NAG Library Routine Document

F07AJF (DGETRI)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F07AJF (DGETRI) computes the inverse of a real matrix

A

, where

A

has been factorized by F07ADF (DGETRF).

2 Specification

SUBROUTINE F07AJF (

N, A, LDA, IPIV, WORK, LWORK, INFO)

INTEGER	N, LDA, IPIV(*), LWORK, INFO
REAL (KIND=nag_wp)	A(LDA,*), WORK(max(1,LWORK))

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 Parameters

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

Errors or warnings detected by the routine:

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

$INFO > 0$: If $INFO = i$ , the $i$ th diagonal element of the factor $U$ 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 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).

9 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 DocumentF07AJF (DGETRI)