nag_dtrcon (f07tgc) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_dtrcon (f07tgc)

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dtrcon (f07tgc) estimates the condition number of a real triangular matrix.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dtrcon (Nag_OrderType order, Nag_NormType norm, Nag_UploType uplo, Nag_DiagType diag, Integer n, const double a[], Integer pda, double *rcond, NagError *fail)

3  Description

nag_dtrcon (f07tgc) estimates the condition number of a real triangular matrix A, in either the 1-norm or the -norm:
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 function actually returns an estimate of the reciprocal of the condition number.
The function 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  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:     norm Nag_NormTypeInput
On entry: indicates whether κ1A or κA is estimated.
norm=Nag_OneNorm
κ1A is estimated.
norm=Nag_InfNorm
κA is estimated.
Constraint: norm=Nag_OneNorm or Nag_InfNorm.
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:     a[dim] const doubleInput
Note: the dimension, dim, of the array a must be at least max1,pda×n.
On entry: the n by n triangular matrix A.
If order=Nag_ColMajor, Aij is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[i-1×pda+j-1].
If uplo=Nag_Upper, the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
If diag=Nag_UnitDiag, the diagonal elements of A are assumed to be 1, and are not referenced.
7:     pda IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array a.
Constraint: pdamax1,n.
8:     rcond double *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 if the estimate underflows. If rcond is less than machine precision, then A is singular to working precision.
9:     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.
On entry, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,n.
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.

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  Parallelism and Performance

nag_dtrcon (f07tgc) 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

A call to nag_dtrcon (f07tgc) 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 nag_dtrtrs (f07tec) with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The complex analogue of this function is nag_ztrcon (f07tuc).

10  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 .  
The true condition number in the 1-norm is 116.41.

10.1  Program Text

Program Text (f07tgce.c)

10.2  Program Data

Program Data (f07tgce.d)

10.3  Program Results

Program Results (f07tgce.r)


nag_dtrcon (f07tgc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

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