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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_wav_3d_coeff_ext (c09fy)

▸▿ Contents

    1  Purpose
    2  Syntax
    3  Description
    4  References
▸▿ 5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
    8  Further Comments
    9  Example

Purpose

nag_wav_3d_coeff_ext (c09fy) extracts a selected set of discrete wavelet transform (DWT) coefficients from the full set of coefficients stored in compact form, as computed by nag_wav_3d_sngl_fwd (c09fa) (single level three-dimensional DWT) or nag_wav_3d_multi_fwd (c09fc) (multi-level three-dimensional DWT).

Syntax

[d, icomm, ifail] = c09fy(ilev, cindex, c, icomm, 'lenc', lenc)
[d, icomm, ifail] = nag_wav_3d_coeff_ext(ilev, cindex, c, icomm, 'lenc', lenc)

Description

nag_wav_3d_coeff_ext (c09fy) is intended to be used after a call to either nag_wav_3d_sngl_fwd (c09fa) (single level three-dimensional DWT) or nag_wav_3d_multi_fwd (c09fc) (multi-level three-dimensional DWT), either of which must be preceded by a call to nag_wav_3d_init (c09ac) (three-dimensional wavelet filter initialization). Given an initial three-dimensional data set A, a prior call to nag_wav_3d_sngl_fwd (c09fa) or nag_wav_3d_multi_fwd (c09fc) computes the approximation coefficients (at the highest requested level in the case of nag_wav_3d_multi_fwd (c09fc)) and seven sets of detail coefficients (at all levels in the case of nag_wav_3d_multi_fwd (c09fc)) and stores these in compact form in a one-dimensional array c. nag_wav_3d_coeff_ext (c09fy) can then extract either the approximation coefficients or one of the sets of detail coefficients (at one of the levels following nag_wav_3d_multi_fwd (c09fc)) into a three-dimensional data set stored in d.
If a multi-level DWT was performed by a prior call to nag_wav_3d_multi_fwd (c09fc) then the dimensions of the three-dimensional data stored in d depend on the level extracted and are available from the arrays dwtlvm, dwtlvn and dwtlvfr as returned by nag_wav_3d_multi_fwd (c09fc) which contain the first, second and third dimensions respectively.
If a single level DWT was performed by a prior call to nag_wav_3d_sngl_fwd (c09fa) then the dimensions of the three-dimensional data stored in d can be determined from nwct, nwcn and nwcfr as returned by the setup function nag_wav_3d_init (c09ac).
See Multiresolution and higher dimensional DWT in the C09 Chapter Introduction for a discussion of the three-dimensional DWT.

References

None.

Parameters

Note: the following notation is used in this section:

Compulsory Input Parameters

1:     ilev – int64int32nag_int scalar
The level at which coefficients are to be extracted.
If ilev=0, it is assumed that the coefficient array c was produced by a preceding call to the single level function nag_wav_3d_sngl_fwd (c09fa).
If ilev>0, it is assumed that the coefficient array c was produced by a preceding call to the multi-level function nag_wav_3d_multi_fwd (c09fc).
Constraints:
2:     cindex – int64int32nag_int scalar
Identifies which coefficients to extract. The coefficients are identified as follows:
cindex=0
The approximation coefficients, produced by application of the low pass filter over columns, rows and frames of A (LLL). After a call to the multi-level transform function nag_wav_3d_multi_fwd (c09fc) (which implies that ilev>0) the approximation coefficients are available only for ilev=nwl, where nwl is the value used in a preceding call to nag_wav_3d_multi_fwd (c09fc).
cindex=1
The detail coefficients produced by applying the low pass filter over columns and rows of A and the high pass filter over frames (LLH).
cindex=2
The detail coefficients produced by applying the low pass filter over columns, high pass filter over rows and low pass filter over frames of A (LHL).
cindex=3
The detail coefficients produced by applying the low pass filter over columns of A and high pass filter over rows and frames (LHH).
cindex=4
The detail coefficients produced by applying the high pass filter over columns of A and low pass filter over rows and frames (HLL).
cindex=5
The detail coefficients produced by applying the high pass filter over columns, low pass filter over rows and high pass filter over frames of A (HLH).
cindex=6
The detail coefficients produced by applying the high pass filter over columns and rows of A and the low pass filter over frames (HHL).
cindex=7
The detail coefficients produced by applying the high pass filter over columns, rows and frames of A (HHH).
Constraints:
  • if ilev=0, 0≤cindex≤7;
  • if ilev=nwl, following a call to nag_wav_3d_multi_fwd (c09fc) transforming nwl levels, 0≤cindex≤7;
  • otherwise 1≤cindex≤7.
3:     clenc – double array
DWT coefficients, as computed by nag_wav_3d_sngl_fwd (c09fa) or nag_wav_3d_multi_fwd (c09fc).
4:     icomm260 – int64int32nag_int array
Contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization function nag_wav_3d_init (c09ac).

