Mercurial > hg > octave-nkf
annotate scripts/special-matrix/hadamard.m @ 15536:2e8eb9ac43a5 stable rc-3-6-4-0
3.6.4-rc0 release candidate
* configure.ac (AC_INIT): Version is now 3.6.2-rc0.
(OCTAVE_RELEASE_DATE): Now 2012-05-11.
| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Wed, 17 Oct 2012 10:05:44 -0400 |
| parents | 72c96de7a403 |
| children | f3d52523cde1 |
| rev | line source |
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1 ## Copyright (C) 1993-2012 Paul Kienzle |
| 5827 | 2 ## |
| 3 ## This file is part of Octave. | |
| 4 ## | |
| 5 ## Octave is free software; you can redistribute it and/or modify it | |
| 6 ## under the terms of the GNU General Public License as published by | |
| 7016 | 7 ## the Free Software Foundation; either version 3 of the License, or (at |
| 8 ## your option) any later version. | |
| 5827 | 9 ## |
| 10 ## Octave is distributed in the hope that it will be useful, but | |
| 11 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 13 ## General Public License for more details. | |
| 14 ## | |
| 15 ## You should have received a copy of the GNU General Public License | |
| 7016 | 16 ## along with Octave; see the file COPYING. If not, see |
| 17 ## <http://www.gnu.org/licenses/>. | |
| 5827 | 18 ## |
| 19 ## Original version by Paul Kienzle distributed as free software in the | |
| 20 ## public domain. | |
| 21 | |
| 22 ## -*- texinfo -*- | |
| 23 ## @deftypefn {Function File} {} hadamard (@var{n}) | |
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24 ## Construct a Hadamard matrix (@nospell{Hn}) of size @var{n}-by-@var{n}. The |
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25 ## size @var{n} must be of the form @math{2^k * p} in which |
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26 ## p is one of 1, 12, 20 or 28. The returned matrix is normalized, |
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27 ## meaning @w{@code{Hn(:,1) == 1}} and @w{@code{Hn(1,:) == 1}}. |
| 5827 | 28 ## |
| 29 ## Some of the properties of Hadamard matrices are: | |
| 30 ## | |
| 31 ## @itemize @bullet | |
| 32 ## @item | |
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33 ## @code{kron (Hm, Hn)} is a Hadamard matrix of size @var{m}-by-@var{n}. |
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34 ## |
| 5827 | 35 ## @item |
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36 ## @code{Hn * Hn' = @var{n} * eye (@var{n})}. |
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37 ## |
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38 ## @item |
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39 ## The rows of @nospell{Hn} are orthogonal. |
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40 ## |
| 5827 | 41 ## @item |
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42 ## @code{det (@var{A}) <= abs (det (Hn))} for all @var{A} with |
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43 ## @w{@code{abs (@var{A}(i, j)) <= 1}}. |
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44 ## |
| 5827 | 45 ## @item |
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46 ## Multiplying any row or column by -1 and the matrix will remain a Hadamard |
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47 ## matrix. |
| 5827 | 48 ## @end itemize |
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49 ## @seealso{compan, hankel, toeplitz} |
| 5827 | 50 ## @end deftypefn |
| 51 | |
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52 |
| 5827 | 53 ## Reference [1] contains a list of Hadamard matrices up to n=256. |
| 54 ## See code for h28 in hadamard.m for an example of how to extend | |
| 55 ## this function for additional p. | |
| 56 ## | |
| 57 ## References: | |
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58 ## [1] A Library of Hadamard Matrices, N. J. A. Sloane |
| 5827 | 59 ## http://www.research.att.com/~njas/hadamard/ |
| 60 | |
| 61 function h = hadamard (n) | |
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62 |
| 5827 | 63 if (nargin != 1) |
| 64 print_usage (); | |
| 65 endif | |
| 66 | |
| 8507 | 67 ## Find k if n = 2^k*p. |
| 5827 | 68 k = 0; |
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69 while (n > 1 && fix (n/2) == n/2) |
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70 k++; |
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71 n /= 2; |
| 5827 | 72 endwhile |
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73 |
| 8507 | 74 ## Find base hadamard. |
| 75 ## Except for n=2^k, need a multiple of 4. | |
| 5827 | 76 if (n != 1) |
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77 k -= 2; |
| 5827 | 78 endif |
| 79 | |
| 8507 | 80 ## Trigger error if not a multiple of 4. |
| 5827 | 81 if (k < 0) |
| 82 n =- 1; | |
| 83 endif | |
| 84 | |
| 85 switch (n) | |
| 86 case 1 | |
| 87 h = 1; | |
| 88 case 3 | |
| 89 h = h12 (); | |
| 90 case 5 | |
| 91 h = h20 (); | |
| 92 case 7 | |
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93 h = h28 (); |
| 5827 | 94 otherwise |
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95 error ("hadamard: N must be 2^k*p, for p = 1, 12, 20 or 28"); |
| 5827 | 96 endswitch |
| 97 | |
| 8507 | 98 ## Build H(2^k*n) from kron(H(2^k),H(n)). |
| 5827 | 99 h2 = [1,1;1,-1]; |
| 100 while (true) | |
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101 if (fix (k/2) != k/2) |
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102 h = kron (h2, h); |
| 5827 | 103 endif |
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104 k = fix (k/2); |
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105 if (k == 0) |
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106 break; |
| 5827 | 107 endif |
| 108 h2 = kron (h2, h2); | |
| 109 endwhile | |
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110 |
| 5827 | 111 endfunction |
| 112 | |
| 113 function h = h12 () | |
| 8507 | 114 tu = [-1,+1,-1,+1,+1,+1,-1,-1,-1,+1,-1]; |
| 115 tl = [-1,-1,+1,-1,-1,-1,+1,+1,+1,-1,+1]; | |
| 116 ## Note: assert (tu(2:end), tl(end:-1:2)). | |
| 5827 | 117 h = ones (12); |
| 118 h(2:end,2:end) = toeplitz (tu, tl); | |
| 119 endfunction | |
| 120 | |
| 121 function h = h20 () | |
| 8507 | 122 tu = [+1,-1,-1,+1,+1,+1,+1,-1,+1,-1,+1,-1,-1,-1,-1,+1,+1,-1,-1]; |
| 123 tl = [+1,-1,-1,+1,+1,-1,-1,-1,-1,+1,-1,+1,-1,+1,+1,+1,+1,-1,-1]; | |
