nag_zpocon (f07fuc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zpocon (f07fuc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zpocon (f07fuc) estimates the condition number of a complex Hermitian positive definite matrix A, where A has been factorized by nag_zpotrf (f07frc).

2  Specification

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

3  Description

nag_zpocon (f07fuc) estimates the condition number (in the 1-norm) of a complex Hermitian positive definite matrix A:
κ1A=A1A-11 .
Since A is Hermitian, κ1A=κA=AA-1.
Because κ1A is infinite if A is singular, the function actually returns an estimate of the reciprocal of κ1A.
The function should be preceded by a call to nag_zhe_norm (f16ucc) to compute A1 and a call to nag_zpotrf (f07frc) to compute the Cholesky factorization of A. The function then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate A-11.

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 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uploNag_UploTypeInput
On entry: specifies how A has been factorized.
A=UHU, where U is upper triangular.
A=LLH, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
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.
On entry: the Cholesky factor of A, as returned by nag_zpotrf (f07frc).
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array a.
Constraint: pdamax1,n.
6:     anormdoubleInput
On entry: the 1-norm of the original matrix A, which may be computed by calling nag_zhe_norm (f16ucc) with its argument norm=Nag_OneNorm. anorm must be computed either before calling nag_zpotrf (f07frc) or else from a copy of the original matrix A.
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

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
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.
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_zpocon (f07fuc) involves solving a number of systems of linear equations of the form Ax=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_zpotrs (f07fsc) 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_dpocon (f07fgc).

9  Example

This example estimates the condition number in the 1-norm (or -norm) of the matrix A, where
A= 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i .
Here A is Hermitian positive definite and must first be factorized by nag_zpotrf (f07frc). The true condition number in the 1-norm is 201.92.

9.1  Program Text

Program Text (f07fuce.c)

9.2  Program Data

Program Data (f07fuce.d)

9.3  Program Results

Program Results (f07fuce.r)

nag_zpocon (f07fuc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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