Mercurial > octave-antonio
annotate scripts/specfun/expint.m @ 20162:2645f9ef8c88 stable
doc: Update more docstrings to have one sentence summary as first line.
Reviewed specfun, special-matrix, testfun, and time script directories.
* scripts/specfun/expint.m, scripts/specfun/isprime.m,
scripts/specfun/legendre.m, scripts/specfun/primes.m,
scripts/specfun/reallog.m, scripts/specfun/realsqrt.m,
scripts/special-matrix/gallery.m, scripts/special-matrix/hadamard.m,
scripts/special-matrix/hankel.m, scripts/special-matrix/hilb.m,
scripts/special-matrix/invhilb.m, scripts/special-matrix/magic.m,
scripts/special-matrix/pascal.m, scripts/special-matrix/rosser.m,
scripts/special-matrix/toeplitz.m, scripts/special-matrix/vander.m,
scripts/special-matrix/wilkinson.m, scripts/testfun/assert.m,
scripts/testfun/demo.m, scripts/testfun/example.m, scripts/testfun/fail.m,
scripts/testfun/rundemos.m, scripts/testfun/runtests.m,
scripts/testfun/speed.m, scripts/time/asctime.m, scripts/time/calendar.m,
scripts/time/clock.m, scripts/time/ctime.m, scripts/time/datenum.m,
scripts/time/datestr.m, scripts/time/datevec.m, scripts/time/etime.m,
scripts/time/is_leap_year.m, scripts/time/now.m, scripts/time/weekday.m:
Update more docstrings to have one sentence summary as first line.
| author | Rik <rik@octave.org> |
|---|---|
| date | Sun, 03 May 2015 17:00:11 -0700 |
| parents | 446c46af4b42 |
| children |
| rev | line source |
|---|---|
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1 ## Copyright (C) 2006, 2013 Sylvain Pelissier |
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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 |
| 16768 | 19 ## Author: Sylvain Pelissier <sylvain.pelissier@gmail.com> |
| 20 | |
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21 ## -*- texinfo -*- |
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22 ## @deftypefn {Function File} {} expint (@var{x}) |
| 17363 | 23 ## Compute the exponential integral: |
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24 ## @tex |
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25 ## $$ |
| 17363 | 26 ## {\rm E_1} (x) = \int_x^\infty {e^{-t} \over t} dt |
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27 ## $$ |
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28 ## @end tex |
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29 ## @ifnottex |
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30 ## |
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31 ## @example |
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32 ## @group |
| 17363 | 33 ## infinity |
| 34 ## / | |
| 35 ## E_1 (x) = | exp (-t)/t dt | |
| 36 ## / | |
| 37 ## x | |
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38 ## @end group |
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39 ## @end example |
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40 ## |
| 17363 | 41 ## @end ifnottex |
| 42 ## Note: For compatibility, this functions uses the @sc{matlab} definition | |
| 43 ## of the exponential integral. Most other sources refer to this particular | |
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44 ## value as @math{E_1 (x)}, and the exponential integral as |
| 17363 | 45 ## @tex |
| 46 ## $$ | |
| 47 ## {\rm Ei} (x) = - \int_{-x}^\infty {e^{-t} \over t} dt. | |
| 48 ## $$ | |
| 49 ## @end tex | |
| 50 ## @ifnottex | |
| 51 ## | |
| 52 ## @example | |
| 53 ## @group | |
| 54 ## infinity | |
| 55 ## / | |
| 56 ## Ei (x) = - | exp (-t)/t dt | |
| 57 ## / | |
| 58 ## -x | |
| 59 ## @end group | |
| 60 ## @end example | |
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61 ## |
| 17363 | 62 ## @end ifnottex |
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63 ## The two definitions are related, for positive real values of @var{x}, by |
| 17363 | 64 ## @tex |
| 65 ## $ | |
| 66 ## E_1 (-x) = -{\rm Ei} (x) - i\pi. | |
| 67 ## $ | |
| 68 ## @end tex | |
| 69 ## @ifnottex | |
| 70 ## @w{@code{E_1 (-x) = -Ei (x) - i*pi}}. | |
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71 ## @end ifnottex |
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72 ## @end deftypefn |
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73 |
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74 function y = expint (x) |
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75 |
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76 if (nargin != 1) |
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77 print_usage (); |
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78 endif |
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79 |
| 17363 | 80 y = x; # Copy over all values, including NaNs |
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81 |
| 17363 | 82 if (isreal (x)) |
| 83 idx = (x >= 0); | |
| 84 y(idx) = -expint_Ei (-x(idx)); | |
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85 |
| 17363 | 86 idx = (x < 0); |
| 87 y(idx) = -expint_Ei (-x(idx)) - i*pi; | |
| 88 else | |
| 89 idx = (imag (x) > 0); | |
