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doc: Use Octave citation style for references in ellipj, ellipke. * libinterp/corefcn/ellipj.cc(Fellipj): Redo docstring. Use Octave citation style for references. * scripts/specfun/ellipke.m: Redo docstring. Use Octave citation style for references.
author Rik <rik@octave.org>
date Wed, 04 Sep 2013 10:13:53 -0700
parents b81b9d079515
children f9e8544ce66d
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1 ## Copyright (C) 2001 David Billinghurst
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2 ## Copyright (C) 2001 Paul Kienzle
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3 ## Copyright (C) 2003 Jaakko Ruohio
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4 ##
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5 ## This file is part of Octave.
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6 ##
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7 ## Octave is free software; you can redistribute it and/or modify it
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8 ## under the terms of the GNU General Public License as published by
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9 ## the Free Software Foundation; either version 3 of the License, or (at
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10 ## your option) any later version.
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11 ##
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12 ## Octave is distributed in the hope that it will be useful, but
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13 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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14 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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15 ## General Public License for more details.
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16 ##
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17 ## You should have received a copy of the GNU General Public License
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18 ## along with Octave; see the file COPYING. If not, see
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19 ## <http://www.gnu.org/licenses/>.
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20
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21 ## -*- texinfo -*-
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22 ## @deftypefn {Function File} {@var{k} =} ellipke (@var{m})
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23 ## @deftypefnx {Function File} {@var{k} =} ellipke (@var{m}, @var{tol})
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24 ## @deftypefnx {Function File} {[@var{k}, @var{e}] =} ellipke (@dots{})
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25 ## Compute complete elliptic integrals of the first K(@var{m}) and second
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26 ## E(@var{m}) kind.
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27 ##
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28 ## @var{m} must be a scalar or real array with 0 @leq{} @var{m} @leq{} 1.
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29 ##
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30 ## The optional input @var{tol} is currently ignored (@sc{matlab} uses this
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31 ## to allow a faster, less accurate approximation).
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32 ##
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33 ## Called with only one output, elliptic integrals of the first kind are
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34 ## returned.
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35 ##
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36 ## Reference: Milton Abramowitz and Irene A. Stegun,
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37 ## @cite{Handbook of Mathematical Functions}, Chapter 17, Dover, 1965.
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38 ## @seealso{ellipj}
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39 ## @end deftypefn
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40
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41 ## Author: David Billinghurst <David.Billinghurst@riotinto.com>
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42 ## Author: Paul Kienzle <pkienzle@users.sf.net>
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43 ## Author: Jaakko Ruohio
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44
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45 function [k, e] = ellipke (m)
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46
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47 if (nargin < 1 || nargin > 2)
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48 print_usage ();
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49 endif
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50
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51 k = e = zeros (size (m));
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52 m = m(:);
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53 if (any (!isreal (m)))
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54 error ("ellipke must have real m");
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55 endif
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56 if (any (m > 1))
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57 error ("ellipke must have m <= 1");
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58 endif
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59
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60 Nmax = 16;
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61 idx = find (m == 1);
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62 if (!isempty (idx))
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63 k(idx) = Inf;
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64 e(idx) = 1;
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65 endif
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66
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67 idx = find (m == -Inf);
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68 if (!isempty (idx))
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69 k(idx) = 0;
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70 e(idx) = Inf;
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71 endif
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72
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73 ## Arithmetic-Geometric Mean (AGM) algorithm
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74 ## ( Abramowitz and Stegun, Section 17.6 )
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75 idx = find (m != 1 & m != -Inf);
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76 if (!isempty (idx))
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77 idx_neg = find (m < 0 & m != -Inf);
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78 mult_k = 1./sqrt (1 - m(idx_neg));
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79 mult_e = sqrt (1 - m(idx_neg));
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80 m(idx_neg) = -m(idx_neg)./(1 - m(idx_neg));
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81 a = ones (length (idx), 1);
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82 b = sqrt (1 - m(idx));
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83 c = sqrt (m(idx));
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84 f = 0.5;
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85 sum = f*c.*c;
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86 for n = 2:Nmax
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87 t = (a + b)/2;
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88 c = (a - b)/2;
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89 b = sqrt (a.*b);
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90 a = t;
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91 f = f * 2;
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92 sum = sum + f*c.*c;
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93 if (all (c./a < eps)) break; endif
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94 endfor
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95 if (n >= Nmax) error ("ellipke: not enough workspace"); endif
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96 k(idx) = 0.5*pi./a;
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97 e(idx) = 0.5*pi.*(1 - sum)./a;
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98 k(idx_neg) = mult_k.*k(idx_neg);
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99 e(idx_neg) = mult_e.*e(idx_neg);
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100 endif
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101
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102 endfunction
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103
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104
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105 ## Test complete elliptic functions of first and second kind
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106 ## against "exact" solution from Mathematica 3.0
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107 %!test
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108 %! m = [0.0; 0.01; 0.1; 0.5; 0.9; 0.99; 1.0 ];
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109 %! [k,e] = ellipke (m);
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110 %!
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111 %! ## K(1.0) is really infinity - see below
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112 %! k_exp = [1.5707963267948966192;
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113 %! 1.5747455615173559527;
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114 %! 1.6124413487202193982;
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115 %! 1.8540746773013719184;
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116 %! 2.5780921133481731882;
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117 %! 3.6956373629898746778;
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118 %! 0.0 ];
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119 %! e_exp = [1.5707963267948966192;
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120 %! 1.5668619420216682912;
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121 %! 1.5307576368977632025;
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122 %! 1.3506438810476755025;
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123 %! 1.1047747327040733261;
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124 %! 1.0159935450252239356;
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125 %! 1.0 ];
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126 %! if (k(7)==Inf), k(7)=0; endif;
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127 %! assert (k, k_exp, 8*eps);
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128 %! assert (e, e_exp, 8*eps);
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129
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130 ## Test against A&S Table 17.1
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131 %!test
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132 %! m = [0:5:50]'/100;
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133 %! k_exp = [1.570796326794897;
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134 %! 1.591003453790792;
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135 %! 1.612441348720219;
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136 %! 1.635256732264580;
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137 %! 1.659623598610528;
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138 %! 1.685750354812596;
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139 %! 1.713889448178791;
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140 %! 1.744350597225613;
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141 %! 1.777519371491253;
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142 %! 1.813883936816983;
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143 %! 1.854074677301372 ];
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144 %! e_exp = [1.570796327;
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145 %! 1.550973352;
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146 %! 1.530757637;
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147 %! 1.510121831;
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148 %! 1.489035058;
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149 %! 1.467462209;
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150 %! 1.445363064;
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151 %! 1.422691133;
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152 %! 1.399392139;
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153 %! 1.375401972;
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154 %! 1.350643881 ];
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155 %! [k,e] = ellipke (m);
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156 %! assert (k, k_exp, 1e-15);
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157 %! assert (e, e_exp, 1e-8);
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158
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159 ## Test input validation
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160 %!error ellipke ()
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161 %!error ellipke (1,2,3)
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162