NAG FL Interface
f07fff (dpoequ)

Settings help

FL Name Style:

FL Specification Language:

1 Purpose

f07fff computes a diagonal scaling matrix S intended to equilibrate a real n × 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 <nag.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 the names f07fff, nagf_lapacklin_dpoequ or its LAPACK name dpoequ.

3 Description

f07fff 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 , κ2(B) , 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 Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
2: a(lda,*) Real (Kind=nag_wp) array Input
Note: the second dimension of the array a must be at least max(1,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 Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f07fff is called.
Constraint: ldamax(1,n).
4: s(n) Real (Kind=nag_wp) array Output
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: max|aij|. If amax is very close to overflow or underflow, the matrix A should be scaled.
7: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

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

Background information to multithreading can be found in the Multithreading documentation.
f07fff is not threaded in any implementation.

9 Further Comments

The complex analogue of this routine is f07ftf.

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)