| 90 y(idx) = -expint_Ei (-x(idx)) - i*pi; | |
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91 |
| 17363 | 92 idx = (imag (x) < 0); |
| 93 y(idx) = -expint_Ei (-x(idx)) + i*pi; | |
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94 |
| 17363 | 95 isreal_idx = (imag (x) == 0); |
| 96 idx = (isreal_idx & real (x) >= 0); | |
| 97 y(idx) = -expint_Ei (-x(idx)); | |
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98 |
| 17363 | 99 idx = (isreal_idx & real (x) < 0); |
| 100 y(idx) = -expint_Ei (-x(idx)) - i*pi; | |
| 101 endif | |
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102 |
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103 endfunction |
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104 |
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105 ## -*- texinfo -*- |
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106 ## @deftypefn {Function File} {@var{y} =} expint_Ei (@var{x}) |
| 17363 | 107 ## Compute the exponential integral: |
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108 ## @verbatim |
| 17363 | 109 ## infinity |
| 110 ## / | |
| 111 ## expint_Ei (x) = - | exp(-t)/t dt | |
| 112 ## / | |
| 113 ## -x | |
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114 ## @end verbatim |
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115 ## @end deftypefn |
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116 |
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117 function y = expint_Ei (x) |
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118 |
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119 y = zeros (size (x)); |
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120 F = @(x) exp (-x)./x; |
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121 |
| 17363 | 122 for t = 1:numel (x) |
| 123 xt = x(t); | |
| 124 if (xt < 0 && imag (xt) == 0) | |
| 125 ## Direct integration for most real inputs | |
| 126 y(t) = -quad (F, -xt, Inf, [0, 1e-10]); | |
| 127 elseif (xt > 2 && imag (xt) == 0) | |
| 128 persistent Ei_2 = 4.954234356001890; | |
| 129 y(t) = Ei_2 - quad (F, -xt, -2); | |
| 130 elseif (abs (xt) < 10) | |
| 131 ## Series Expansion for real (range [0,2]) or complex inputs (r < 10) | |
| 132 k = 1; | |
| 133 do | |
| 134 term = xt^k / (k*factorial (k)); | |
| 135 y(t) += term; | |
| 136 until (abs (term) < eps (abs (y(t))) / 2 || k++ >= 100) | |
| 137 y(t) = 0.57721566490153286 + log (xt) + y(t); | |
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138 else |
| 17363 | 139 ## FIXME: This expansion is accurate to only 1e-13 at the beginning |
| 140 ## near 10+i, although it becomes more accurate as the magnitude | |
| 141 ## of xt grows. | |
| 142 if (imag (xt) <= 0) | |
| 143 persistent a1 = 4.03640; | |
| 144 persistent a2 = 1.15198; | |
| 145 persistent b1 = 5.03637; | |
| 146 persistent b2 = 4.19160; | |
| 147 y(t) = -(xt^2 - a1*xt + a2) ... | |
| 148 / ((xt^2 - b1*xt + b2) * (-xt) * exp (-xt)) ... | |
| 149 - i*pi; | |
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150 else |
| 17363 | 151 y(t) = conj (expint_Ei (conj (xt))); |
| 152 endif; | |
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153 endif |
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154 endfor |
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155 endfunction |
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156 |
| 17363 | 157 |
| 158 ## Test against A&S Table 5.1 | |
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159 %!test |
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160 %! x = [5:5:50]'/100; |
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161 %! gamma = 0.5772156649; |
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162 %! y_exp = [0.9876375971; |
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163 %! 0.9755453033; |
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164 %! 0.9637156702; |
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165 %! 0.9521414833; |
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166 %! 0.9408157528; |
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167 %! 0.9297317075; |
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168 %! 0.9188827858; |
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169 %! 0.9082626297; |
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170 %! 0.8978650778; |
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171 %! 0.8876841584 ]; |
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172 %! y = (expint (x) + log(x) + gamma) ./ x; |
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173 %! assert (y, y_exp, 1e-9); |
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174 %!test |
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175 %! x = [50:5:95]'/100; |
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176 %! y_exp = [0.559773595; |
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177 %! 0.503364081; |
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178 %! 0.454379503; |
