NAG CL Interface
f07ajc (dgetri)

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1 Purpose

f07ajc computes the inverse of a real matrix A, where A has been factorized by f07adc.

2 Specification

#include <nag.h>
void  f07ajc (Nag_OrderType order, Integer n, double a[], Integer pda, const Integer ipiv[], NagError *fail)
The function may be called by the names: f07ajc, nag_lapacklin_dgetri or nag_dgetri.

3 Description

f07ajc is used to compute the inverse of a real matrix A, the function must be preceded by a call to f07adc, which computes the LU factorization of A as A=PLU. The inverse of A is computed by forming U-1 and then solving the equation XPL=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: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least max(1,pda×n).
The (i,j)th element of the matrix A is stored in
  • a[(j-1)×pda+i-1] when order=Nag_ColMajor;
  • a[(i-1)×pda+j-1] when order=Nag_RowMajor.
On entry: the LU factorization of A, as returned by f07adc.
On exit: the factorization is overwritten by the n×n matrix A-1.
4: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax(1,n).
5: ipiv[dim] const Integer Input
Note: the dimension, dim, of the array ipiv must be at least max(1,n).
On entry: the pivot indices, as returned by f07adc.
6: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
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:
|XA-I|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 |AX-I| cannot be guaranteed, although it is almost always satisfied. See Du Croz and Higham (1992).

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07ajc 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 function. 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 43n3.
The complex analogue of this function is f07awc.

10 Example

This example computes the inverse of the matrix A, where
A= ( 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 ) .  
Here A is nonsymmetric and must first be factorized by f07adc.

10.1 Program Text

Program Text (f07ajce.c)

10.2 Program Data

Program Data (f07ajce.d)

10.3 Program Results

Program Results (f07ajce.r)