nag_dpotri (f07fjc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_dpotri (f07fjc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dpotri (f07fjc) computes the inverse of a real symmetric positive definite matrix A, where A has been factorized by nag_dpotrf (f07fdc).

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dpotri (Nag_OrderType order, Nag_UploType uplo, Integer n, double a[], Integer pda, NagError *fail)

3  Description

nag_dpotri (f07fjc) is used to compute the inverse of a real symmetric positive definite matrix A, the function must be preceded by a call to nag_dpotrf (f07fdc), which computes the Cholesky factorization of A.
If uplo=Nag_Upper, A=UTU and A-1 is computed by first inverting U and then forming U-1U-T.
If uplo=Nag_Lower, A=LLT and A-1 is computed by first inverting L and then forming L-TL-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:     orderNag_OrderTypeInput
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 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uploNag_UploTypeInput
On entry: specifies how A has been factorized.
A=UTU, where U is upper triangular.
A=LLT, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
On entry: the upper triangular matrix U if uplo=Nag_Upper or the lower triangular matrix L if uplo=Nag_Lower, as returned by nag_dpotrf (f07fdc).
On exit: U is overwritten by the upper triangle of A-1 if uplo=Nag_Upper; L is overwritten by the lower triangle of A-1 if uplo=Nag_Lower.
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array a.
Constraint: pdamax1,n.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
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.
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-I2cnεκ2A   and   AX-I2cnεκ2A ,
where cn is a modest function of n, ε is the machine precision and κ2A is the condition number of A defined by
κ2A=A2A-12 .

8  Further Comments

The total number of floating point operations is approximately 23n3.
The complex analogue of this function is nag_zpotri (f07fwc).

9  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 nag_dpotrf (f07fdc).

9.1  Program Text

Program Text (f07fjce.c)

9.2  Program Data

Program Data (f07fjce.d)

9.3  Program Results

Program Results (f07fjce.r)

nag_dpotri (f07fjc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012