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annotate scripts/general/quadv.m @ 20039:eba80000fa0d rc-4-0-0-2

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maint: Bump version number for 4.0.0-rc2.
author John W. Eaton <jwe@octave.org>
date Fri, 20 Mar 2015 16:19:38 -0400
parents 9fc020886ae9
children 7503499a252b
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1 ## Copyright (C) 2008-2015 David Bateman
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2 ## Copyright (C) 2012 Alexander Klein
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3 ##
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4 ## This file is part of Octave.
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5 ##
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6 ## Octave is free software; you can redistribute it and/or modify it
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7 ## under the terms of the GNU General Public License as published by
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8 ## the Free Software Foundation; either version 3 of the License, or (at
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9 ## your option) any later version.
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10 ##
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11 ## Octave is distributed in the hope that it will be useful, but
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12 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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14 ## General Public License for more details.
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15 ##
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16 ## You should have received a copy of the GNU General Public License
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17 ## along with Octave; see the file COPYING. If not, see
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18 ## <http://www.gnu.org/licenses/>.
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19
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20 ## -*- texinfo -*-
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21 ## @deftypefn {Function File} {@var{q} =} quadv (@var{f}, @var{a}, @var{b})
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22 ## @deftypefnx {Function File} {@var{q} =} quadv (@var{f}, @var{a}, @var{b}, @var{tol})
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23 ## @deftypefnx {Function File} {@var{q} =} quadv (@var{f}, @var{a}, @var{b}, @var{tol}, @var{trace})
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24 ## @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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25 ## @deftypefnx {Function File} {[@var{q}, @var{nfun}] =} quadv (@dots{})
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26 ##
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27 ## Numerically evaluate the integral of @var{f} from @var{a} to @var{b}
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28 ## using an adaptive Simpson's rule.
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29 ## @var{f} is a function handle, inline function, or string
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30 ## containing the name of the function to evaluate.
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31 ## @code{quadv} is a vectorized version of @code{quad} and the function
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32 ## defined by @var{f} must accept a scalar or vector as input and return a
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33 ## scalar, vector, or array as output.
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34 ##
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35 ## @var{a} and @var{b} are the lower and upper limits of integration. Both
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36 ## limits must be finite.
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37 ##
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38 ## The optional argument @var{tol} defines the tolerance used to stop
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39 ## the adaptation procedure. The default value is @math{1e^{-6}}.
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40 ##
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41 ## The algorithm used by @code{quadv} involves recursively subdividing the
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42 ## integration interval and applying Simpson's rule on each subinterval.
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43 ## If @var{trace} is true then after computing each of these partial
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44 ## integrals display: (1) the total number of function evaluations,
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45 ## (2) the left end of the subinterval, (3) the length of the subinterval,
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46 ## (4) the approximation of the integral over the subinterval.
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47 ##
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48 ## Additional arguments @var{p1}, etc., are passed directly to the function
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49 ## @var{f}. To use default values for @var{tol} and @var{trace}, one may pass
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50 ## empty matrices ([]).
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51 ##
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52 ## The result of the integration is returned in @var{q}. @var{nfun} indicates
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53 ## the number of function evaluations that were made.
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54 ##
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55 ## Note: @code{quadv} is written in Octave's scripting language and can be
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56 ## used recursively in @code{dblquad} and @code{triplequad}, unlike the
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57 ## similar @code{quad} function.
