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doc: Update docstrings to have one sentence summary as first line. Update scripts in audio, elfun, general, geometry, and image directories. * scripts/audio/@audioplayer/__get_properties__.m, scripts/audio/@audioplayer/audioplayer.m, scripts/audio/@audioplayer/get.m, scripts/audio/@audioplayer/isplaying.m, scripts/audio/@audioplayer/play.m, scripts/audio/@audioplayer/playblocking.m, scripts/audio/@audioplayer/set.m, scripts/audio/@audioplayer/subsasgn.m, scripts/audio/@audioplayer/subsref.m, scripts/audio/@audiorecorder/audiorecorder.m, scripts/audio/@audiorecorder/get.m, scripts/audio/@audiorecorder/getaudiodata.m, scripts/audio/@audiorecorder/getplayer.m, scripts/audio/@audiorecorder/isrecording.m, scripts/audio/@audiorecorder/play.m, scripts/audio/@audiorecorder/record.m, scripts/audio/@audiorecorder/recordblocking.m, scripts/audio/@audiorecorder/set.m, scripts/audio/@audiorecorder/stop.m, scripts/audio/@audiorecorder/subsasgn.m, scripts/audio/@audiorecorder/subsref.m, scripts/audio/lin2mu.m, scripts/audio/mu2lin.m, scripts/audio/record.m, scripts/audio/sound.m, scripts/audio/soundsc.m, scripts/audio/wavread.m, scripts/audio/wavwrite.m, scripts/elfun/cosd.m, scripts/elfun/sind.m, scripts/elfun/tand.m, scripts/general/accumarray.m, scripts/general/accumdim.m, scripts/general/bitcmp.m, scripts/general/bitget.m, scripts/general/bitset.m, scripts/general/blkdiag.m, scripts/general/cart2pol.m, scripts/general/cart2sph.m, scripts/general/cell2mat.m, scripts/general/celldisp.m, scripts/general/chop.m, scripts/general/circshift.m, scripts/general/common_size.m, scripts/general/cplxpair.m, scripts/general/cumtrapz.m, scripts/general/dblquad.m, scripts/general/deal.m, scripts/general/del2.m, scripts/general/display.m, scripts/general/divergence.m, scripts/general/fieldnames.m, scripts/general/flip.m, scripts/general/flipdim.m, scripts/general/fliplr.m, scripts/general/flipud.m, scripts/general/gradient.m, scripts/general/interp3.m, scripts/general/interpft.m, scripts/general/interpn.m, scripts/general/loadobj.m, scripts/general/logspace.m, scripts/general/methods.m, scripts/general/nargchk.m, scripts/general/narginchk.m, scripts/general/nargoutchk.m, scripts/general/nextpow2.m, scripts/general/nthargout.m, scripts/general/num2str.m, scripts/general/pol2cart.m, scripts/general/polyarea.m, scripts/general/postpad.m, scripts/general/prepad.m, scripts/general/profile.m, scripts/general/quadgk.m, scripts/general/quadl.m, scripts/general/quadv.m, scripts/general/randi.m, scripts/general/rat.m, scripts/general/repmat.m, scripts/general/rot90.m, scripts/general/rotdim.m, scripts/general/saveobj.m, scripts/general/shift.m, scripts/general/shiftdim.m, scripts/general/sortrows.m, scripts/general/sph2cart.m, scripts/general/structfun.m, scripts/general/subsindex.m, scripts/general/trapz.m, scripts/general/triplequad.m, scripts/geometry/delaunayn.m, scripts/geometry/dsearch.m, scripts/geometry/dsearchn.m, scripts/geometry/griddata.m, scripts/geometry/griddata3.m, scripts/geometry/griddatan.m, scripts/geometry/inpolygon.m, scripts/geometry/rectint.m, scripts/geometry/tsearchn.m, scripts/geometry/voronoi.m, scripts/geometry/voronoin.m, scripts/help/__unimplemented__.m, scripts/help/doc.m, scripts/help/doc_cache_create.m, scripts/help/get_first_help_sentence.m, scripts/help/help.m, scripts/help/lookfor.m, scripts/help/print_usage.m, scripts/help/type.m, scripts/help/which.m, scripts/image/autumn.m, scripts/image/bone.m, scripts/image/brighten.m, scripts/image/cmpermute.m, scripts/image/colorcube.m, scripts/image/contrast.m, scripts/image/cool.m, scripts/image/copper.m, scripts/image/cubehelix.m, scripts/image/flag.m, scripts/image/gmap40.m, scripts/image/gray.m, scripts/image/gray2ind.m, scripts/image/hot.m, scripts/image/hsv.m, scripts/image/image.m, scripts/image/imagesc.m, scripts/image/imfinfo.m, scripts/image/imformats.m, scripts/image/imread.m, scripts/image/imshow.m, scripts/image/imwrite.m, scripts/image/iscolormap.m, scripts/image/jet.m, scripts/image/lines.m, scripts/image/ntsc2rgb.m, scripts/image/ocean.m, scripts/image/pink.m, scripts/image/prism.m, scripts/image/rainbow.m, scripts/image/rgb2ntsc.m, scripts/image/spinmap.m, scripts/image/spring.m, scripts/image/summer.m, scripts/image/white.m, scripts/image/winter.m: Update docstrings to have one sentence summary as first line. Re-structure to have line lengths <= 80 chars.
