F07HFF (DPBEQU) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

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

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
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  Parameters

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:     AB(LDAB,*) – 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:     S(N) – 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

Errors or warnings detected by the routine:
If INFO=-i, the ith argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
If INFO=i, the ith 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  Further Comments

The complex analogue of this routine is F07HTF (ZPBEQU).

9  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.

9.1  Program Text

Program Text (f07hffe.f90)

9.2  Program Data

Program Data (f07hffe.d)

9.3  Program Results

Program Results (f07hffe.r)

F07HFF (DPBEQU) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012