Mercurial > hg > octave-nkf
annotate scripts/specfun/ellipke.m @ 20818:9d2023d1a63c
binoinv.m: Implement binary search algorithm for 28X performance increase (bug #34363).
* binoinv.m: Call new functions scalar_binoinv or vector_binoinv to calculate
binoinv. If there are still uncalculated values then call bin_search_binoinv
to perform binary search for remaining values. Add more BIST tests.
* binoinv.m (scalar_binoinv): New subfunction to calculate binoinv for scalar x.
Stops when x > 1000.
* binoinv.m (vector_binoinv): New subfunction to calculate binoinv for scalar x.
Stops when x > 1000.
| author | Lachlan Andrew <lachlanbis@gmail.com> |
|---|---|
| date | Sun, 11 Oct 2015 19:49:40 -0700 |
| parents | 8c4317b8f7c5 |
| children |
| rev | line source |
|---|---|
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1 ## Copyright (C) 2001, 2013 David Billinghurst |
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2 ## Copyright (C) 2001, 2013 Paul Kienzle |
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3 ## Copyright (C) 2003, 2013 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 ## |
| 17371 | 28 ## @var{m} must be a scalar or real array with -Inf @leq{} @var{m} @leq{} 1. |
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29 ## |
| 19287 | 30 ## The optional input @var{tol} controls the stopping tolerance of the |
| 31 ## algorithm and defaults to @code{eps (class (@var{m}))}. The tolerance can | |
| 32 ## be increased to compute a faster, less accurate approximation. | |
| 33 ## | |
| 34 ## When called with one output only elliptic integrals of the first kind are | |
| 35 ## returned. | |
| 36 ## | |
| 37 ## Mathematical Note: | |
| 38 ## | |
| 39 ## Elliptic integrals of the first kind are defined as | |
| 40 ## | |
| 41 ## @tex | |
| 42 ## $$ | |
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43 ## {\rm K} (m) = \int_0^1 {dt \over \sqrt{(1 - t^2) (1 - m t^2)}} |
| 19287 | 44 ## $$ |
| 45 ## @end tex | |
| 46 ## @ifnottex | |
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47 ## |
| 19287 | 48 ## @example |
| 49 ## @group | |
| 50 ## 1 | |
| 51 ## / dt | |
| 52 ## K (m) = | ------------------------------ | |
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53 ## / sqrt ((1 - t^2)*(1 - m*t^2)) |
| 19287 | 54 ## 0 |
| 55 ## @end group | |
| 56 ## @end example | |
| 57 ## | |
| 58 ## @end ifnottex | |
| 59 ## | |
| 60 ## Elliptic integrals of the second kind are defined as | |
| 61 ## | |
| 62 ## @tex | |
| 63 ## $$ | |
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64 ## {\rm E} (m) = \int_0^1 {\sqrt{1 - m t^2} \over \sqrt{1 - t^2}} dt |
| 19287 | 65 ## $$ |
| 66 ## @end tex | |
| 67 ## @ifnottex | |
| 68 ## | |
| 69 ## @example | |
| 70 ## @group | |
| 71 ## 1 | |
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72 ## / sqrt (1 - m*t^2) |
| 19287 | 73 ## E (m) = | ------------------ dt |
| 74 ## / sqrt (1 - t^2) | |
| 75 ## 0 | |
| 76 ## @end group | |
| 77 ## @end example | |
| 78 ## | |
| 79 ## @end ifnottex | |
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80 ## |
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81 ## Reference: Milton @nospell{Abramowitz} and Irene A. @nospell{Stegun}, |
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82 ## @cite{Handbook of Mathematical Functions}, Chapter 17, Dover, 1965. |
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83 ## @seealso{ellipj} |
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84 ## @end deftypefn |
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85 |
| 16768 | 86 ## Author: David Billinghurst <David.Billinghurst@riotinto.com> |
| 87 ## Author: Paul Kienzle <pkienzle@users.sf.net> | |
| 88 ## Author: Jaakko Ruohio | |
| 89 | |
| 19287 | 90 function [k, e] = ellipke (m, tol = []) |
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91 |
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92 if (nargin < 1 || nargin > 2) |
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93 print_usage (); |
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94 endif |
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95 |
| 19287 | 96 sz = size (m); |
| 17371 | 97 if (! isreal (m)) |
| 98 error ("ellipke: M must be real"); | |
| 99 elseif (any (m > 1)) | |
| 100 error ("ellipke: M must be <= 1"); | |
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101 endif |
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102 |
| 19287 | 103 if (isempty (tol)) |
| 104 tol = eps (class (m)); | |
| 105 elseif (! (isreal (tol) && isscalar (tol) && tol > 0)) | |
| 106 error ("ellipke: TOL must be a real scalar > 0") | |
| 107 endif | |
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108 |
| 19287 | 109 k = e = zeros (sz); |
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110 |
| 17371 | 111 ## Handle extreme values |
| 112 idx_1 = (m == 1); | |
| 113 k(idx_1) = Inf; | |
| 114 e(idx_1) = 1; | |
| 115 | |
| 116 idx_neginf = (m == -Inf); | |
| 117 k(idx_neginf) = 0; | |
| 118 e(idx_neginf) = Inf; | |
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119 |
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120 ## Arithmetic-Geometric Mean (AGM) algorithm |
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121 ## ( Abramowitz and Stegun, Section 17.6 ) |
| 17371 | 122 Nmax = 16; |
| 123 idx = !idx_1 & !idx_neginf; | |
| 124 if (any (idx)) | |
| 125 idx_neg = find (m < 0 & !idx_neginf); | |
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126 mult_k = 1./sqrt (1 - m(idx_neg)); |
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127 mult_e = sqrt (1 - m(idx_neg)); |
| 17371 | 128 m(idx_neg) = -m(idx_neg) ./ (1 - m(idx_neg)); |
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129 b = sqrt (1 - m(idx)); |
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130 a = ones (size (b)); |
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131 c = sqrt (m(idx)); |
