nag_zpoequ (f07ftc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_zpoequ (f07ftc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zpoequ (f07ftc) computes a diagonal scaling matrix S  intended to equilibrate a complex n  by n  Hermitian positive definite matrix A  and reduce its condition number.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zpoequ (Nag_OrderType order, Integer n, const Complex a[], Integer pda, double s[], double *scond, double *amax, NagError *fail)

3  Description

nag_zpoequ (f07ftc) computes a diagonal scaling matrix S  chosen so that
sj=1 / ajj .
This means that the matrix B  given by
has diagonal elements equal to unity. This in turn means that the condition number of B , κ2B , is within a factor n  of the matrix of smallest possible condition number over all possible choices of diagonal scalings (see Corollary 7.6 of Higham (2002)).

4  References

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

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:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
3:     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 matrix A whose scaling factors are to be computed. Only the diagonal elements of the array a are referenced.
4:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
5:     s[n]doubleOutput
On exit: if fail.code= NE_NOERROR, s contains the diagonal elements of the scaling matrix S.
6:     sconddouble *Output
On exit: if fail.code= NE_NOERROR, scond contains the ratio of the smallest value of s to the largest value of s. If scond0.1 and amax is neither too large nor too small, it is not worth scaling by S.
7:     amaxdouble *Output
On exit: maxaij. If amax is very close to overflow or underflow, the matrix A should be scaled.
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.
The valueth diagonal element of A is not positive (and hence A cannot be positive definite).

7  Accuracy

The computed scale factors will be close to the exact scale factors.

8  Further Comments

The real analogue of this function is nag_dpoequ (f07ffc).

9  Example

This example equilibrates the Hermitian positive definite matrix A  given by
A = (3.23 -(1.51-1.92i 1.90+0.84i×1050 -0.42+2.50i (1.51+1.92i -(3.58 -0.23+1.11i×1050 -1.18+1.37i 1.90-0.84i×105 -0.23-1.11i×105 -4.09×1010 (2.33-0.14i×105 (0.42-2.50i (-1.18-1.37i 2.33+0.14i×1050 -4.29 .
Details of the scaling factors and the scaled matrix are output.

9.1  Program Text

Program Text (f07ftce.c)

9.2  Program Data

Program Data (f07ftce.d)

9.3  Program Results

Program Results (f07ftce.r)

nag_zpoequ (f07ftc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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