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annotate scripts/general/quadv.m @ 8920:eb63fbe60fab

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update copyright notices
author John W. Eaton <jwe@octave.org>
date Sat, 07 Mar 2009 10:41:27 -0500
parents cadc73247d65
children 1bf0ce0930be
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1 ## Copyright (C) 2008, 2009 David Bateman
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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
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19 ## -*- texinfo -*-
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20 ## @deftypefn {Function File} {@var{q} =} quadv (@var{f}, @var{a}, @var{b})
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21 ## @deftypefnx {Function File} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}, @var{tol})
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22 ## @deftypefnx {Function File} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}, @var{tol}, @var{trace})
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23 ## @deftypefnx {Function File} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}, @var{tol}, @var{trace}, @var{p1}, @var{p2}, @dots{})
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24 ## @deftypefnx {Function File} {[@var{q}, @var{fcnt}] =} quadl (@dots{})
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25 ##
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26 ## Numerically evaluate integral using adaptive Simpson's rule.
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27 ## @code{quadv (@var{f}, @var{a}, @var{b})} approximates the integral of
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28 ## @code{@var{f}(@var{x})} to the default absolute tolerance of @code{1e-6}.
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29 ## @var{f} is either a function handle, inline function or string
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30 ## containing the name of the function to evaluate. The function @var{f}
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31 ## must accept a string, and can return a vector representing the
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32 ## approximation to @var{n} different sub-functions.
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33 ##
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34 ## If defined, @var{tol} defines the absolute tolerance to which to
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35 ## which to integrate each sub-interval of @code{@var{f}(@var{x})}.
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36 ## While if @var{trace} is defined, displays the left end point of the
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37 ## current interval, the interval length, and the partial integral.
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38 ##
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39 ## Additional arguments @var{p1}, etc, are passed directly to @var{f}.
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40 ## To use default values for @var{tol} and @var{trace}, one may pass
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41 ## empty matrices.
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42 ## @seealso{triplequad, dblquad, quad, quadl, quadgk, trapz}
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43 ## @end deftypefn
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44
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45 function [q, fcnt] = quadv (f, a, b, tol, trace, varargin)
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46 if (nargin < 3)
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47 print_usage ();
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48 endif
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49 if (nargin < 4)
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50 tol = [];
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51 endif
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52 if (nargin < 5)
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53 trace = [];
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54 endif
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55 if (isa (a, "single") || isa (b, "single"))
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56 myeps = eps ("single");
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57 else
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58 myeps = eps;
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59 endif
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60 if (isempty (tol))
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61 tol = 1e-6;
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62 endif
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63 if (isempty (trace))
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64 trace = 0;
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65 endif
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66
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67 ## Split the interval into 3 abscissa, and apply a 3 point Simpson's rule
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68 c = (a + b) / 2;
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69 fa = feval (f, a, varargin{:});
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70 fc = feval (f, c, varargin{:});
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71 fb = feval (f, b, varargin{:});
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72 fcnt = 3;
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73
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74 ## If have edge singularities, move edge point by eps*(b-a) as
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75 ## discussed in Shampine paper used to implement quadgk
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76 if (isinf (fa))
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77 fa = feval (f, a + myeps * (b-a), varargin{:});
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78 endif
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79 if (isinf (fb))
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80 fb = feval (f, b - myeps * (b-a), varargin{:});
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81 endif
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82
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83 h = (b - a) / 2;
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84 q = (b - a) / 6 * (fa + 4 * fc + fb);
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85
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86 [q, fcnt, hmin] = simpsonstp (f, a, b, c, fa, fb, fc, q, fcnt, abs (b - a),
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87 tol, trace, varargin{:});
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88
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89 if (fcnt > 10000)
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90 warning ("maximum iteration count reached");
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91 elseif (isnan (q) || isinf (q))
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92 warning ("infinite or NaN function evaluations were returned");
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93 elseif (hmin < (b - a) * myeps)
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94 warning ("minimum step size reached -- possibly singular integral");
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95 endif
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96 endfunction
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97
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98 function [q, fcnt, hmin] = simpsonstp (f, a, b, c, fa, fb, fc, q0,
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99 fcnt, hmin, tol, trace, varargin)
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100 if (fcnt > 10000)
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101 q = q0;
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102 else
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103 d = (a + c) / 2;
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104 e = (c + b) / 2;
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105 fd = feval (f, d, varargin{:});
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106 fe = feval (f, e, varargin{:});
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107 fcnt += 2;
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108 q1 = (c - a) / 6 * (fa + 4 * fd + fc);
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109 q2 = (b - c) / 6 * (fc + 4 * fe + fb);
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110 q = q1 + q2;
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111
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112 if (abs(a - c) < hmin)
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113 hmin = abs (a - c);
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114 endif
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115
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116 if (trace)
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117 disp ([fcnt, a, b-a, q]);
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118 endif
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119
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120 ## Force at least one adpative step.
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121 if (fcnt == 5 || abs (q - q0) > tol)
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122 [q1, fcnt, hmin] = simpsonstp (f, a, c, d, fa, fc, fd, q1, fcnt, hmin,
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123 tol, trace, varargin{:});
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124 [q2, fcnt, hmin] = simpsonstp (f, c, b, e, fc, fb, fe, q2, fcnt, hmin,
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125 tol, trace, varargin{:});
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126 q = q1 + q2;
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127 endif
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128 endif
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129 endfunction
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130
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131 %!assert (quadv (@sin, 0, 2 * pi), 0, 1e-5)
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132 %!assert (quadv (@sin, 0, pi), 2, 1e-5)
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133
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134 %% Handles weak singularities at the edge
680631e787aa Add quadv, quadgk, dblquad and triplequad functions
David Bateman <dbateman@free.fr>
parents:
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135 %!assert (quadv (@(x) 1 ./ sqrt(x), 0, 1), 2, 1e-5)
680631e787aa Add quadv, quadgk, dblquad and triplequad functions
David Bateman <dbateman@free.fr>
parents:
diff changeset
136