Optional Input Parameters

1:     lenc – int64int32nag_int scalar
Default: the dimension of the array c.
The dimension of the array c.
Constraint: lenc must be unchanged from the value used in the preceding call to either nag_wav_3d_sngl_fwd (c09fa) or nag_wav_3d_multi_fwd (c09fc)..

Output Parameters

1:     dlddsdd: – double array
The last dimension of the array d will be ncfr
sdd=ncn.
The requested coefficients.
If the DWT coefficients were computed by nag_wav_3d_sngl_fwd (c09fa) then
  • if cindex=0, the approximation coefficients are stored in dijk, for i=1,2,…,ncm, j=1,2,…,ncn and k=1,2,…,ncfr;
  • if 1≤cindex≤7, the detail coefficients, as indicated by cindex, are stored in dijk, for i=1,2,…,ncm, j=1,2,…,ncn and k=1,2,…,ncfr.
If the DWT coefficients were computed by nag_wav_3d_multi_fwd (c09fc) then
  • if cindex=0 and ilev=nwl, the approximation coefficients are stored in dijk, for i=1,2,…,ncm, j=1,2,…,ncn and k=1,2,…,ncfr;
  • if 1≤cindex≤7, the detail coefficients, as indicated by cindex, for level ilev are stored in dijk, for i=1,2,…,ncm, j=1,2,…,ncn and k=1,2,…,ncfr.
2:     icomm260 – int64int32nag_int array
Communication array, used to store information between calls to nag_wav_3d_coeff_ext (c09fy).
3:     ifail – int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
Constraint: ilev=0 following a call to the single level function nag_wav_3d_sngl_fwd (c09fa).
Constraint: ilev>0 following a call to the multi-level function nag_wav_3d_multi_fwd (c09fc).
Constraint: ilev≤nwl, where nwl is the number of levels used in the call to nag_wav_3d_multi_fwd (c09fc).
   ifail=2
Constraint: cindex≤7.
Constraint: cindex≥0.
   ifail=3
Constraint: lenc≥nct, where nct is the number of DWT coefficients computed in the preceding call to nag_wav_3d_sngl_fwd (c09fa).
Constraint: lenc≥nct, where nct is the number of DWT coefficients computed in the preceding call to nag_wav_3d_multi_fwd (c09fc).
   ifail=4
Constraint: ldd≥ncm, where ncm is the number of DWT coefficients in the first dimension at the selected level ilev.
Constraint: ldd≥ncm, where ncm is the number of DWT coefficients in the first dimension following the single level transform.
Constraint: sdd≥ncn, where ncn is the number of DWT coefficients in the second dimension at the selected level ilev.
Constraint: sdd≥ncn, where ncn is the number of DWT coefficients in the second dimension following the single level transform.
   ifail=5
Constraint: cindex>0 when ilev<nwl in the preceding call to nag_wav_3d_multi_fwd (c09fc).
   ifail=6
Either the initialization function has not been called first or icomm has been corrupted.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

Not applicable.

Further Comments

None.

