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:     order Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uplo Nag_UploTypeInput
On entry: specifies how A has been factorized.
uplo=Nag_Upper
A=UTU, where U is upper triangular.
uplo=Nag_Lower
A=LLT, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     n IntegerInput
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:     pda IntegerInput
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:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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: pdamax1,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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_SINGULAR
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
Parallelism and Performance

nag_dpotri (f07fjc) 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 23n3.
The complex analogue of this function is nag_zpotri (f07fwc).

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 nag_dpotrf (f07fdc).

10.1
Program Text

Program Text (f07fjce.c)

10.2
Program Data

Program Data (f07fjce.d)

10.3
Program Results

Program Results (f07fjce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017