NAG Library Routine Document

f07fff (dpoequ)

1
Purpose

f07fff (dpoequ) 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

Fortran Interface
Subroutine f07fff ( n, a, lda, s, scond, amax, info)
Integer, Intent (In):: n, lda
Integer, Intent (Out):: info
Real (Kind=nag_wp), Intent (In):: a(lda,*)
Real (Kind=nag_wp), Intent (Out):: s(n), scond, amax
C Header Interface
#include <nagmk26.h>
void  f07fff_ (const Integer *n, const double a[], const Integer *lda, double s[], double *scond, double *amax, Integer *info)
The routine may be called by its LAPACK name dpoequ.

3
Description

f07fff (dpoequ) 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:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
2:     alda* – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array a must be at least max1,n.
On entry: the matrix A whose scaling factors are to be computed. Only the diagonal elements of the array a are referenced.
3:     lda – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07fff (dpoequ) is called.
Constraint: ldamax1,n.
4:     sn – Real (Kind=nag_wp) arrayOutput
On exit: if info=0, s contains the diagonal elements of the scaling matrix S.
5:     scond – Real (Kind=nag_wp)Output
On exit: if info=0, 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.
6:     amax – Real (Kind=nag_wp)Output
On exit: maxaij. If amax is very close to overflow or underflow, the matrix A should be scaled.
7:     info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

6
Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
info>0
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
Parallelism and Performance

f07fff (dpoequ) is not threaded in any implementation.

9
Further Comments

The complex analogue of this routine is f07ftf (zpoequ).

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 (f07fffe.f90)

10.2
Program Data

Program Data (f07fffe.d)

10.3
Program Results

Program Results (f07fffe.r)