Example

See Example in nag_wav_3d_init (c09ac), nag_wav_3d_sngl_fwd (c09fa), nag_wav_3d_multi_fwd (c09fc) and nag_wav_3d_coeff_ins (c09fz).

Open in the MATLAB editor: c09fy_example

function c09fy_example


fprintf('c09fy example results\n\n');

m  = int64(8);
n  = int64(8);
fr = int64(8);
wavnam = 'DB4';
mode = 'zero';
wtrans = 'Multilevel';
a = zeros(m, n, fr);
a(:, :, 1) = [10, 31, 04, 10, 13, 15, 04, 06;
              26, 24, 03, 18, 17, 22, 20, 05;
              06, 05, 06, 11, 22, 23, 23, 01;
              09, 15, 18, 01, 30, 24, 08, 01;
              18, 04, 26, 20, 31, 21, 04, 06;
              25, 23, 25, 14, 13, 03, 03, 29;
              22, 29, 07, 29, 13, 31, 03, 12;
              22, 03, 30, 05, 10, 04, 01, 19];
a(:, :, 2) = [01, 02, 14, 31, 19, 28, 06, 15;
              26, 25, 25, 04, 05, 15, 24, 05;
              01, 29, 08, 18, 22, 18, 31, 23;
              08, 04, 16, 21, 14, 02, 02, 21;
              10, 03, 14, 03, 25, 10, 24, 15;
              03, 16, 26, 21, 16, 19, 25, 27;
              28, 29, 01, 20, 03, 24, 31, 28;
              31, 28, 14, 30, 13, 29, 20, 04];
a(:, :, 3) = [31, 26, 23, 05, 22, 01, 16, 08;
              21, 01, 29, 10, 23, 14, 09, 03;
              20, 10, 11, 22, 26, 31, 03, 21;
              09, 24, 19, 03, 04, 01, 13, 29;
              18, 16, 05, 06, 09, 16, 08, 16;
              32, 19, 32, 01, 06, 04, 01, 17;
              29, 29, 02, 29, 27, 25, 31, 06;
              28, 15, 15, 22, 18, 01, 18, 14];
a(:, :, 4) = [15, 09, 04, 14, 26, 10, 03, 28;
              21, 24, 32, 27, 01, 27, 08, 16;
              10, 27, 29, 15, 13, 01, 05, 16;
              04, 01, 08, 31, 14, 06, 05, 27;
              01, 19, 11, 31, 12, 31, 17, 26;
              27, 01, 16, 06, 18, 02, 17, 17;
              30, 09, 15, 32, 32, 29, 16, 02;
              03, 11, 26, 02, 23, 08, 10, 31];
a(:, :, 5) = [12, 07, 06, 12, 01, 13, 30, 26;
              27, 27, 20, 16, 30, 28, 13, 30;
              29, 15, 15, 05, 01, 13, 31, 02;
              31, 21, 27, 30, 08, 07, 11, 03;
              17, 04, 06, 01, 09, 25, 03, 15;
              12, 18, 16, 05, 09, 16, 06, 13;
              03, 05, 26, 30, 19, 11, 32, 24;
              06, 16, 07, 15, 31, 10, 20, 14];
a(:, :, 6) = [20, 07, 17, 11, 04, 21, 25, 17;
              18, 22, 22, 06, 01, 05, 15, 17;
              25, 24, 16, 13, 19, 16, 23, 10;
              01, 31, 05, 13, 11, 12, 01, 18;
              01, 27, 09, 05, 29, 26, 23, 13;
              02, 17, 17, 14, 31, 21, 16, 05;
              26, 21, 10, 21, 09, 11, 01, 15;
              08, 15, 18, 04, 16, 09, 03, 29];
a(:, :, 7) = [26, 02, 30, 26, 07, 04, 09, 01;
              15, 02, 10, 22, 16, 15, 04, 03;
              04, 07, 32, 27, 07, 05, 17, 04;
              22, 30, 06, 18, 32, 02, 01, 31;
              15, 19, 20, 12, 10, 28, 27, 03;
              26, 31, 21, 02, 27, 10, 22, 13;
              32, 03, 27, 23, 01, 11, 04, 26;
              03, 01, 31, 21, 27, 21, 14, 09];
a(:, :, 8) = [02, 16, 16, 23, 23, 09, 27, 12;
              15, 17, 20, 27, 05, 04, 18, 16;
              29, 32, 20, 08, 14, 32, 11, 04;
              28, 01, 15, 19, 14, 09, 30, 18;
              20, 02, 08, 11, 20, 24, 14, 03;
              18, 15, 16, 03, 23, 01, 19, 31;
              32, 27, 28, 09, 15, 23, 09, 13;
              01, 24, 30, 04, 18, 11, 01, 22];


