NAG Library Routine Document

f07hff (dpbequ)

1
Purpose

f07hff (dpbequ) computes a diagonal scaling matrix S  intended to equilibrate a real n  by n  symmetric positive definite band matrix A , with bandwidth 2kd+1 , and reduce its condition number.

2
Specification

Fortran Interface
Subroutine f07hff ( uplo, n, kd, ab, ldab, s, scond, amax, info)
Integer, Intent (In):: n, kd, ldab
Integer, Intent (Out):: info
Real (Kind=nag_wp), Intent (In):: ab(ldab,*)
Real (Kind=nag_wp), Intent (Out):: s(n), scond, amax
Character (1), Intent (In):: uplo
C Header Interface
#include <nagmk26.h>
void  f07hff_ (const char *uplo, const Integer *n, const Integer *kd, const double ab[], const Integer *ldab, double s[], double *scond, double *amax, Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name dpbequ.

3
Description

f07hff (dpbequ) 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:     uplo – Character(1)Input
On entry: indicates whether the upper or lower triangular part of A is stored in the array ab, as follows:
uplo='U'
The upper triangle of A is stored.
uplo='L'
The lower triangle of A is stored.
Constraint: uplo='U' or 'L'.
2:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
3:     kd – IntegerInput
On entry: kd, the number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'.
Constraint: kd0.
4:     abldab* – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab must be at least max1,n.
On entry: the upper or lower triangle of the symmetric positive definite band matrix A whose scaling factors are to be computed.
The matrix is stored in rows 1 to kd+1, more precisely,
  • if uplo='U', the elements of the upper triangle of A within the band must be stored with element Aij in abkd+1+i-jj​ for ​max1,j-kdij;
  • if uplo='L', the elements of the lower triangle of A within the band must be stored with element Aij in ab1+i-jj​ for ​jiminn,j+kd.
Only the elements of the array ab corresponding to the diagonal elements of A are referenced. (Row kd+1 of ab when uplo='U', row 1 of ab when uplo='L'.)
5:     ldab – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07hff (dpbequ) is called.
Constraint: ldabkd+1.
6:     sn – Real (Kind=nag_wp) arrayOutput
On exit: if info=0, s contains the diagonal elements of the scaling matrix S.
7:     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.
8:     amax – Real (Kind=nag_wp)Output
On exit: maxaij. If amax is very close to overflow or underflow, the matrix A should be scaled.
9:     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

f07hff (dpbequ) is not threaded in any implementation.

9
Further Comments

The complex analogue of this routine is f07htf (zpbequ).

10
Example

This example equilibrates the symmetric positive definite matrix A  given by
A = 5.49 -2.68×1010 -0 -0 2.68×1010 -5.63×1020 -2.39×1010 -0 0 -2.39×1010 -2.60 -2.22 0 -0 -2.22 -5.17 .  
Details of the scaling factors and the scaled matrix are output.

10.1
Program Text

Program Text (f07hffe.f90)

10.2
Program Data

Program Data (f07hffe.d)

10.3
Program Results

Program Results (f07hffe.r)