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annotate scripts/general/quadv.m @ 11563:3c6e8aaa9555

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Grammarcheck m-files before 3.4 release.
author Rik <octave@nomad.inbox5.com>
date Tue, 18 Jan 2011 20:55:01 -0800
parents 1811f4f8b3b5
children c792872f8942
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1 ## Copyright (C) 2008-2011 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} =} quadv (@var{f}, @var{a}, @var{b}, @var{tol})
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22 ## @deftypefnx {Function File} {@var{q} =} quadv (@var{f}, @var{a}, @var{b}, @var{tol}, @var{trace})
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23 ## @deftypefnx {Function File} {@var{q} =} quadv (@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}] =} quadv (@dots{})
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25 ##
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26 ## Numerically evaluate the integral of @var{f} from @var{a} to @var{b}
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27 ## using adaptive Simpson's rule.
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28 ## @var{f} is either a function handle, inline function or string
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29 ## containing the name of the function to evaluate.
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30 ## The function defined by @var{f} may be a scalar, vector or array-valued.
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31 ##
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32 ## If a value for @var{tol} is given, it defines the tolerance used to stop
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33 ## the adaptation procedure, otherwise the default value of 1e-6 is used.
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34 ##
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35 ## The algorithm used by @code{quadv}, involves recursively subdividing the
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36 ## integration interval and applying Simpson's rule on each sub-interval.
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37 ## If @var{trace} is @var{true}, after computing each of these partial
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38 ## integrals, display the total number of function evaluations, the left end
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39 ## of the sub-interval, the length of the sub-interval and the approximation
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40 ## of the integral over the sub-interval.
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41 ##
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42 ## Additional arguments @var{p1}, etc., are passed directly to @var{f}.
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43 ## To use default values for @var{tol} and @var{trace}, one may pass
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44 ## empty matrices.
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45 ##
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46 ##@seealso{quad, quadl, quadgk, quadcc, trapz, dblquad, triplequad}
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47 ## @end deftypefn
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48
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49 function [q, fcnt] = quadv (f, a, b, tol, trace, varargin)
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50 if (nargin < 3)
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51 print_usage ();
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52 endif
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53 if (nargin < 4)
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54 tol = [];
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55 endif
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56 if (nargin < 5)
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57 trace = [];
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58 endif
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59 if (isa (a, "single") || isa (b, "single"))
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60 myeps = eps ("single");
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61 else
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62 myeps = eps;
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63 endif
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64 if (isempty (tol))
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65 tol = 1e-6;
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66 endif
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67 if (isempty (trace))
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68 trace = 0;
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69 endif
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70
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71 ## Split the interval into 3 abscissa, and apply a 3 point Simpson's rule
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72 c = (a + b) / 2;
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73 fa = feval (f, a, varargin{:});
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74 fc = feval (f, c, varargin{:});
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75 fb = feval (f, b, varargin{:});
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76 fcnt = 3;
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77
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78 ## If have edge singularities, move edge point by eps*(b-a) as
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79 ## discussed in Shampine paper used to implement quadgk
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80 if (isinf (fa))
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81 fa = feval (f, a + myeps * (b-a), varargin{:});
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82 endif
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83 if (isinf (fb))
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84 fb = feval (f, b - myeps * (b-a), varargin{:});
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85 endif
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86
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87 h = (b - a);
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88 q = (b - a) / 6 * (fa + 4 * fc + fb);
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89
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90 [q, fcnt, hmin] = simpsonstp (f, a, b, c, fa, fb, fc, q, fcnt, abs (h),
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91 tol, trace, varargin{:});
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92
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93 if (fcnt > 10000)
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94 warning ("maximum iteration count reached");
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95 elseif (isnan (q) || isinf (q))
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96 warning ("infinite or NaN function evaluations were returned");
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97 elseif (hmin < (b - a) * myeps)
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98 warning ("minimum step size reached -- possibly singular integral");
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99 endif
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100 endfunction
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101
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102 function [q, fcnt, hmin] = simpsonstp (f, a, b, c, fa, fb, fc, q0,
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103 fcnt, hmin, tol, trace, varargin)
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104 if (fcnt > 10000)
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105 q = q0;
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106 else
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107 d = (a + c) / 2;
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108 e = (c + b) / 2;
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109 fd = feval (f, d, varargin{:});
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110 fe = feval (f, e, varargin{:});
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111 fcnt += 2;
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112 q1 = (c - a) / 6 * (fa + 4 * fd + fc);
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113 q2 = (b - c) / 6 * (fc + 4 * fe + fb);
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114 q = q1 + q2;
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115
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116 if (abs(a - c) < hmin)
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117 hmin = abs (a - c);
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118 endif
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119
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120 if (trace)
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121 disp ([fcnt, a, b-a, q]);
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122 endif
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123
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124 ## Force at least one adpative step.
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125 if (fcnt == 5 || abs (q - q0) > tol)
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126 [q1, fcnt, hmin] = simpsonstp (f, a, c, d, fa, fc, fd, q1, fcnt, hmin,
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127 tol, trace, varargin{:});
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128 [q2, fcnt, hmin] = simpsonstp (f, c, b, e, fc, fb, fe, q2, fcnt, hmin,
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129 tol, trace, varargin{:});
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130 q = q1 + q2;
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131 endif
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132 endif
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133 endfunction
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134
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135 %!assert (quadv (@sin, 0, 2 * pi), 0, 1e-5)
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136 %!assert (quadv (@sin, 0, pi), 2, 1e-5)
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137
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138 %% Handles weak singularities at the edge
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139 %!assert (quadv (@(x) 1 ./ sqrt(x), 0, 1), 2, 1e-5)
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140
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141 %% Handles vector-valued functions
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142 %!assert (quadv (@(x) [(sin (x)), (sin (2 * x))], 0, pi), [2, 0], 1e-5)
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143