Mercurial > octave
annotate scripts/general/quadl.m @ 27919:1891570abac8
update Octave Project Developers copyright for the new year
In files that have the "Octave Project Developers" copyright notice,
update for 2020.
author | John W. Eaton <jwe@octave.org> |
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date | Mon, 06 Jan 2020 22:29:51 -0500 |
parents | b442ec6dda5c |
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1 ## Copyright (C) 1998-2020 The Octave Project Developers |
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2 ## |
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3 ## See the file COPYRIGHT.md in the top-level directory of this distribution |
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4 ## or <https://octave.org/COPYRIGHT.html/>. |
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5 ## |
5837 | 6 ## |
7 ## This file is part of Octave. | |
8 ## | |
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9 ## Octave is free software: you can redistribute it and/or modify it |
5837 | 10 ## under the terms of the GNU General Public License as published by |
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11 ## the Free Software Foundation, either version 3 of the License, or |
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12 ## (at your option) any later version. |
5837 | 13 ## |
14 ## Octave is distributed in the hope that it will be useful, but | |
15 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
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16 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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17 ## GNU General Public License for more details. |
5837 | 18 ## |
19 ## You should have received a copy of the GNU General Public License | |
7016 | 20 ## along with Octave; see the file COPYING. If not, see |
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21 ## <https://www.gnu.org/licenses/>. |
5837 | 22 |
23 ## -*- texinfo -*- | |
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24 ## @deftypefn {} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}) |
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25 ## @deftypefnx {} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}, @var{tol}) |
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26 ## @deftypefnx {} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}, @var{tol}, @var{trace}) |
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27 ## @deftypefnx {} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}, @var{tol}, @var{trace}, @var{p1}, @var{p2}, @dots{}) |
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28 ## @deftypefnx {} {[@var{q}, @var{nfun}] =} quadl (@dots{}) |
5837 | 29 ## |
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30 ## Numerically evaluate the integral of @var{f} from @var{a} to @var{b} using |
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31 ## an adaptive @nospell{Lobatto} rule. |
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32 ## |
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33 ## @var{f} is a function handle, inline function, or string containing the name |
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34 ## of the function to evaluate. The function @var{f} must be vectorized and |
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35 ## return a vector of output values when given a vector of input values. |
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36 ## |
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37 ## @var{a} and @var{b} are the lower and upper limits of integration. Both |
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38 ## limits must be finite. |
5837 | 39 ## |
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40 ## The optional argument @var{tol} defines the absolute tolerance with which |
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41 ## to perform the integration. The default value is 1e-6. |
5837 | 42 ## |
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43 ## The algorithm used by @code{quadl} involves recursively subdividing the |
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44 ## integration interval. If @var{trace} is defined then for each subinterval |
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45 ## display: (1) the total number of function evaluations, (2) the left end of |
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46 ## the subinterval, (3) the length of the subinterval, (4) the approximation of |
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47 ## the integral over the subinterval. |
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48 ## |
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49 ## Additional arguments @var{p1}, etc., are passed directly to the function |
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50 ## @var{f}. To use default values for @var{tol} and @var{trace}, one may pass |
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51 ## empty matrices ([]). |
5837 | 52 ## |
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53 ## The result of the integration is returned in @var{q}. |
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54 ## |
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55 ## The optional output @var{nfun} indicates the total number of function |
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56 ## evaluations performed. |
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57 ## |
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58 ## Reference: @nospell{W. Gander and W. Gautschi}, @cite{Adaptive Quadrature - |
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59 ## Revisited}, BIT Vol.@: 40, No.@: 1, March 2000, pp.@: 84--101. |
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60 ## @url{https://www.inf.ethz.ch/personal/gander/} |
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61 ## @seealso{quad, quadv, quadgk, quadcc, trapz, dblquad, triplequad, integral, |
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62 ## integral2, integral3} |
5837 | 63 ## @end deftypefn |
64 | |
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65 ## Original Author: Walter Gautschi |
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66 ## Date: 08/03/98 |
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67 ## Reference: Gander, Computermathematik, Birkhaeuser, 1992. |
5837 | 68 |
69 ## 2003-08-05 Shai Ayal | |
70 ## * permission from author to release as GPL | |
71 | |
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72 function [q, nfun] = quadl (f, a, b, tol = [], trace = false, varargin) |
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73 |
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74 if (nargin < 3) |
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75 print_usage (); |
5837 | 76 endif |
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77 |
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78 if (isa (a, "single") || isa (b, "single")) |
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79 eps = eps ("single"); |
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80 else |
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81 eps = eps ("double"); |
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82 endif |
5838 | 83 if (isempty (tol)) |
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84 tol = 1e-6; |
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85 elseif (! isscalar (tol) || tol < 0) |
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86 error ("quadl: TOL must be a scalar >=0"); |
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87 elseif (tol < eps) |
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88 warning ("quadl: TOL specified is smaller than machine precision, using %g", |
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89 tol); |
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90 tol = eps; |
5837 | 91 endif |
5838 | 92 if (isempty (trace)) |
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93 trace = false; |
5837 | 94 endif |
95 | |
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96 y = feval (f, [a, b], varargin{:}); |
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97 nfun = 1; |
5838 | 98 |
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99 fa = y(1); |
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100 fb = y(2); |