% Query wavelet filter dimensions
[lmax, nf, nwct, nwcn, nwcfr, icomm, ifail] = ...
c09ac(...
      wavnam, wtrans, mode, m, n, fr);

% Transform one less than the max possible number of levels.
nwl = lmax - 1;

% Perform Discrete Wavelet transform
[c, dwtlvm, dwtlvn, dwtlvfr, icomm, ifail] = ...
c09fc(...
       n, fr, a, nwct, nwl, icomm);

% c09ac returns nwct based on max levels, so recalculate.
nwct = sum(7*dwtlvm(1:nwl).*dwtlvn(1:nwl).*dwtlvfr(1:nwl)) + ...
       dwtlvm(1)*dwtlvn(1)*dwtlvfr(1);

fprintf(' Number of Levels :                     %10d\n\n', nwl);
fprintf(' Length of wavelet filter :             %10d\n', nf);
fprintf(' Total number of wavelet coefficients : %10d\n\n', nwct);
fprintf(' Number of coefficients in 1st dimension for each level:\n');
fprintf(' %8d\n', dwtlvm(1:nwl));
fprintf(' Number of coefficients in 2nd dimension for each level:\n');
fprintf(' %8d\n', dwtlvn(1:nwl));
fprintf(' Number of coefficients in 3rd dimension for each level:\n');
fprintf(' %8d\n', dwtlvfr(1:nwl));

% Select the deepest level and approximation coefficients.
want_level = int64(nwl);
want_coeffs = int64(0);

% Dimensions for this set of coefficients.
nwcm = dwtlvm(1);
nwcn = dwtlvn(1);
nwcfr = dwtlvfr(1);

fprintf('\n--------------------------------\n');
fprintf(' Level %d output is %d by %d by %d.\n', nwl, nwcm, nwcn, nwcfr);
fprintf('--------------------------------\n\n');
fprintf('Approximation coefficients (LLL)\n');

%  Extract the required coefficients
[d, icomm, ifail] = c09fy(...
                          want_level, want_coeffs, c, icomm);

% Print out the selected set of coefficients
fprintf('Level %d, Coefficients %d:\n', want_level, want_coeffs);
matrix = 'General'; diag   = 'Non-unit'; fmt = 'F9.4';
labrow = 'Integer'; labcol = labrow;
rlabs  = {' '};     clabs  = rlabs;
ncols  = int64(80); indent = int64(0);

for k = 1:nwcfr
  fprintf('\n');
  title = sprintf('Frame: %3d',k);
  [ifail] =  x04cb(...
                   matrix, diag, d(:,:,k), fmt, title, labrow, ...
                   rlabs, labcol, clabs, ncols, indent);
end

% Reconstruct original data
[b, ifail] = c09fd(nwl, c, m, n, fr, icomm);

fprintf('\n Reconstruction       b : \n');
% Result should be integers so use more compact output
fmt = 'F6.1';
for k = 1:fr
  fprintf('\n');
  title = sprintf('Frame: %3d',k);
  [ifail] =  x04cb(...
                   matrix, diag, b(:,:,k), fmt, title, labrow, ...
                   rlabs, labcol, clabs, ncols, indent);
end



c09fy example results

 Number of Levels :                              2

 Length of wavelet filter :                      8
 Total number of wavelet coefficients :       5145

