NAG Library Routine Document

f07hff  (dpbequ)


    1  Purpose
    7  Accuracy


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.


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.


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)).


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


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:
The upper triangle of A is stored.
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).

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

f07hff (dpbequ) is not threaded in any implementation.

Further Comments

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


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.

Program Text

Program Text (f07hffe.f90)

Program Data

Program Data (f07hffe.d)

Program Results

Program Results (f07hffe.r)

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