F07UJF (DTPTRI) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F07UJF (DTPTRI) computes the inverse of a real triangular matrix, using packed storage.

2  Specification

REAL (KIND=nag_wp)  AP(*)
The routine may be called by its LAPACK name dtptri.

3  Description

F07UJF (DTPTRI) forms the inverse of a real triangular matrix A, using packed storage. 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

5  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: specifies whether A is upper or lower triangular.
A is upper triangular.
A is lower triangular.
Constraint: UPLO='U' or 'L'.
2:     DIAG – CHARACTER(1)Input
On entry: indicates whether A is a nonunit or unit triangular matrix.
A is a nonunit triangular matrix.
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: DIAG='N' or 'U'.
3:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
4:     AP(*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array AP must be at least max1,N×N+1/2.
On entry: the n by n triangular matrix A, packed by columns.
More precisely,
  • if UPLO='U', the upper triangle of A must be stored with element Aij in APi+jj-1/2 for ij;
  • if UPLO='L', the lower triangle of A must be stored with element Aij in APi+2n-jj-1/2 for ij.
If DIAG='U', the diagonal elements of A are assumed to be 1, and are not referenced; the same storage scheme is used whether DIAG='N' or ‘U’.
On exit: A is overwritten by A-1, using the same storage format as described above.
5:     INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
If INFO=-i, the ith parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
If INFO=i, ai,i is exactly zero; A is singular and 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  Further Comments

The total number of floating point operations is approximately 13n3.
The complex analogue of this routine is F07UWF (ZTPTRI).

9  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 ,
using packed storage.

9.1  Program Text

Program Text (f07ujfe.f90)

9.2  Program Data

Program Data (f07ujfe.d)

9.3  Program Results

Program Results (f07ujfe.r)

F07UJF (DTPTRI) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

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