F07GTF (ZPPEQU) (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

F07GTF (ZPPEQU) computes a diagonal scaling matrix S  intended to equilibrate a complex n  by n  Hermitian positive definite matrix A , stored in packed format, and reduce its condition number.

2  Specification

COMPLEX (KIND=nag_wp)  AP(*)
The routine may be called by its LAPACK name zppequ.

3  Description

F07GTF (ZPPEQU) 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 AP, 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:     AP(*) – COMPLEX (KIND=nag_wp) arrayInput
Note: the dimension of the array AP must be at least max1,N×N+1/2.
On entry: the n by n Hermitian matrix A, packed by columns.
More precisely,
  • if UPLO='U', the upper triangle of A must be stored with element Aij in APi+jj-1/2 for ij;
  • if UPLO='L', the lower triangle of A must be stored with element Aij in APi+2n-jj-1/2 for ij.
Only the elements of AP corresponding to the diagonal elements A are referenced.
4:     S(N) – 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).

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 real analogue of this routine is F07GFF (DPPEQU).

9  Example

This example equilibrates the Hermitian positive definite matrix A  given by
A = (3.23 ((1.51-1.92i 1.90+0.84i×105 ((0.42+2.50i (1.51+1.92i ((3.58 -0.23+1.11i×105 -1.18+1.37i 1.90-0.84i×105 -0.23-1.11i×105 4.09×1010 (2.33-0.14i×105 (0.42-2.50i (-1.18-1.37i 2.33+0.14i×105 ((4.29 .
Details of the scaling factors and the scaled matrix are output.

9.1  Program Text

Program Text (f07gtfe.f90)

9.2  Program Data

Program Data (f07gtfe.d)

9.3  Program Results

Program Results (f07gtfe.r)

F07GTF (ZPPEQU) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

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