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annotate scripts/special-matrix/hadamard.m @ 10821:693e22af08ae
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| author | Rik <octave@nomad.inbox5.com> |
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| date | Mon, 26 Jul 2010 21:25:36 -0700 |
| parents | 3140cb7a05a1 |
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| 8920 | 1 ## Copyright (C) 1993, 1994, 1995, 1996, 1997, 1998, 2004, 2006, 2007, 2009 |
| 7017 | 2 ## Paul Kienzle |
| 5827 | 3 ## |
| 4 ## This file is part of Octave. | |
| 5 ## | |
| 6 ## Octave is free software; you can redistribute it and/or modify it | |
| 7 ## under the terms of the GNU General Public License as published by | |
| 7016 | 8 ## the Free Software Foundation; either version 3 of the License, or (at |
| 9 ## your option) any later version. | |
| 5827 | 10 ## |
| 11 ## Octave is distributed in the hope that it will be useful, but | |
| 12 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 14 ## General Public License for more details. | |
| 15 ## | |
| 16 ## You should have received a copy of the GNU General Public License | |
| 7016 | 17 ## along with Octave; see the file COPYING. If not, see |
| 18 ## <http://www.gnu.org/licenses/>. | |
| 5827 | 19 ## |
| 20 ## Original version by Paul Kienzle distributed as free software in the | |
| 21 ## public domain. | |
| 22 | |
| 23 ## -*- texinfo -*- | |
| 24 ## @deftypefn {Function File} {} hadamard (@var{n}) | |
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25 ## Construct a Hadamard matrix @var{Hn} of size @var{n}-by-@var{n}. The |
| 6516 | 26 ## size @var{n} must be of the form @code{2 ^ @var{k} * @var{p}} in which |
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27 ## @var{p} is one of 1, 12, 20 or 28. The returned matrix is normalized, |
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28 ## meaning @code{Hn(:,1) == 1} and @code{Hn(1,:) == 1}. |
| 5827 | 29 ## |
| 30 ## Some of the properties of Hadamard matrices are: | |
| 31 ## | |
| 32 ## @itemize @bullet | |
| 33 ## @item | |
| 34 ## @code{kron (@var{Hm}, @var{Hn})} is a Hadamard matrix of size | |
| 35 ## @var{m}-by-@var{n}. | |
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36 ## |
| 5827 | 37 ## @item |
| 6248 | 38 ## @code{Hn * Hn' == @var{n} * eye (@var{n})}. |
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39 ## |
| 5827 | 40 ## @item |
| 41 ## The rows of @var{Hn} are orthogonal. | |
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42 ## |
| 5827 | 43 ## @item |
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44 ## @code{det (@var{A}) <= abs(det (@var{Hn}))} for all @var{A} with |
| 5827 | 45 ## @code{abs (@var{A} (@var{i}, @var{j})) <= 1}. |
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46 ## |
| 5827 | 47 ## @item |
| 48 ## Multiply any row or column by -1 and still have a Hadamard matrix. | |
| 49 ## @end itemize | |
| 50 ## | |
| 51 ## @end deftypefn | |
| 52 | |
| 53 | |
| 54 ## Reference [1] contains a list of Hadamard matrices up to n=256. | |
| 55 ## See code for h28 in hadamard.m for an example of how to extend | |
| 56 ## this function for additional p. | |
| 57 ## | |
| 58 ## References: | |
| 59 ## [1] A Library of Hadamard Matrices, N. J. A. Sloane | |
| 60 ## http://www.research.att.com/~njas/hadamard/ | |
| 61 | |
| 62 function h = hadamard (n) | |
| 63 | |
| 64 if (nargin != 1) | |
| 65 print_usage (); | |
| 66 endif | |
| 67 | |
| 8507 | 68 ## Find k if n = 2^k*p. |
| 5827 | 69 k = 0; |
