NAG Library Function Document

nag_dpftri (f07wjc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_dpftri (f07wjc) computes the inverse of a real symmetric positive definite matrix using the Cholesky factorization computed by nag_dpftrf (f07wdc) stored in Rectangular Full Packed (RFP) format.

2
Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dpftri (Nag_OrderType order, Nag_RFP_Store transr, Nag_UploType uplo, Integer n, double ar[], NagError *fail)

3
Description

nag_dpftri (f07wjc) is used to compute the inverse of a real symmetric positive definite matrix A, stored in RFP format. The RFP storage format is described in Section 3.3.3 in the f07 Chapter Introduction. The function must be preceded by a call to nag_dpftrf (f07wdc), 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
Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

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:     transr Nag_RFP_StoreInput
On entry: specifies whether the RFP representation of A is normal or transposed.
transr=Nag_RFP_Normal
The matrix A is stored in normal RFP format.
transr=Nag_RFP_Trans
The matrix A is stored in transposed RFP format.
Constraint: transr=Nag_RFP_Normal or Nag_RFP_Trans.
3:     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.
4:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
5:     ar[n×n+1/2] doubleInput/Output
On entry: the Cholesky factorization of A stored in RFP format, as returned by nag_dpftrf (f07wdc).
On exit: the factorization is overwritten by the n by n matrix A-1 stored in RFP format.
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.
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_MAT_NOT_POS_DEF
The leading minor of order value is not positive definite and the factorization could not be completed. Hence A itself is not positive definite. This may indicate an error in forming the matrix A. There is no function specifically designed to invert a symmetric matrix stored in RFP format which is not positive definite; the matrix must be treated as a full symmetric matrix, by calling nag_dsytri (f07mjc).
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.

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_dpftri (f07wjc) 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_zpftri (f07wwc).

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, stored in RFP format, and must first be factorized by nag_dpftrf (f07wdc).

10.1
Program Text

Program Text (f07wjce.c)

10.2
Program Data

Program Data (f07wjce.d)

10.3
Program Results

Program Results (f07wjce.r)

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