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58 ## @seealso{quad, quadl, quadgk, quadcc, trapz, dblquad, triplequad}
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59 ## @end deftypefn
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60
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61 function [q, nfun] = quadv (f, a, b, tol, trace, varargin)
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62 ## TODO: Make norm for convergence testing configurable
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63
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64 if (nargin < 3)
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65 print_usage ();
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66 endif
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67 if (nargin < 4)
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68 tol = [];
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69 endif
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70 if (nargin < 5)
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71 trace = [];
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72 endif
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73 if (isa (a, "single") || isa (b, "single"))
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74 myeps = eps ("single");
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75 else
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76 myeps = eps;
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77 endif
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78 if (isempty (tol))
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79 tol = 1e-6;
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80 endif
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81 if (isempty (trace))
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82 trace = 0;
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83 endif
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84
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85 ## Split the interval into 3 abscissa, and apply a 3 point Simpson's rule
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86 c = (a + b) / 2;
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87 fa = feval (f, a, varargin{:});
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88 fc = feval (f, c, varargin{:});
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89 fb = feval (f, b, varargin{:});
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90 nfun = 3;
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91
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92 ## If have edge singularities, move edge point by eps*(b-a) as
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93 ## discussed in Shampine paper used to implement quadgk
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94 if (any (isinf (fa(:))))
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95 fa = feval (f, a + myeps * (b-a), varargin{:});
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96 endif
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97 if (any (isinf (fb(:))))
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98 fb = feval (f, b - myeps * (b-a), varargin{:});
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99 endif
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100
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101 h = (b - a);
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102 q = (b - a) / 6 * (fa + 4 * fc + fb);
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103
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104 [q, nfun, hmin] = simpsonstp (f, a, b, c, fa, fb, fc, q, nfun, abs (h),
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105 tol, trace, varargin{:});
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106
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107 if (nfun > 10000)
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108 warning ("maximum iteration count reached");
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109 elseif (any (! isfinite (q(:))))
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110 warning ("infinite or NaN function evaluations were returned");
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111 elseif (hmin < (b - a) * myeps)
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112 warning ("minimum step size reached -- possibly singular integral");
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113 endif
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114 endfunction
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115
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116 function [q, nfun, hmin] = simpsonstp (f, a, b, c, fa, fb, fc, q0,
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117 nfun, hmin, tol, trace, varargin)
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118 if (nfun > 10000)
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119 q = q0;
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120 else
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121 d = (a + c) / 2;
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122 e = (c + b) / 2;
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123 fd = feval (f, d, varargin{:});
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124 fe = feval (f, e, varargin{:});
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125 nfun += 2;
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126 q1 = (c - a) / 6 * (fa + 4 * fd + fc);
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127 q2 = (b - c) / 6 * (fc + 4 * fe + fb);
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128 q = q1 + q2;
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129
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130 if (abs(a - c) < hmin)
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131 hmin = abs (a - c);
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132 endif
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133
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134 if (trace)
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135 disp ([nfun, a, b-a, q]);
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136 endif
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137
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138 ## Force at least one adpative step.
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139 ## Not vectorizing q-q0 in the norm provides a more rigid criterion for
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140 ## matrix-valued functions.
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141 if (nfun == 5 || norm (q - q0, Inf) > tol)
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142 [q1, nfun, hmin] = simpsonstp (f, a, c, d, fa, fc, fd, q1, nfun, hmin,
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143 tol, trace, varargin{:});
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144 [q2, nfun, hmin] = simpsonstp (f, c, b, e, fc, fb, fe, q2, nfun, hmin,
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145 tol, trace, varargin{:});
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146 q = q1 + q2;
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147 endif
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148 endif
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149 endfunction
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150
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151
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152 %!assert (quadv (@sin, 0, 2 * pi), 0, 1e-5)
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153 %!assert (quadv (@sin, 0, pi), 2, 1e-5)
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154
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155 ## Handles weak singularities at the edge
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156 %!assert (quadv (@(x) 1 ./ sqrt (x), 0, 1), 2, 1e-5)
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157
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158 ## Handles vector-valued functions
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159 %!assert (quadv (@(x) [(sin (x)), (sin (2 * x))], 0, pi), [2, 0], 1e-5)
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160
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161 ## Handles matrix-valued functions
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162 %!assert (quadv (@(x) [ x, x, x; x, 1./sqrt(x), x; x, x, x ], 0, 1 ), [0.5, 0.5, 0.5; 0.5, 2, 0.5; 0.5, 0.5, 0.5], 1e-5)
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163