nag_zppequ (f07gtc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zppequ (f07gtc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zppequ (f07gtc) computes a diagonal scaling matrix S  intended to equilibrate a complex n  by n  Hermitian positive definite matrix A , stored in packed format, and reduce its condition number.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zppequ (Nag_OrderType order, Nag_UploType uplo, Integer n, const Complex ap[], double s[], double *scond, double *amax, NagError *fail)

3  Description

nag_zppequ (f07gtc) computes a diagonal scaling matrix S  chosen so that
sj=1 / ajj .
This means that the matrix B  given by
B=SAS ,
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 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:     uploNag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of A is stored in the array ap, as follows:
uplo=Nag_Upper
The upper triangle of A is stored.
uplo=Nag_Lower
The lower triangle of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     ap[dim]const ComplexInput
Note: the dimension, dim, of the array ap must be at least max1,n×n+1/2.
On entry: the n by n Hermitian matrix A, packed by rows or columns.
The storage of elements Aij depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ap[j-1×j/2+i-1], for ij;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ap[2n-j×j-1/2+i-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ap[2n-i×i-1/2+j-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ap[i-1×i/2+j-1], for ij.
Only the elements of ap corresponding to the diagonal elements A are referenced.
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

NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
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_MAT_NOT_POS_DEF
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_dppequ (f07gfc).

9  Example

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

9.1  Program Text

Program Text (f07gtce.c)

9.2  Program Data

Program Data (f07gtce.d)

9.3  Program Results

Program Results (f07gtce.r)


nag_zppequ (f07gtc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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