 Number of coefficients in 1st dimension for each level:
        7
        7
 Number of coefficients in 2nd dimension for each level:
        7
        7
 Number of coefficients in 3rd dimension for each level:
        7
        7

--------------------------------
 Level 2 output is 7 by 7 by 7.
--------------------------------

Approximation coefficients (LLL)
Level 2, Coefficients 0:

 Frame:   1
           1        2        3        4        5        6        7
 1   -0.0000  -0.0000   0.0000   0.0000   0.0001   0.0000   0.0000
 2   -0.0000  -0.0000   0.0000  -0.0001   0.0000  -0.0007  -0.0000
 3    0.0000   0.0000  -0.0001  -0.0002  -0.0020   0.0036  -0.0002
 4   -0.0000  -0.0000  -0.0002   0.0021   0.0025  -0.0124   0.0010
 5    0.0001  -0.0000  -0.0017   0.0009   0.0928   0.1155   0.0004
 6    0.0002  -0.0007   0.0013  -0.0063   0.1584   0.0931   0.0096
 7    0.0000  -0.0001   0.0003  -0.0006   0.0123   0.0061   0.0014

 Frame:   2
           1        2        3        4        5        6        7
 1   -0.0000   0.0000   0.0000  -0.0000  -0.0010  -0.0005  -0.0000
 2    0.0000  -0.0000   0.0001  -0.0006   0.0026   0.0035   0.0004
 3    0.0001  -0.0000  -0.0008   0.0027   0.0133  -0.0064  -0.0032
 4   -0.0002   0.0000   0.0032  -0.0067  -0.0708   0.0073   0.0148
 5   -0.0003   0.0035  -0.0155   0.0406  -0.3676  -0.3434  -0.0682
 6   -0.0011   0.0004   0.0241  -0.0866  -0.4993  -0.5807  -0.0674
 7   -0.0002  -0.0003   0.0048  -0.0128  -0.0800  -0.0731  -0.0045

 Frame:   3
           1        2        3        4        5        6        7
 1    0.0000   0.0000  -0.0002   0.0005   0.0006   0.0027   0.0005
 2   -0.0000   0.0002  -0.0012   0.0037  -0.0224   0.0005  -0.0006
 3   -0.0002  -0.0011   0.0067  -0.0126   0.0447  -0.0734   0.0068
 4    0.0008   0.0025  -0.0141  -0.0008   0.0872   0.3261  -0.0494
 5    0.0012  -0.0173   0.0687  -0.0681   0.5915  -0.1717   0.3943
 6    0.0016   0.0123  -0.1221   0.4190  -0.5269   1.2295   0.1617
 7    0.0003   0.0028  -0.0182   0.0396   0.1154   0.2823   0.0102

 Frame:   4
           1        2        3        4        5        6        7
 1   -0.0000  -0.0002   0.0011  -0.0030   0.0059  -0.0102  -0.0026
 2    0.0000  -0.0010   0.0042  -0.0106   0.0948  -0.0180  -0.0005
 3    0.0004   0.0061  -0.0296   0.0586  -0.3921   0.3650   0.0134
 4   -0.0018  -0.0155   0.0684  -0.0636   0.5365  -1.4566   0.0298
 5   -0.0070   0.0592  -0.1486  -0.1055  -2.9693   0.1109  -1.4193
 6   -0.0017  -0.0424   0.2595  -0.7280   2.4682  -4.1771  -0.5119
 7    0.0003  -0.0079   0.0273  -0.0205  -0.1224  -0.9982  -0.0710

 Frame:   5
           1        2        3        4        5        6        7
 1    0.0001  -0.0000  -0.0005  -0.0015   0.0804   0.1009   0.0139
 2   -0.0006   0.0033  -0.0017  -0.0019  -0.5303  -0.5712  -0.0438
 3   -0.0014  -0.0157   0.0800  -0.1856   0.4182   0.4931   0.0090
 4    0.0099   0.0522  -0.4140   1.1260   0.6111  -0.0042  -0.1288
 5    0.0831  -0.4718   0.9591  -2.9510  84.8494  91.3686  10.1751
 6    0.1599  -0.3194  -0.8962   1.8546 106.1903 117.2751  12.9904
 7    0.0213  -0.0211  -0.2179   0.4955  12.5323  12.9746   1.3422