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179 %! 0.411516976; |
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180 %! 0.373768843; |
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181 %! 0.340340813; |
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182 %! 0.310596579; |
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183 %! 0.284019269; |
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184 %! 0.260183939; |
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185 %! 0.238737524 ]; |
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186 %! y = expint (x); |
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187 %! assert (y, y_exp, 1e-9); |
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188 %!test |
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189 %! x = [100:5:145]'/100; |
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190 %! y_exp = [0.219383934; |
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191 %! 0.201872813; |
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192 %! 0.185990905; |
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193 %! 0.171555354; |
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194 %! 0.158408437; |
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195 %! 0.146413373; |
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196 %! 0.135450958; |
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197 %! 0.125416844; |
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198 %! 0.116219313; |
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199 %! 0.107777440 ]; |
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200 %! y = expint (x); |
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201 %! assert (y, y_exp, 1e-9); |
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202 %!test |
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203 %! x = [150:5:200]'/100; |
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204 %! y_exp = [0.100019582; |
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205 %! 0.092882108; |
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206 %! 0.086308334; |
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207 %! 0.080247627; |
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208 %! 0.074654644; |
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209 %! 0.069488685; |
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210 %! 0.064713129; |
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211 %! 0.060294967; |
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212 %! 0.056204378; |
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213 %! 0.052414380; |
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214 %! 0.048900511 ]; |
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215 %! y = expint (x); |
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216 %! assert (y, y_exp, 1e-9); |
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217 |
| 17363 | 218 ## Series expansion (-2 < x < 0) |
| 219 ## Expected values from Mathematica | |
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220 %!test |
| 17363 | 221 %! x = [-0.1; -0.5; -1; -1.5; -2]; |
| 222 %! y_exp = [ 1.6228128139692767 - i*pi; | |
| 223 %! -0.45421990486317358 - i*pi; | |
| 224 %! -1.8951178163559368 - i*pi; | |
| 225 %! -3.3012854491297978 - i*pi; | |
| 226 %! -4.9542343560018902 - i*pi]; | |
| 227 %! y = expint (x); | |
| 228 %! assert (y, y_exp, eps (real (y_exp))); | |
| 229 | |
| 230 ## (x < -2, x real) | |
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231 %!test |
| 17363 | 232 %! x = [-2.5; -3; -10;-15; -25]; |
| 233 %! y_exp = [-7.0737658945786007 - i*pi; | |
| 234 %! -9.9338325706254165 - i*pi; | |
| 235 %! -2492.2289762418777 - i*pi; | |
| 236 %! -234955.85249076830 - i*pi; | |
| 237 %! -3.0059509065255486e9 - i*pi]; | |
| 238 %! y = expint (x); | |
| 239 %! assert (y, y_exp, 8*eps (real (y_exp))); | |
| 240 | |
| 241 ## Complex values | |
| 242 %!test | |
| 243 %! x = [i; -1-i; 10-i; 10+i]; | |
| 244 %! y_exp = [-0.33740392290096813 - i*0.62471325642771360; | |
| 245 %! -1.7646259855638540 + i*0.75382280207927082; | |
| 246 %! 1.90746381979783120e-6 + i*3.67354374003294739e-6; | |
| 247 %! 1.90746381979783120e-6 - i*3.67354374003294739e-6]; | |
| 248 %! y = expint (x); | |
| 249 %! assert (y, y_exp, 1e-12); | |
| 250 | |
| 251 ## Exceptional values (-Inf, Inf, NaN, 0, 0.37250741078) | |
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252 %!test |
| 17363 | 253 %! x = [-Inf; Inf; NaN; 0; -0.3725074107813668]; |
| 254 %! y_exp = [-Inf - i*pi; | |
| 255 %! -Inf; # should be 0; | |
| 256 %! NaN; | |
| 257 %! Inf; | |
| 258 %! 0 - i*pi]; | |
| 259 %! y = expint (x); | |
| 260 %! assert (y, y_exp, 5*eps); | |
| 261 | |
| 262 ## Test input validation | |
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263 %!error expint () |
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264 %!error expint (1,2) |
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265 |
| 17363 | 266 |