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102 h = b - a; |
5838 | 103 |
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104 [q, nfun, hmin] = adaptlobstp (f, a, b, fa, fb, Inf, nfun, abs (h), |
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105 tol, trace, varargin{:}); |
5838 | 106 |
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107 if (nfun > 10_000) |
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108 warning ("quadl: maximum iteration count reached -- possible singular integral"); |
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109 elseif (any (! isfinite (q(:)))) |
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110 warning ("quadl: infinite or NaN function evaluations were returned"); |
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111 elseif (hmin < (b - a) * eps) |
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112 warning ("quadl: minimum step size reached -- possible singular integral"); |
5837 | 113 endif |
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114 |
5837 | 115 endfunction |
116 | |
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117 function [q, nfun, hmin] = adaptlobstp (f, a, b, fa, fb, q0, nfun, hmin, |
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118 tol, trace, varargin) |
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119 persistent alpha = sqrt (2/3); |
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120 persistent beta = 1 / sqrt (5); |
5837 | 121 |
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122 if (nfun > 10_000) |
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123 q = q0; |
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124 return; |
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125 endif |
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126 |
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127 h = (b - a) / 2; |
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128 m = (a + b) / 2; |
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129 mll = m - alpha*h; |
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130 ml = m - beta*h; |
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131 mr = m + beta*h; |
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132 mrr = m + alpha*h; |
5838 | 133 x = [mll, ml, m, mr, mrr]; |
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134 y = feval (f, x, varargin{:}); |
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135 nfun += 1; |
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136 fmll = y(1); |
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137 fml = y(2); |
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138 fm = y(3); |
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139 fmr = y(4); |
5837 | 140 fmrr = y(5); |
5838 | 141 i2 = (h/6)*(fa + fb + 5*(fml+fmr)); |
142 i1 = (h/1470)*(77*(fa+fb) + 432*(fmll+fmrr) + 625*(fml+fmr) + 672*fm); | |
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143 |
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144 if (abs (b - a) < hmin) |
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145 hmin = abs (b - a); |
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146 endif |
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147 |
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148 if (trace) |
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149 disp ([nfun, a, b-a, i1]); |
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150 endif |
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151 |
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152 ## Force at least one adaptive step (nfun > 2 test). |
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153 if ((abs (i1-i2) < tol || mll <= a || b <= mrr) && nfun > 2) |
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154 q = i1; |
5837 | 155 else |
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156 q = zeros (6, 1, class (x)); |
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157 [q(1), nfun, hmin] = adaptlobstp (f, a , mll, fa , fmll, q0/6, nfun, hmin, |
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158 tol, trace, varargin{:}); |
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159 [q(2), nfun, hmin] = adaptlobstp (f, mll, ml , fmll, fml , q0/6, nfun, hmin, |
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160 tol, trace, varargin{:}); |
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161 [q(3), nfun, hmin] = adaptlobstp (f, ml , m , fml , fm , q0/6, nfun, hmin, |
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162 tol, trace, varargin{:}); |
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163 [q(4), nfun, hmin] = adaptlobstp (f, m , mr , fm , fmr , q0/6, nfun, hmin, |
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164 tol, trace, varargin{:}); |
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165 [q(5), nfun, hmin] = adaptlobstp (f, mr , mrr, fmr , fmrr, q0/6, nfun, hmin, |
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166 tol, trace, varargin{:}); |
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167 [q(6), nfun, hmin] = adaptlobstp (f, mrr, b , fmrr, fb , q0/6, nfun, hmin, |
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168 tol, trace, varargin{:}); |
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169 q = sum (q); |
5837 | 170 endif |
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171 |
5837 | 172 endfunction |
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173 |
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174 |
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175 ## basic functionality |
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176 %!assert (quadl (@(x) sin (x), 0, pi), 2, 1e-6) |
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177 |
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178 ## the values here are very high so it may be unavoidable that this fails |
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179 %!assert (quadl (@(x) sin (3*x).*cosh (x).*sinh (x),10,15, 1e-3), |
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180 %! 2.588424538641647e+10, 1e-3) |
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181 |
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182 ## extra parameters |
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183 %!assert (quadl (@(x,a,b) sin (a + b*x), 0, 1, [], [], 2, 3), |
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184 %! cos(2)/3 - cos(5)/3, 1e-6) |
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185 |
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186 ## test different tolerances. |
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187 %!test |
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188 %! [q, nfun1] = quadl (@(x) sin (2 + 3*x).^2, 0, 10, 0.5, []); |
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189 %! assert (q, (60 + sin(4) - sin(64))/12, 0.5); |
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190 %! [q, nfun2] = quadl (@(x) sin (2 + 3*x).^2, 0, 10, 0.1, []); |
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191 %! assert (q, (60 + sin(4) - sin(64))/12, 0.1); |
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192 %! assert (nfun2 > nfun1); |
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193 |
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194 %!test # test single input/output |
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195 %! assert (class (quadl (@sin, 0, 1)), "double"); |
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196 %! assert (class (quadl (@sin, single (0), 1)), "single"); |
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197 %! assert (class (quadl (@sin, 0, single (1))), "single"); |
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198 |
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199 ## Test input validation |
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200 %!error quadl () |
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201 %!error quadl (@sin) |
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202 %!error quadl (@sin,1) |
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203 %!error <TOL must be a scalar> quadl (@sin,0,1, ones (2,2)) |
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204 %!error <TOL must be .* .=0> quadl (@sin,0,1, -1) |