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NAG Toolbox: nag_lapack_dgbequ (f07bf)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dgbequ (f07bf) computes diagonal scaling matrices DR  and DC  intended to equilibrate a real m  by n  band matrix A  of band width kl + ku + 1 , and reduce its condition number.

Syntax

[r, c, rowcnd, colcnd, amax, info] = f07bf(m, kl, ku, ab, 'n', n)
[r, c, rowcnd, colcnd, amax, info] = nag_lapack_dgbequ(m, kl, ku, ab, 'n', n)

Description

nag_lapack_dgbequ (f07bf) computes the diagonal scaling matrices. The diagonal scaling matrices are chosen to try to make the elements of largest absolute value in each row and column of the matrix B  given by
B = DR A DC  
have absolute value 1 . The diagonal elements of DR  and DC  are restricted to lie in the safe range δ,1/δ , where δ  is the value returned by function nag_machine_real_safe (x02am). Use of these scaling factors is not guaranteed to reduce the condition number of A  but works well in practice.

References

None.

Parameters

Compulsory Input Parameters

1:     m int64int32nag_int scalar
m, the number of rows of the matrix A.
Constraint: m0.
2:     kl int64int32nag_int scalar
kl, the number of subdiagonals of the matrix A.
Constraint: kl0.
3:     ku int64int32nag_int scalar
ku, the number of superdiagonals of the matrix A.
Constraint: ku0.
4:     abldab: – double array
The first dimension of the array ab must be at least kl+ku+1.
The second dimension of the array ab must be at least max1,n.
The m by n band matrix A whose scaling factors are to be computed.
The matrix is stored in rows 1 to kl+ku+1, more precisely, the element Aij must be stored in
abku+1+i-jj  for ​max1,j-kuiminm,j+kl. 
See Further Comments in nag_lapack_dgbsv (f07ba) for further details.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the second dimension of the array ab.
n, the number of columns of the matrix A.
Constraint: n0.

Output Parameters

1:     rm – double array
If info=0 or info>m, r contains the row scale factors, the diagonal elements of DR. The elements of r will be positive.
2:     cn – double array
If info=0, c contains the column scale factors, the diagonal elements of DC. The elements of c will be positive.
3:     rowcnd – double scalar
If info=0 or info>m, rowcnd contains the ratio of the smallest value of ri to the largest value of ri. If rowcnd0.1 and amax is neither too large nor too small, it is not worth scaling by DR.
4:     colcnd – double scalar
If info=0, colcnd contains the ratio of the smallest value of ci to the largest value of ci.
If colcnd0.1, it is not worth scaling by DC.
5:     amax – double scalar
maxaij. If amax is very close to overflow or underflow, the matrix A should be scaled.
6:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  info>0andinfom
Row _ of A is exactly zero.
W  info>m
Column _ of A is exactly zero.

Accuracy

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

Further Comments

The complex analogue of this function is nag_lapack_zgbequ (f07bt).

Example

This example equilibrates the band matrix A  given by
A = -0.23 2.54 -3.66×10-10 -0 -6.98×1010 2.46×1010 -2.73 -2.13×1010 -0 2.56 -2.46×10-10 -4.07 -0 0 -4.78×10-10 -3.82 .  
Details of the scaling factors, and the scaled matrix are output.
function f07bf_example


fprintf('f07bf example results\n\n');

m  = int64(4);
kl = int64(1);
ku = int64(2);
a = [-2.30e-01,  2.54e+00, -3.66e-10,  0;
     -6.98e+10,  2.46e+10, -2.73e+00  -2.13e+10;
      0,         2.56e+00,  2.46e-10,  4.07e+00;
      0,         0,        -4.78e-10, -3.82e+00];

% Convert a to packed representation
[~, ab, ifail] = f01zc(...
                       'p', kl, ku, a, zeros(m, m));

% Compute row and column scaling factors
[r, c, rowcnd, colcnd, amax, info] = ...
  f07bf(m, kl, ku, ab);

format shorte;
fprintf('\nrowcnd = %8.1e colcnd = %8.1e amax = %8.1e\n\n', ...
        rowcnd, colcnd, amax);
fprintf('Row scale factors:\n');
disp(r');
fprintf('Column scale factors:\n');
disp(c');

% Compute values close to underflow and overflow
small = x02am/(x02aj*double(x02bh));
big = 1/small;
thresh = 0.1;

if (rowcnd >= thresh) && (amax >= small) && (amax <= big)
  if colcnd<thresh
    % Just column scale A
    as = a*diag(c);
  end
elseif colcnd>=thresh
  % Just row scale A
  as = diag(r)*a;
else
  % Row and column scale A
  as = diag(r)*a*diag(c);
end

format short;
fprintf('\nScaled Matrix:\n');
disp(as);


f07bf example results


rowcnd =  3.6e-11 colcnd =  1.4e-10 amax =  7.0e+10

Row scale factors:
   3.9370e-01   1.4327e-11   2.4570e-01   2.6178e-01

Column scale factors:
   1.0000e+00   1.0000e+00   6.9399e+09   1.0000e+00


Scaled Matrix:
   -0.0906    1.0000   -1.0000         0
   -1.0000    0.3524   -0.2714   -0.3052
         0    0.6290    0.4195    1.0000
         0         0   -0.8684   -1.0000


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