Mercurial > octave-nkf
annotate scripts/general/quadl.m @ 20614:10ec79b47808
use new string_value method to handle value extraction errors
* __voronoi__.cc, chol.cc, colamd.cc, fftw.cc:
Use new string_value method.
author | John W. Eaton <jwe@octave.org> |
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date | Thu, 08 Oct 2015 18:15:56 -0400 |
parents | 83792dd9bcc1 |
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1 ## Copyright (C) 1998-2015 Walter Gautschi |
5837 | 2 ## |
3 ## This file is part of Octave. | |
4 ## | |
5 ## Octave is free software; you can redistribute it and/or modify it | |
6 ## under the terms of the GNU General Public License as published by | |
7016 | 7 ## the Free Software Foundation; either version 3 of the License, or (at |
8 ## your option) any later version. | |
5837 | 9 ## |
10 ## Octave is distributed in the hope that it will be useful, but | |
11 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
13 ## General Public License for more details. | |
14 ## | |
15 ## You should have received a copy of the GNU General Public License | |
7016 | 16 ## along with Octave; see the file COPYING. If not, see |
17 ## <http://www.gnu.org/licenses/>. | |
5837 | 18 |
19 ## -*- texinfo -*- | |
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20 ## @deftypefn {Function File} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}) |
5837 | 21 ## @deftypefnx {Function File} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}, @var{tol}) |
22 ## @deftypefnx {Function File} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}, @var{tol}, @var{trace}) | |
23 ## @deftypefnx {Function File} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}, @var{tol}, @var{trace}, @var{p1}, @var{p2}, @dots{}) | |
24 ## | |
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25 ## Numerically evaluate the integral of @var{f} from @var{a} to @var{b} using |
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26 ## an adaptive Lobatto rule. |
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27 ## |
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28 ## @var{f} is a function handle, inline function, or string containing the name |
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29 ## of the function to evaluate. The function @var{f} must be vectorized and |
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30 ## return a vector of output values when given a vector of input values. |
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31 ## |
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32 ## @var{a} and @var{b} are the lower and upper limits of integration. Both |
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33 ## limits must be finite. |
5837 | 34 ## |
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35 ## The optional argument @var{tol} defines the relative tolerance with which |
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36 ## to perform the integration. The default value is @code{eps}. |
5837 | 37 ## |
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38 ## The algorithm used by @code{quadl} involves recursively subdividing the |
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39 ## integration interval. If @var{trace} is defined then for each subinterval |
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40 ## display: (1) the left end of the subinterval, (2) the length of the |
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41 ## subinterval, (3) the approximation of the integral over the subinterval. |
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42 ## |
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43 ## Additional arguments @var{p1}, etc., are passed directly to the function |
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44 ## @var{f}. To use default values for @var{tol} and @var{trace}, one may pass |
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45 ## empty matrices ([]). |
5837 | 46 ## |
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47 ## Reference: @nospell{W. Gander and W. Gautschi}, @cite{Adaptive Quadrature - |
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48 ## Revisited}, BIT Vol. 40, No. 1, March 2000, pp. 84--101. |
5837 | 49 ## @url{http://www.inf.ethz.ch/personal/gander/} |
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50 ## @seealso{quad, quadv, quadgk, quadcc, trapz, dblquad, triplequad} |
5837 | 51 ## @end deftypefn |
52 | |
53 ## Author: Walter Gautschi | |
54 ## Date: 08/03/98 | |
55 ## Reference: Gander, Computermathematik, Birkhaeuser, 1992. | |
56 | |
57 ## 2003-08-05 Shai Ayal | |
58 ## * permission from author to release as GPL | |
59 ## 2004-02-10 Paul Kienzle | |
60 ## * renamed to quadl for compatibility | |
61 ## * replace global variable terminate2 with local function need_warning | |
62 ## * add paper ref to docs | |
63 | |
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64 function q = quadl (f, a, b, tol = [], trace = false, varargin) |
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65 |
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66 if (nargin < 3) |
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67 print_usage (); |
5837 | 68 endif |
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69 |
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70 if (isa (a, "single") || isa (b, "single")) |
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71 myeps = eps ("single"); |
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72 else |
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73 myeps = eps; |
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74 endif |
5838 | 75 if (isempty (tol)) |
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76 tol = myeps; |
5837 | 77 endif |
5838 | 78 if (isempty (trace)) |
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79 trace = false; |
5837 | 80 endif |
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81 if (tol < myeps) |
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82 tol = myeps; |
5837 | 83 endif |
84 | |
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85 ## Track whether recursion has occurred |
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86 global __quadl_recurse_done__; |
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87 __quadl_recurse_done__ = false; |
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88 ## Track whether warning about machine precision has been issued |
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89 global __quadl_need_warning__; |
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90 __quadl_need_warning__ = true; |
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91 |
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92 m = (a+b)/2; |
5837 | 93 h = (b-a)/2; |
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94 alpha = sqrt (2/3); |
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95 beta = 1/sqrt (5); |
5838 | 96 |
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97 x1 = .942882415695480; |
5837 | 98 x2 = .641853342345781; |
99 x3 = .236383199662150; | |
5838 | 100 |
