NAG FL Interface
f07fjf (dpotri)

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1 Purpose

f07fjf computes the inverse of a real symmetric positive definite matrix A, where A has been factorized by f07fdf.

2 Specification

Fortran Interface
Subroutine f07fjf ( uplo, n, a, lda, info)
Integer, Intent (In) :: n, lda
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*)
Character (1), Intent (In) :: uplo
C Header Interface
#include <nag.h>
void  f07fjf_ (const char *uplo, const Integer *n, double a[], const Integer *lda, Integer *info, const Charlen length_uplo)
The routine may be called by the names f07fjf, nagf_lapacklin_dpotri or its LAPACK name dpotri.

3 Description

f07fjf is used to compute the inverse of a real symmetric positive definite matrix A, the routine must be preceded by a call to f07fdf, which computes the Cholesky factorization of A.
If uplo='U', A=UTU and A-1 is computed by first inverting U and then forming (U-1)U-T.
If uplo='L', A=LLT and A-1 is computed by first inverting L and then forming L-T(L-1).

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: uplo Character(1) Input
On entry: specifies how A has been factorized.
A=UTU, where U is upper triangular.
A=LLT, where L is lower triangular.
Constraint: uplo='U' or 'L'.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: the upper triangular matrix U if uplo='U' or the lower triangular matrix L if uplo='L', as returned by f07fdf.
On exit: U is overwritten by the upper triangle of A-1 if uplo='U'; L is overwritten by the lower triangle of A-1 if uplo='L'.
4: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f07fjf is called.
Constraint: ldamax(1,n).
5: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
Diagonal element value of the Cholesky factor is zero; the Cholesky factor is singular and the inverse of A cannot be computed.

7 Accuracy

The computed inverse X satisfies
XA-I2c(n)εκ2(A)   and   AX-I2c(n)εκ2(A) ,  
where c(n) is a modest function of n, ε is the machine precision and κ2(A) is the condition number of A defined by
κ2(A)=A2A-12 .  

8 Parallelism and Performance

f07fjf 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 23n3.
The complex analogue of this routine is f07fwf.

10 Example

This example computes the inverse of the matrix A, where
A= ( 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 ) .  
Here A is symmetric positive definite and must first be factorized by f07fdf.

10.1 Program Text

Program Text (f07fjfe.f90)

10.2 Program Data

Program Data (f07fjfe.d)

10.3 Program Results

Program Results (f07fjfe.r)