 Frame:   6
           1        2        3        4        5        6        7
 1    0.0002  -0.0004  -0.0006   0.0005   0.0945   0.1342   0.0157
 2   -0.0008   0.0048  -0.0052   0.0013  -0.7012  -0.3668  -0.0231
 3   -0.0006  -0.0125   0.0347  -0.0396   1.3945  -0.2227  -0.1395
 4    0.0034   0.0166  -0.0246  -0.0495  -3.2417  -0.3508   0.3284
 5    0.1373  -0.4804  -0.1436   0.6068 105.5811 101.7766  10.0719
 6    0.1359  -0.6132   0.8736  -2.8616 121.1074 124.4215  13.7050
 7    0.0068  -0.0939   0.4312  -1.4152  12.9366  13.1259   1.6024

 Frame:   7
           1        2        3        4        5        6        7
 1    0.0000  -0.0001   0.0006  -0.0024   0.0134   0.0160   0.0014
 2   -0.0001   0.0006   0.0003  -0.0044  -0.0813  -0.0377  -0.0021
 3    0.0006   0.0002  -0.0206   0.0816   0.0851  -0.0274  -0.0148
 4   -0.0028  -0.0074   0.1035  -0.3488   0.0136  -0.1313   0.0288
 5    0.0177  -0.0358  -0.0968   0.1416  11.4442  11.6279   0.9779
 6    0.0187  -0.0759   0.0227   0.1041  13.7268  13.3069   1.5629
 7    0.0002  -0.0164   0.0748  -0.2042   1.6290   1.2827   0.1547

 Reconstruction       b : 

 Frame:   1
        1     2     3     4     5     6     7     8
 1   10.0  31.0   4.0  10.0  13.0  15.0   4.0   6.0
 2   26.0  24.0   3.0  18.0  17.0  22.0  20.0   5.0
 3    6.0   5.0   6.0  11.0  22.0  23.0  23.0   1.0
 4    9.0  15.0  18.0   1.0  30.0  24.0   8.0   1.0
 5   18.0   4.0  26.0  20.0  31.0  21.0   4.0   6.0
 6   25.0  23.0  25.0  14.0  13.0   3.0   3.0  29.0
 7   22.0  29.0   7.0  29.0  13.0  31.0   3.0  12.0
 8   22.0   3.0  30.0   5.0  10.0   4.0   1.0  19.0

 Frame:   2
        1     2     3     4     5     6     7     8
 1    1.0   2.0  14.0  31.0  19.0  28.0   6.0  15.0
 2   26.0  25.0  25.0   4.0   5.0  15.0  24.0   5.0
 3    1.0  29.0   8.0  18.0  22.0  18.0  31.0  23.0
 4    8.0   4.0  16.0  21.0  14.0   2.0   2.0  21.0
 5   10.0   3.0  14.0   3.0  25.0  10.0  24.0  15.0
 6    3.0  16.0  26.0  21.0  16.0  19.0  25.0  27.0
 7   28.0  29.0   1.0  20.0   3.0  24.0  31.0  28.0
 8   31.0  28.0  14.0  30.0  13.0  29.0  20.0   4.0

 Frame:   3
        1     2     3     4     5     6     7     8
 1   31.0  26.0  23.0   5.0  22.0   1.0  16.0   8.0
 2   21.0   1.0  29.0  10.0  23.0  14.0   9.0   3.0
 3   20.0  10.0  11.0  22.0  26.0  31.0   3.0  21.0
 4    9.0  24.0  19.0   3.0   4.0   1.0  13.0  29.0
 5   18.0  16.0   5.0   6.0   9.0  16.0   8.0  16.0
 6   32.0  19.0  32.0   1.0   6.0   4.0   1.0  17.0
 7   29.0  29.0   2.0  29.0  27.0  25.0  31.0   6.0
 8   28.0  15.0  15.0  22.0  18.0   1.0  18.0  14.0

