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1 ## Copyright (C) 2000 Kai Habel |
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2 ## |
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3 ## This file is part of Octave. |
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4 ## |
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5 ## Octave is free software; you can redistribute it and/or modify it |
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6 ## under the terms of the GNU General Public License as published by |
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7 ## the Free Software Foundation; either version 3 of the License, or (at |
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8 ## your option) any later version. |
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9 ## |
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10 ## Octave is distributed in the hope that it will be useful, but |
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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13 ## General Public License for more details. |
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14 ## |
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15 ## You should have received a copy of the GNU General Public License |
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16 ## along with Octave; see the file COPYING. If not, see |
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17 ## <http://www.gnu.org/licenses/>. |
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18 |
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19 ## -*- texinfo -*- |
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20 ## @deftypefn {Function File} {@var{L} =} legendre (@var{n}, @var{X}) |
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21 ## |
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22 ## Legendre Function of degree n and order m |
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23 ## where all values for m = 0..@var{n} are returned. |
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24 ## @var{n} must be a scalar in the range [0..255]. |
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25 ## The return value has one dimension more than @var{x}. |
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26 ## |
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27 ## @example |
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28 ## The Legendre Function of degree n and order m |
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29 ## |
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30 ## @group |
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31 ## m m 2 m/2 d^m |
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32 ## P(x) = (-1) * (1-x ) * ---- P (x) |
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33 ## n dx^m n |
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34 ## @end group |
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35 ## |
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36 ## with: |
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37 ## Legendre polynomial of degree n |
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38 ## |
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39 ## @group |
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40 ## 1 d^n 2 n |
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41 ## P (x) = ------ [----(x - 1) ] |
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42 ## n 2^n n! dx^n |
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43 ## @end group |
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44 ## |
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45 ## legendre(3,[-1.0 -0.9 -0.8]) returns the matrix |
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46 ## |
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47 ## @group |
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48 ## x | -1.0 | -0.9 | -0.8 |
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49 ## ------------------------------------ |
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50 ## m=0 | -1.00000 | -0.47250 | -0.08000 |
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51 ## m=1 | 0.00000 | -1.99420 | -1.98000 |
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52 ## m=2 | 0.00000 | -2.56500 | -4.32000 |
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53 ## m=3 | 0.00000 | -1.24229 | -3.24000 |
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54 ## @end group |
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55 ## @end example |
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56 ## @end deftypefn |
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57 |
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58 ## FIXME Add Schmidt semi-normalized and fully normalized legendre functions |
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59 |
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60 ## Author: Kai Habel <kai.habel@gmx.de> |
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61 |
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62 function L = legendre (n, x) |
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63 |
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64 warning ("legendre is unstable for higher orders"); |
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65 |
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66 if (nargin != 2) |
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67 print_usage (); |
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68 endif |
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69 |
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70 if (! isscalar (n) || n < 0 || n > 255 || n != fix (n)) |
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71 error ("n must be a integer between 0 and 255]"); |
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72 endif |
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73 |
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74 if (! isvector (x) || any (x < -1 || x > 1)) |
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75 error ("x must be vector in range -1 <= x <= 1"); |
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76 endif |
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77 |
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78 if (n == 0) |
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79 L = ones (size (x)); |
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80 elseif (n == 1) |
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81 L = [x; -sqrt(1 - x .^ 2)]; |
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82 else |
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83 i = 0:n; |
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84 a = (-1) .^ i .* bincoeff (n, i); |
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85 p = [a; zeros(size (a))]; |
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86 p = p(:); |
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87 p(length (p)) = []; |
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88 #p contains the polynom (x^2-1)^n |
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89 |
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90 #now create a vector with 1/(2.^n*n!)*(d/dx).^n |
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91 d = [((n + rem(n, 2)):-1:(rem (n, 2) + 1)); 2 * ones(fix (n / 2), n)]; |
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92 d = cumsum (d); |
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93 d = fliplr (prod (d')); |
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94 d = [d; zeros(1, length (d))]; |
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95 d = d(1:n + 1) ./ (2 ^ n *prod (1:n)); |
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96 |
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97 Lp = d' .* p(1:length (d)); |
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98 #Lp contains the Legendre Polynom of degree n |
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99 |
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100 # now create a polynom matrix with d/dx^m for m=0..n |
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101 d2 = flipud (triu (ones (n))); |
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102 d2 = cumsum (d2); |
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103 d2 = fliplr (cumprod (flipud (d2))); |
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104 d3 = fliplr (triu (ones (n + 1))); |
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105 d3(2:n + 1, 1:n) = d2; |
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106 |
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107 # multiply for each m (d/dx)^m with Lp(n,x) |
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108 # and evaluate at x |
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109 Y = zeros(n + 1, length (x)); |
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110 [dr, dc] = size (d3); |
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111 for m = 0:dr - 1 |
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112 Y(m + 1, :) = polyval (d3(m + 1, 1:(dc - m)) .* Lp(1:(dc - m))', x)(:)'; |
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113 endfor |
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114 |
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115 # calculate (-1)^m*(1-x^2)^(m/2) for m=0..n at x |
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116 # and multiply with (d/dx)^m(Pnx) |
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117 l = length (x); |
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118 X = kron ((1 - x(:) .^ 2)', ones (n + 1, 1)); |
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119 M = kron ((0:n)', ones (1, l)); |
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120 L = X .^ (M / 2) .* (-1) .^ M .* Y; |
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121 endif |
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122 endfunction |
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123 |
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124 %!test |
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125 %! result=legendre(3,[-1.0 -0.9 -0.8]); |
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126 %! expected = [ |
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127 %! -1.00000 -0.47250 -0.08000 |
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128 %! 0.00000 -1.99420 -1.98000 |
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129 %! 0.00000 -2.56500 -4.32000 |
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130 %! 0.00000 -1.24229 -3.24000 |
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131 %! ]; |
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132 %! assert(result,expected,1e-5); |
