F07UGF (DTPCON) (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

F07UGF (DTPCON) estimates the condition number of a real triangular matrix, using packed storage.

2  Specification

REAL (KIND=nag_wp)  AP(*), RCOND, WORK(3*N)
The routine may be called by its LAPACK name dtpcon.

3  Description

F07UGF (DTPCON) estimates the condition number of a real triangular matrix A, in either the 1-norm or the -norm, using packed storage:
κ1 A = A1 A-11   or   κ A = A A-1 .
Note that κA=κ1AT.
Because the condition number is infinite if A is singular, the routine actually returns an estimate of the reciprocal of the condition number.
The routine computes A1 or A exactly, and uses Higham's implementation of Hager's method (see Higham (1988)) to estimate A-11 or A-1.

4  References

Higham N J (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396

5  Parameters

1:     NORM – CHARACTER(1)Input
On entry: indicates whether κ1A or κA is estimated.
NORM='1' or 'O'
κ1A is estimated.
κA is estimated.
Constraint: NORM='1', 'O' or 'I'.
2:     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'.
3:     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'.
4:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
5:     AP(*) – REAL (KIND=nag_wp) arrayInput
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’.
6:     RCOND – REAL (KIND=nag_wp)Output
On exit: an estimate of the reciprocal of the condition number of A. RCOND is set to zero if exact singularity is detected or the estimate underflows. If RCOND is less than machine precision, A is singular to working precision.
7:     WORK(3×N) – REAL (KIND=nag_wp) arrayWorkspace
8:     IWORK(N) – INTEGER arrayWorkspace
9:     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.

7  Accuracy

The computed estimate RCOND is never less than the true value ρ, and in practice is nearly always less than 10ρ, although examples can be constructed where RCOND is much larger.

8  Further Comments

A call to F07UGF (DTPCON) involves solving a number of systems of linear equations of the form Ax=b or ATx=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately n2 floating point operations but takes considerably longer than a call to F07UEF (DTPTRS) with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The complex analogue of this routine is F07UUF (ZTPCON).

9  Example

This example estimates the condition number in the 1-norm 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. The true condition number in the 1-norm is 116.41.

9.1  Program Text

Program Text (f07ugfe.f90)

9.2  Program Data

Program Data (f07ugfe.d)

9.3  Program Results

Program Results (f07ugfe.r)

F07UGF (DTPCON) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

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