| 124 ## Note: assert (tu(2:end), tl(end:-1:2)). | |
| 5827 | 125 h = ones (20); |
| 126 h(2:end,2:end) = fliplr (toeplitz (tu, tl)); | |
| 127 endfunction | |
| 128 | |
| 129 function h = h28 () | |
| 130 ## Williamson matrix construction from | |
| 131 ## http://www.research.att.com/~njas/hadamard/had.28.will.txt | |
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132 ## Normalized so that each row and column starts with +1 |
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133 h = [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
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134 1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 1 1 1 1 -1 1 1 1 1 -1 1 -1 1 -1 |
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135 1 -1 1 -1 -1 -1 -1 1 -1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 |
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136 1 -1 -1 1 -1 -1 -1 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 |
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137 1 -1 -1 -1 1 -1 -1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 |
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138 1 -1 -1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 1 1 1 1 -1 |
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139 1 -1 -1 -1 -1 -1 1 -1 1 -1 -1 -1 1 -1 1 1 1 -1 1 1 1 1 -1 1 -1 1 -1 1 |
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140 1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 |
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141 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 1 -1 1 |
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142 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 |
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143 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 1 -1 -1 1 |
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144 1 -1 -1 1 1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 |
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145 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 |
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146 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 |
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147 1 1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 -1 |
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148 1 1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 1 1 1 1 -1 -1 |
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149 1 1 -1 1 -1 1 1 1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 1 1 |
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150 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 -1 -1 |
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151 1 -1 1 -1 1 -1 1 1 1 1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 -1 -1 -1 1 -1 |
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152 1 1 1 -1 1 -1 1 1 -1 1 -1 -1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 1 -1 -1 |
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153 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 1 |
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154 1 1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 |
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155 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 -1 1 -1 -1 1 1 |
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156 1 -1 1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 |
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157 1 1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 -1 -1 |
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158 1 -1 1 1 -1 1 1 -1 -1 -1 1 -1 1 1 1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 -1 |
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159 1 1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 -1 -1 1 |
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160 1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 -1]; |
| 5827 | 161 endfunction |
| 162 | |
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163 |
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164 %!assert (hadamard (1), 1) |
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165 %!assert (hadamard (2), [1,1;1,-1]) |
| 5827 | 166 %!test |
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167 %! for n = [1,2,4,8,12,24,48,20,28,2^9] |
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168 %! h = hadamard(n); |
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521adfd775be
hadamard.m: Fix failing %!tests
Rik <octave@nomad.inbox5.com>
parents:
13893
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169 %! assert (norm (h*h' - n*eye (n)), 0); |
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13893
5b1ddcf5ede1
hadamard.m: Embed hadamard28 matrix as constant rather than computed value.
Rik <octave@nomad.inbox5.com>
parents:
12639
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170 %! endfor |
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5b1ddcf5ede1
hadamard.m: Embed hadamard28 matrix as constant rather than computed value.
Rik <octave@nomad.inbox5.com>
parents:
12639
diff
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171 |
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5b1ddcf5ede1
hadamard.m: Embed hadamard28 matrix as constant rather than computed value.
Rik <octave@nomad.inbox5.com>
parents:
12639
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172 %!error hadamard () |
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5b1ddcf5ede1
hadamard.m: Embed hadamard28 matrix as constant rather than computed value.
Rik <octave@nomad.inbox5.com>
parents:
12639
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173 %!error hadamard (1,2) |
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13913
521adfd775be
hadamard.m: Fix failing %!tests
Rik <octave@nomad.inbox5.com>
parents:
13893
diff
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174 %!error <N must be 2\^k\*p> hadamard (5) |
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13893
5b1ddcf5ede1
hadamard.m: Embed hadamard28 matrix as constant rather than computed value.
Rik <octave@nomad.inbox5.com>
parents:
12639
diff
changeset
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175 |