: the dimension of the array work as declared in the (sub)program from which f07ajf (dgetri) is called, unless $lwork = - 1$ , in which case a workspace query is assumed and the routine only calculates the optimal dimension of work (using the formula given below).

Suggested value: for optimum performance lwork should be at least $n \times nb$ , where $nb$ is the block size.

Constraint: $lwork \geq \max (1, n)$ or $lwork = - 1$ .
7: $info$ – IntegerOutput: On exit: $info = 0$ unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0$: Element $〈value〉$ of the diagonal is zero. $U$ is singular, and the inverse of $A$ cannot be computed.

7

Accuracy

The computed inverse

X

satisfies a bound of the form:

|X A - I| \leq c (n) ε |X| P |L| |U|,

where

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

Note that a similar bound for

|A X - I|

cannot be guaranteed, although it is almost always satisfied. See Du Croz and Higham (1992).

8

Parallelism and Performance

f07ajf (dgetri) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

The total number of floating-point operations is approximately

\frac{4}{3} n^{3}

The complex analogue of this routine is f07awf (zgetri).

10

Example

This example computes the inverse of the matrix

A

, where

A = (\begin{array}{r} 1.80 & 2.88 & 2.05 & - 0.89 \\ 5.25 & - 2.95 & - 0.95 & - 3.80 \\ 1.58 & - 2.69 & - 2.90 & - 1.04 \\ - 1.11 & - 0.66 & - 0.59 & 0.80 \end{array}) .

Here

A

is nonsymmetric and must first be factorized by f07adf (dgetrf).

NAG Library Routine Document

f07ajf (dgetri)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification