NAG FL Interface
f07hff (dpbequ)

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1 Purpose

f07hff computes a diagonal scaling matrix S intended to equilibrate a real n × 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 <nag.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 the names f07hff, nagf_lapacklin_dpbequ or its LAPACK name dpbequ.

3 Description

f07hff 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: 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 Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: kd Integer Input
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: ab(ldab,*) Real (Kind=nag_wp) array Input
Note: the second dimension of the array ab must be at least max(1,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 ab(kd+1+i-j,j)​ for ​max(1,j-kd)ij;
  • if uplo='L', the elements of the lower triangle of A within the band must be stored with element Aij in ab(1+i-j,j)​ for ​jimin(n,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 Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f07hff is called.
Constraint: ldabkd+1.
6: s(n) Real (Kind=nag_wp) array Output
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: max|aij|. If amax is very close to overflow or underflow, the matrix A should be scaled.
9: 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.
f07hff is not threaded in any implementation.

9 Further Comments

The complex analogue of this routine is f07htf.

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)