101 x = [a, m-x1*h, m-alpha*h, m-x2*h, m-beta*h, m-x3*h, m, m+x3*h, ... | |
102 m+beta*h, m+x2*h, m+alpha*h, m+x1*h, b]; | |
103 | |
104 y = feval (f, x, varargin{:}); | |
105 | |
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106 fa = y(1); |
5837 | 107 fb = y(13); |
5838 | 108 |
109 i2 = (h/6)*(y(1) + y(13) + 5*(y(5)+y(9))); | |
110 | |
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111 i1 = (h/1470)*( 77*(y(1)+y(13)) |
10549 | 112 + 432*(y(3)+y(11)) |
113 + 625*(y(5)+y(9)) | |
114 + 672*y(7)); | |
5838 | 115 |
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116 is = h*( .0158271919734802*(y(1)+y(13)) |
10549 | 117 +.0942738402188500*(y(2)+y(12)) |
118 + .155071987336585*(y(3)+y(11)) | |
119 + .188821573960182*(y(4)+y(10)) | |
120 + .199773405226859*(y(5)+y(9)) | |
121 + .224926465333340*(y(6)+y(8)) | |
122 + .242611071901408*y(7)); | |
5838 | 123 |
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124 s = sign (is); |
5838 | 125 if (s == 0) |
126 s = 1; | |
5837 | 127 endif |
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128 erri1 = abs (i1-is); |
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129 erri2 = abs (i2-is); |
5837 | 130 if (erri2 != 0) |
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131 R = erri1/erri2; |
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132 else |
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133 R = 1; |
5837 | 134 endif |
135 if (R > 0 && R < 1) | |
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136 tol /= R; |
5837 | 137 endif |
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138 is = s * abs (is) * tol/myeps; |
5837 | 139 if (is == 0) |
140 is = b-a; | |
141 endif | |
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142 |
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143 q = adaptlobstp (f, a, b, fa, fb, is, trace, varargin{:}); |
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144 |
5837 | 145 endfunction |
146 | |
147 ## ADAPTLOBSTP Recursive function used by QUADL. | |
148 ## | |
5838 | 149 ## Q = ADAPTLOBSTP('F', A, B, FA, FB, IS, TRACE) tries to |
5837 | 150 ## approximate the integral of F(X) from A to B to |
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151 ## an appropriate relative error. The argument 'F' is |
5837 | 152 ## a string containing the name of f. The remaining |
153 ## arguments are generated by ADAPTLOB or by recursion. | |
154 ## | |
155 ## Walter Gautschi, 08/03/98 | |
156 | |
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157 function q = adaptlobstp (f, a, b, fa, fb, is, trace, varargin) |
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158 global __quadl_recurse_done__; |
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159 global __quadl_need_warning__; |
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160 |
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161 h = (b-a)/2; |
5837 | 162 m = (a+b)/2; |
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163 alpha = sqrt (2/3); |
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164 beta = 1 / sqrt (5); |
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165 mll = m-alpha*h; |
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166 ml = m-beta*h; |
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167 mr = m+beta*h; |
5837 | 168 mrr = m+alpha*h; |
5838 | 169 x = [mll, ml, m, mr, mrr]; |
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170 y = feval (f, x, varargin{:}); |
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171 fmll = y(1); |
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172 fml = y(2); |
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173 fm = y(3); |
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174 fmr = y(4); |
5837 | 175 fmrr = y(5); |
5838 | 176 i2 = (h/6)*(fa + fb + 5*(fml+fmr)); |
177 i1 = (h/1470)*(77*(fa+fb) + 432*(fmll+fmrr) + 625*(fml+fmr) + 672*fm); | |
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178 if ((is+(i1-i2) == is || mll <= a || b <= mrr) && __quadl_recurse_done__) |
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179 if ((m <= a || b <= m) && __quadl_need_warning__) |
5838 | 180 warning ("quadl: interval contains no more machine number"); |
181 warning ("quadl: required tolerance may not be met"); | |
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182 __quadl_need_warning__ = false; |
5837 | 183 endif |
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184 q = i1; |
5837 | 185 if (trace) |
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186 disp ([a, b-a, q]); |
5837 | 187 endif |
188 else | |
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189 __quadl_recurse_done__ = true; |
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190 q = ( adaptlobstp (f, a , mll, fa , fmll, is, trace, varargin{:}) |
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191 + adaptlobstp (f, mll, ml , fmll, fml , is, trace, varargin{:}) |
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192 + adaptlobstp (f, ml , m , fml , fm , is, trace, varargin{:}) |
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193 + adaptlobstp (f, m , mr , fm , fmr , is, trace, varargin{:}) |
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194 + adaptlobstp (f, mr , mrr, fmr , fmrr, is, trace, varargin{:}) |
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195 + adaptlobstp (f, mrr, b , fmrr, fb , is, trace, varargin{:})); |
5837 | 196 endif |
197 endfunction | |
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198 |
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199 |
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200 ## basic functionality |
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201 %!assert (quadl (@(x) sin (x), 0, pi, [], []), 2, -3e-16) |
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202 |
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203 ## the values here are very high so it may be unavoidable that this fails |
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204 %!assert (quadl (@(x) sin (3*x).*cosh (x).*sinh (x),10,15), |
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205 %! 2.588424538641647e+10, -1.1e-14) |
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206 |
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207 ## extra parameters |
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208 %!assert (quadl (@(x,a,b) sin (a + b*x), 0, 1, [], [], 2, 3), |
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209 %! cos(2)/3 - cos(5)/3, -3e-16) |
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210 |
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211 ## test different tolerances. |
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212 %!assert (quadl (@(x) sin (2 + 3*x).^2, 0, 10, 0.3, []), |
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213 %! (60 + sin(4) - sin(64))/12, -0.3) |
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214 %!assert (quadl (@(x) sin (2 + 3*x).^2, 0, 10, 0.1, []), |
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215 %! (60 + sin(4) - sin(64))/12, -0.1) |
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216 |