nag_zgecon (f07auc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_zgecon (f07auc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zgecon (f07auc) estimates the condition number of a complex matrix A, where A has been factorized by nag_zgetrf (f07arc).

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zgecon (Nag_OrderType order, Nag_NormType norm, Integer n, const Complex a[], Integer pda, double anorm, double *rcond, NagError *fail)

3  Description

nag_zgecon (f07auc) estimates the condition number of a complex matrix A, in either the 1-norm or the -norm:
κ1 A = A1 A-11   or   κ A = A A-1 .
Note that κA=κ1AH.
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 should be preceded by a call to nag_zge_norm (f16uac) to compute A1 or A, and a call to nag_zgetrf (f07arc) to compute the LU factorization of A. The function then 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:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     normNag_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:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     a[dim]const ComplexInput
Note: the dimension, dim, of the array a must be at least max1,pda×n.
The i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the LU factorization of A, as returned by nag_zgetrf (f07arc).
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
6:     anormdoubleInput
On entry: if norm=Nag_OneNorm, the 1-norm of the original matrix A.
If norm=Nag_InfNorm, the -norm of the original matrix A.
anorm may be computed by calling nag_zge_norm (f16uac) with the same value for the argument norm.
anorm must be computed either before calling nag_zgetrf (f07arc) or else from a copy of the original matrix A (see Section 9).
Constraint: anorm0.0.
7:     rconddouble *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.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.
NE_REAL
On entry, anorm=value.
Constraint: anorm0.0.

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 nag_zgecon (f07auc) involves solving a number of systems of linear equations of the form Ax=b or AHx=b; the number is usually 5 and never more than 11. Each solution involves approximately 8n2 real floating point operations but takes considerably longer than a call to nag_zgetrs (f07asc) with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The real analogue of this function is nag_dgecon (f07agc).

9  Example

This example estimates the condition number in the 1-norm of the matrix A, where
A= -1.34+2.55i 0.28+3.17i -6.39-2.20i 0.72-0.92i -0.17-1.41i 3.31-0.15i -0.15+1.34i 1.29+1.38i -3.29-2.39i -1.91+4.42i -0.14-1.35i 1.72+1.35i 2.41+0.39i -0.56+1.47i -0.83-0.69i -1.96+0.67i .
Here A is nonsymmetric and must first be factorized by nag_zgetrf (f07arc). The true condition number in the 1-norm is 231.86.

9.1  Program Text

Program Text (f07auce.c)

9.2  Program Data

Program Data (f07auce.d)

9.3  Program Results

Program Results (f07auce.r)


nag_zgecon (f07auc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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