nag_dtftri (f07wkc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dtftri (f07wkc)

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dtftri (f07wkc) computes the inverse of a real triangular matrix stored in Rectangular Full Packed (RFP) format.

2  Specification

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

3  Description

nag_dtftri (f07wkc) forms the inverse of a real triangular matrix A, stored using RFP format. The RFP storage format is described in Section 3.3.3 in the f07 Chapter Introduction. Note that the inverse of an upper (lower) triangular matrix is also upper (lower) triangular.

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 2.3.1.3 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 whether A is upper or lower triangular.
uplo=Nag_Upper
A is upper triangular.
uplo=Nag_Lower
A is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
4:     diag Nag_DiagTypeInput
On entry: indicates whether A is a nonunit or unit triangular matrix.
diag=Nag_NonUnitDiag
A is a nonunit triangular matrix.
diag=Nag_UnitDiag
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag=Nag_NonUnitDiag or Nag_UnitDiag.
5:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
6:     ar[n×n+1/2] doubleInput/Output
On entry: the upper or lower triangular part (as specified by uplo) of the n by n symmetric matrix A, in either normal or transposed RFP format (as specified by transr). The storage format is described in detail in Section 3.3.3 in the f07 Chapter Introduction.
On exit: A is overwritten by A-1, in the same storage format as A.
7:     fail NagError *Input/Output
The NAG error argument (see Section 2.7 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.
An unexpected error has been triggered by this function. Please contact NAG.
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 A is exactly zero. A is singular its inverse cannot be computed.

7  Accuracy

The computed inverse X satisfies
XA-IcnεXA ,  
where cn 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.
The computed inverse satisfies the forward error bound
X-A-1cnεA-1AX .  
See Du Croz and Higham (1992).

8  Parallelism and Performance

nag_dtftri (f07wkc) 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 13n3.
The complex analogue of this function is nag_ztftri (f07wxc).

10  Example

This example computes the inverse of the matrix A, where
A= 4.30 0.00 0.00 0.00 -3.96 -4.87 0.00 0.00 0.40 0.31 -8.02 0.00 -0.27 0.07 -5.95 0.12  
and is stored using RFP format.

10.1  Program Text

Program Text (f07wkce.c)

10.2  Program Data

Program Data (f07wkce.d)

10.3  Program Results

Program Results (f07wkce.r)


nag_dtftri (f07wkc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016