NAG Library Routine Document

f07wxf (ztftri)

1
Purpose

f07wxf (ztftri) computes the inverse of a complex triangular matrix stored in Rectangular Full Packed (RFP) format.

2
Specification

Fortran Interface
Subroutine f07wxf ( transr, uplo, diag, n, ar, info)
Integer, Intent (In):: n
Integer, Intent (Out):: info
Complex (Kind=nag_wp), Intent (Inout):: ar(n*(n+1)/2)
Character (1), Intent (In):: transr, uplo, diag
C Header Interface
#include <nagmk26.h>
void  f07wxf_ (const char *transr, const char *uplo, const char *diag, const Integer *n, Complex ar[], Integer *info, const Charlen length_transr, const Charlen length_uplo, const Charlen length_diag)
The routine may be called by its LAPACK name ztftri.

3
Description

f07wxf (ztftri) forms the inverse of a complex 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:     transr – Character(1)Input
On entry: specifies whether the normal RFP representation of A or its conjugate transpose is stored.
transr='N'
The matrix A is stored in normal RFP format.
transr='C'
The conjugate transpose of the RFP representation of the matrix A is stored.
Constraint: transr='N' or 'C'.
2:     uplo – Character(1)Input
On entry: specifies whether A is upper or lower triangular.
uplo='U'
A is upper triangular.
uplo='L'
A is lower triangular.
Constraint: uplo='U' or 'L'.
3:     diag – Character(1)Input
On entry: indicates whether A is a nonunit or unit triangular matrix.
diag='N'
A is a nonunit triangular matrix.
diag='U'
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag='N' or 'U'.
4:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
5:     arn×n+1/2 – Complex (Kind=nag_wp) arrayInput/Output
On entry: the upper or lower triangular part (as specified by uplo) of the n by n Hermitian 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.
6:     info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

6
Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
info>0
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

f07wxf (ztftri) 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 real floating-point operations is approximately 43n3.
The real analogue of this routine is f07wkf (dtftri).

10
Example

This example computes the inverse of the matrix A, where
A= 4.78+4.56i 0.00+0.00i 0.00+0.00i 0.00+0.00i 2.00-0.30i -4.11+1.25i 0.00+0.00i 0.00+0.00i 2.89-1.34i 2.36-4.25i 4.15+0.80i 0.00+0.00i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33-0.26i  
and is stored using RFP format.

10.1
Program Text

Program Text (f07wxfe.f90)

10.2
Program Data

Program Data (f07wxfe.d)

10.3
Program Results

Program Results (f07wxfe.r)