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132 f = 0.5; |
| 17371 | 133 sum = f*c.^2; |
| 134 n = 2; | |
| 135 do | |
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136 t = (a + b)/2; |
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137 c = (a - b)/2; |
| 19287 | 138 b = sqrt (a .* b); |
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139 a = t; |
| 17371 | 140 f *= 2; |
| 141 sum += f*c.^2; | |
| 19287 | 142 until (all (c./a < tol) || (++n > Nmax)) |
| 17371 | 143 if (n >= Nmax) |
| 144 error ("ellipke: algorithm did not converge in %d iterations", Nmax); | |
| 145 endif | |
| 146 k(idx) = 0.5*pi ./ a; | |
| 147 e(idx) = 0.5*pi*(1 - sum) ./ a; | |
| 148 k(idx_neg) = mult_k .* k(idx_neg); | |
| 149 e(idx_neg) = mult_e .* e(idx_neg); | |
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150 endif |
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151 |
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152 endfunction |
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153 |
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154 |
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155 ## Test complete elliptic functions of first and second kind |
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156 ## against "exact" solution from Mathematica 3.0 |
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157 %!test |
| 19287 | 158 %! m = [0.0, 0.01; 0.1, 0.5; 0.9, 0.99; 1.0, 0.0]; |
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159 %! [k,e] = ellipke (m); |
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160 %! |
| 19287 | 161 %! k_exp = [1.5707963267948966192, 1.5747455615173559527 |
| 162 %! 1.6124413487202193982, 1.8540746773013719184 | |
| 163 %! 2.5780921133481731882, 3.6956373629898746778 | |
| 164 %! Inf , 1.5707963267948966192 ]; | |
| 165 %! e_exp = [1.5707963267948966192, 1.5668619420216682912 | |
| 166 %! 1.5307576368977632025, 1.3506438810476755025 | |
| 167 %! 1.1047747327040733261, 1.0159935450252239356 | |
| 168 %! 1.0 , 1.5707963267948966192 ]; | |
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169 %! assert (k, k_exp, 8*eps); |
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170 %! assert (e, e_exp, 8*eps); |
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171 |
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172 ## Test against A&S Table 17.1 |
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173 %!test |
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174 %! m = [0:5:50]'/100; |
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175 %! k_exp = [1.570796326794897; |
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176 %! 1.591003453790792; |
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177 %! 1.612441348720219; |
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178 %! 1.635256732264580; |
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179 %! 1.659623598610528; |
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180 %! 1.685750354812596; |
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181 %! 1.713889448178791; |
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182 %! 1.744350597225613; |
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183 %! 1.777519371491253; |
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184 %! 1.813883936816983; |
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185 %! 1.854074677301372 ]; |
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186 %! e_exp = [1.570796327; |
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187 %! 1.550973352; |
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188 %! 1.530757637; |
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189 %! 1.510121831; |
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190 %! 1.489035058; |
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191 %! 1.467462209; |
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192 %! 1.445363064; |
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193 %! 1.422691133; |
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194 %! 1.399392139; |
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195 %! 1.375401972; |
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196 %! 1.350643881 ]; |
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197 %! [k,e] = ellipke (m); |
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198 %! assert (k, k_exp, 1e-15); |
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199 %! assert (e, e_exp, 1e-8); |
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200 |
| 17371 | 201 ## Test negative values against "exact" solution from Mathematica. |
| 202 %! m = [-0.01; -1; -5; -100; -1000; -Inf]; | |
| 203 %! [k,e] = ellipke (m); | |
| 204 %! | |
| 205 %! k_exp = [1.5668912730681963584; | |
| 206 %! 1.3110287771460599052; | |
| 207 %! 0.9555039270640439337; | |
| 208 %! 0.3682192486091410329; | |
| 209 %! 0.1530293349884987857; | |
| 210 %! 0]; | |
| 211 %! e_exp = [1.5747159850169884130; | |
| 212 %! 1.9100988945138560089; | |
| 213 %! 2.8301982463458773125; | |
| 214 %! 10.209260919814572009; | |
| 215 %! 31.707204053711259719; | |
| 216 %! Inf ]; | |
| 217 %! assert (k, k_exp, 8*eps); | |
| 218 %! assert (e, e_exp, 8*eps (e_exp)); | |
| 219 | |
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220 ## Test input validation |
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221 %!error ellipke () |
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222 %!error ellipke (1,2,3) |
| 19287 | 223 %!error <M must be real> ellipke (1i) |
| 224 %!error <M must be .= 1> ellipke (2) | |
| 225 %!error <TOL must be a real scalar . 0> ellipke (1, i) | |
| 226 %!error <TOL must be a real scalar . 0> ellipke (1, [1 1]) | |
| 227 %!error <TOL must be a real scalar . 0> ellipke (1, -1) | |
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228 |