 Frame:   4
        1     2     3     4     5     6     7     8
 1   15.0   9.0   4.0  14.0  26.0  10.0   3.0  28.0
 2   21.0  24.0  32.0  27.0   1.0  27.0   8.0  16.0
 3   10.0  27.0  29.0  15.0  13.0   1.0   5.0  16.0
 4    4.0   1.0   8.0  31.0  14.0   6.0   5.0  27.0
 5    1.0  19.0  11.0  31.0  12.0  31.0  17.0  26.0
 6   27.0   1.0  16.0   6.0  18.0   2.0  17.0  17.0
 7   30.0   9.0  15.0  32.0  32.0  29.0  16.0   2.0
 8    3.0  11.0  26.0   2.0  23.0   8.0  10.0  31.0

 Frame:   5
        1     2     3     4     5     6     7     8
 1   12.0   7.0   6.0  12.0   1.0  13.0  30.0  26.0
 2   27.0  27.0  20.0  16.0  30.0  28.0  13.0  30.0
 3   29.0  15.0  15.0   5.0   1.0  13.0  31.0   2.0
 4   31.0  21.0  27.0  30.0   8.0   7.0  11.0   3.0
 5   17.0   4.0   6.0   1.0   9.0  25.0   3.0  15.0
 6   12.0  18.0  16.0   5.0   9.0  16.0   6.0  13.0
 7    3.0   5.0  26.0  30.0  19.0  11.0  32.0  24.0
 8    6.0  16.0   7.0  15.0  31.0  10.0  20.0  14.0

 Frame:   6
        1     2     3     4     5     6     7     8
 1   20.0   7.0  17.0  11.0   4.0  21.0  25.0  17.0
 2   18.0  22.0  22.0   6.0   1.0   5.0  15.0  17.0
 3   25.0  24.0  16.0  13.0  19.0  16.0  23.0  10.0
 4    1.0  31.0   5.0  13.0  11.0  12.0   1.0  18.0
 5    1.0  27.0   9.0   5.0  29.0  26.0  23.0  13.0
 6    2.0  17.0  17.0  14.0  31.0  21.0  16.0   5.0
 7   26.0  21.0  10.0  21.0   9.0  11.0   1.0  15.0
 8    8.0  15.0  18.0   4.0  16.0   9.0   3.0  29.0

 Frame:   7
        1     2     3     4     5     6     7     8
 1   26.0   2.0  30.0  26.0   7.0   4.0   9.0   1.0
 2   15.0   2.0  10.0  22.0  16.0  15.0   4.0   3.0
 3    4.0   7.0  32.0  27.0   7.0   5.0  17.0   4.0
 4   22.0  30.0   6.0  18.0  32.0   2.0   1.0  31.0
 5   15.0  19.0  20.0  12.0  10.0  28.0  27.0   3.0
 6   26.0  31.0  21.0   2.0  27.0  10.0  22.0  13.0
 7   32.0   3.0  27.0  23.0   1.0  11.0   4.0  26.0
 8    3.0   1.0  31.0  21.0  27.0  21.0  14.0   9.0

 Frame:   8
        1     2     3     4     5     6     7     8
 1    2.0  16.0  16.0  23.0  23.0   9.0  27.0  12.0
 2   15.0  17.0  20.0  27.0   5.0   4.0  18.0  16.0
 3   29.0  32.0  20.0   8.0  14.0  32.0  11.0   4.0
 4   28.0   1.0  15.0  19.0  14.0   9.0  30.0  18.0
 5   20.0   2.0   8.0  11.0  20.0  24.0  14.0   3.0
 6   18.0  15.0  16.0   3.0  23.0   1.0  19.0  31.0
 7   32.0  27.0  28.0   9.0  15.0  23.0   9.0  13.0
 8    1.0  24.0  30.0   4.0  18.0  11.0   1.0  22.0
PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox
© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015