| 70 while (n > 1 && floor (n/2) == n/2) | |
| 71 k++; | |
| 72 n = n/2; | |
| 73 endwhile | |
| 74 | |
| 8507 | 75 ## Find base hadamard. |
| 76 ## Except for n=2^k, need a multiple of 4. | |
| 5827 | 77 if (n != 1) |
| 78 k -= 2; | |
| 79 endif | |
| 80 | |
| 8507 | 81 ## Trigger error if not a multiple of 4. |
| 5827 | 82 if (k < 0) |
| 83 n =- 1; | |
| 84 endif | |
| 85 | |
| 86 switch (n) | |
| 87 case 1 | |
| 88 h = 1; | |
| 89 case 3 | |
| 90 h = h12 (); | |
| 91 case 5 | |
| 92 h = h20 (); | |
| 93 case 7 | |
| 94 h = hnormalize (h28 ()); | |
| 95 otherwise | |
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96 error ("hadamard: n must be 2^k*p, for p = 1, 12, 20 or 28"); |
| 5827 | 97 endswitch |
| 98 | |
| 8507 | 99 ## Build H(2^k*n) from kron(H(2^k),H(n)). |
| 5827 | 100 h2 = [1,1;1,-1]; |
| 101 while (true) | |
| 102 if (floor (k/2) != k/2) | |
| 103 h = kron (h2, h); | |
| 104 endif | |
| 105 k = floor (k/2); | |
| 106 if (k == 0) | |
| 107 break; | |
| 108 endif | |
| 109 h2 = kron (h2, h2); | |
| 110 endwhile | |
| 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 | |
| 122 function h = h20 () | |
| 8507 | 123 tu = [+1,-1,-1,+1,+1,+1,+1,-1,+1,-1,+1,-1,-1,-1,-1,+1,+1,-1,-1]; |
| 124 tl = [+1,-1,-1,+1,+1,-1,-1,-1,-1,+1,-1,+1,-1,+1,+1,+1,+1,-1,-1]; | |
| 125 ## Note: assert (tu(2:end), tl(end:-1:2)). | |
| 5827 | 126 h = ones (20); |
| 127 h(2:end,2:end) = fliplr (toeplitz (tu, tl)); | |
| 128 endfunction | |
| 129 | |
| 130 function h = hnormalize (h) | |
| 8507 | 131 ## Make sure each row and column starts with +1. |
| 5827 | 132 h(h(:,1)==-1,:) *= -1; |
| 133 h(:,h(1,:)==-1) *= -1; | |
| 134 endfunction | |
| 135 | |
| 136 function h = h28 () | |
| 137 ## Williamson matrix construction from | |
| 138 ## http://www.research.att.com/~njas/hadamard/had.28.will.txt | |
| 8507 | 139 s = ["+------++----++-+--+-+--++--"; |
| 140 "-+-----+++-----+-+--+-+--++-"; | |
| 141 "--+-----+++---+-+-+----+--++"; | |
| 142 "---+-----+++---+-+-+-+--+--+"; | |
| 143 "----+-----+++---+-+-+++--+--"; | |
| 144 "-----+-----++++--+-+--++--+-"; | |
| 145 "------++----++-+--+-+--++--+"; | |
| 146 "--++++-+-------++--+++-+--+-"; | |
| 147 "---++++-+-----+-++--+-+-+--+"; | |
| 148 "+---+++--+----++-++--+-+-+--"; | |
| 149 "++---++---+----++-++--+-+-+-"; | |
| 150 "+++---+----+----++-++--+-+-+"; | |
| 151 "++++--------+-+--++-++--+-+-"; | |
| 152 "-++++--------+++--++--+--+-+"; | |
| 153 "-+-++-++--++--+--------++++-"; | |
| 154 "+-+-++--+--++--+--------++++"; | |
| 155 "-+-+-++--+--++--+----+---+++"; | |
| 156 "+-+-+-++--+--+---+---++---++"; | |
| 157 "++-+-+-++--+------+--+++---+"; | |
| 158 "-++-+-+-++--+------+-++++---"; | |
| 159 "+-++-+---++--+------+-++++--"; | |
| 160 "-++--++-+-++-+++----++------"; | |
| 161 "+-++--++-+-++-+++-----+-----"; | |
| 162 "++-++---+-+-++-+++-----+----"; | |
| 163 "-++-++-+-+-+-+--+++-----+---"; | |
| 164 "--++-++++-+-+----+++-----+--"; | |
| 165 "+--++-+-++-+-+----+++-----+-"; | |
| 166 "++--++-+-++-+-+----++------+"]; | |
| 5901 | 167 ## Without this, the assignment of -1 will not work properly |
| 168 ## (compatibility forces LHS(idx) = ANY_VAL to keep the LHS logical | |
| 169 ## instead of widening to a type that can represent ANY_VAL). | |
| 8507 | 170 h = double (s == "+"); |
| 5827 | 171 h(!h) = -1; |
| 172 endfunction | |
| 173 | |
| 174 %!assert(hadamard(1),1) | |
| 175 %!assert(hadamard(2),[1,1;1,-1]) | |
| 176 %!test | |
| 177 %! for n=[1,2,4,8,12,24,48,20,28,2^9] | |
| 178 %! h=hadamard(n); assert(norm(h*h'-n*eye(n)),0); | |
| 179 %! end |