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NAG Toolbox: nag_lapack_zppequ (f07gt)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zppequ (f07gt) 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.

Syntax

[s, scond, amax, info] = f07gt(uplo, n, ap)
[s, scond, amax, info] = nag_lapack_zppequ(uplo, n, ap)

Description

nag_lapack_zppequ (f07gt) 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)).

References

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

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Indicates whether the upper or lower triangular part of A is stored in the array ap, 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 int64int32nag_int scalar
n, the order of the matrix A.
Constraint: n0.
3:     ap: – complex array
The dimension of the array ap must be at least max1,n×n+1/2
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.

Optional Input Parameters

None.

Output Parameters

1:     sn – double array
If info=0, s contains the diagonal elements of the scaling matrix S.
2:     scond – double scalar
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.
3:     amax – double scalar
maxaij. If amax is very close to overflow or underflow, the matrix A should be scaled.
4:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

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 _th diagonal element of A is not positive (and hence A cannot be positive definite).

Accuracy

The computed scale factors will be close to the exact scale factors.

Further Comments

The real analogue of this function is nag_lapack_dppequ (f07gf).

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.
function f07gt_example


fprintf('f07gt example results\n\n');

% Upper triangular part of Hermitian matrix A
uplo = 'Upper';
n = int64(4);
ap = [ 3.23   + 0i,       ...
       1.51   - 1.92i,    3.58   + 0i,      ...
       1.90e5 + 0.84e5i, -0.23e5 + 1.11e5i, 4.09e10 + 0i,       ...
       0.42   + 2.50i,   -1.18   + 1.37i,   2.33e5  - 0.14e5i,  4.29 + 0i];

% Scale A
[s, scond, amax, info] = f07gt( ...
                                uplo, n, ap);

fprintf('scond = %8.1e, amax = %8.1e\n\n', scond, amax);
disp('Diagonal scaling factors');
fprintf('%10.1e',s);
fprintf('\n\n');

% Apply scalings
k = 0;
for i = 1:n
  for j = 1:i
    k = k + 1;
    asp(k) = s(i)*ap(k)*s(j);
  end
end

[ifail] = x04dc( ...
                 'Upper', 'Non-unit', n, asp, 'Scaled matrix');


f07gt example results

scond =  8.9e-06, amax =  4.1e+10

Diagonal scaling factors
   5.6e-01   5.3e-01   4.9e-06   4.8e-01

 Scaled matrix
          1       2       3       4
 1   1.0000  0.4441  0.5227  0.1128
     0.0000 -0.5646  0.2311  0.6716

 2           1.0000 -0.0601 -0.3011
             0.0000  0.2901  0.3496

 3                   1.0000  0.5562
                     0.0000 -0.0334

 4                           1.0000
                             0.0000

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