NAG Library Routine Document

f07fff  (dpoequ)


    1  Purpose
    7  Accuracy


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.


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.


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)).


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


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).

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).


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

Parallelism and Performance

f07fff (dpoequ) is not threaded in any implementation.

Further Comments

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


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.

Program Text

Program Text (f07fffe.f90)

Program Data

Program Data (f07fffe.d)

Program Results

Program Results (f07fffe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017