NAG C Library Function Document

nag_dpoequ (f07ffc)

1
Purpose

nag_dpoequ (f07ffc) computes a diagonal scaling matrix S  intended to equilibrate a real n  by n  symmetric positive definite matrix A  and reduce its condition number.

2
Specification

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

3
Description

nag_dpoequ (f07ffc) 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:     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 3.3.1.3 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:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
3:     a[dim] const doubleInput
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:     pda IntegerInput
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:     scond double *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:     amax double *Output
On exit: maxaij. If amax is very close to overflow or underflow, the matrix A should be scaled.
8:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_MAT_NOT_POS_DEF
The valueth diagonal element of A is not positive (and hence A cannot be positive definite).
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 scale factors will be close to the exact scale factors.

8
Parallelism and Performance

nag_dpoequ (f07ffc) is not threaded in any implementation.

9
Further Comments

The complex analogue of this function is nag_zpoequ (f07ftc).

10
Example

This example equilibrates the symmetric positive definite matrix A  given by
A = -4.16 -3.12×105 -0.56 -0.10 -3.12×105 -5.03×1010 -0.83×105 -1.18×105 -0.56 -0.83×105 -0.76 -0.34 -0.10 -1.18×105 -0.34 -1.18 .  
Details of the scaling factors and the scaled matrix are output.

10.1
Program Text

Program Text (f07ffce.c)

10.2
Program Data

Program Data (f07ffce.d)

10.3
Program Results

Program Results (f07ffce.r)