author Rik <rik@octave.org>
date Sun, 03 May 2015 09:36:20 -0700
parents 9fc020886ae9
children
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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 ##
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30 ## @var{f} is a function handle, inline function, or string containing the name
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31 ## of the function to evaluate. @code{quadv} is a vectorized version of
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32 ## @code{quad} and the function defined by @var{f} must accept a scalar or
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33 ## vector as input and return a 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 the
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39 ## adaptation procedure. The default value is 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}
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53 ##
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54 ## @var{nfun} indicates the number of function evaluations that were made.
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55 ##
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56 ## Note: @code{quadv} is written in Octave's scripting language and can be
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57 ## used recursively in @code{dblquad} and @code{triplequad}, unlike the
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58 ## @code{quad} function.
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59 ## @seealso{quad, quadl, quadgk, quadcc, trapz, dblquad, triplequad}
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60 ## @end deftypefn
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61
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62 function [q, nfun] = quadv (f, a, b, tol, trace, varargin)
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63 ## TODO: Make norm for convergence testing configurable
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64
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65 if (nargin < 3)
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66 print_usage ();
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67 endif
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68 if (nargin < 4)
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69 tol = [];
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70 endif
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71 if (nargin < 5)
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72 trace = [];
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73 endif
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74 if (isa (a, "single") || isa (b, "single"))
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75 myeps = eps ("single");
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76 else
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77 myeps = eps;
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78 endif
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79 if (isempty (tol))
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80 tol = 1e-6;
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81 endif
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82 if (isempty (trace))
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83 trace = 0;
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84 endif
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85
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86 ## Split the interval into 3 abscissa, and apply a 3 point Simpson's rule
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87 c = (a + b) / 2;
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88 fa = feval (f, a, varargin{:});
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89 fc = feval (f, c, varargin{:});
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90 fb = feval (f, b, varargin{:});
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91 nfun = 3;
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92
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93 ## If have edge singularities, move edge point by eps*(b-a) as
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94 ## discussed in Shampine paper used to implement quadgk
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95 if (any (isinf (fa(:))))
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96 fa = feval (f, a + myeps * (b-a), varargin{:});
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97 endif
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98 if (any (isinf (fb(:))))
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99 fb = feval (f, b - myeps * (b-a), varargin{:});
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100 endif
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101
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102 h = (b - a);
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103 q = (b - a) / 6 * (fa + 4 * fc + fb);
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104
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105 [q, nfun, hmin] = simpsonstp (f, a, b, c, fa, fb, fc, q, nfun, abs (h),
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106 tol, trace, varargin{:});
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107
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108 if (nfun > 10000)
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109 warning ("maximum iteration count reached");
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110 elseif (any (! isfinite (q(:))))
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111 warning ("infinite or NaN function evaluations were returned");
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112 elseif (hmin < (b - a) * myeps)
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113 warning ("minimum step size reached -- possibly singular integral");
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114 endif
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115 endfunction
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116
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117 function [q, nfun, hmin] = simpsonstp (f, a, b, c, fa, fb, fc, q0,
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118 nfun, hmin, tol, trace, varargin)
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119 if (nfun > 10000)
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120 q = q0;
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121 else
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122 d = (a + c) / 2;
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123 e = (c + b) / 2;
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124 fd = feval (f, d, varargin{:});
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125 fe = feval (f, e, varargin{:});
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126 nfun += 2;
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127 q1 = (c - a) / 6 * (fa + 4 * fd + fc);
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128 q2 = (b - c) / 6 * (fc + 4 * fe + fb);
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129 q = q1 + q2;
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130
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131 if (abs(a - c) < hmin)
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132 hmin = abs (a - c);
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133 endif
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134
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135 if (trace)
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136 disp ([nfun, a, b-a, q]);
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137 endif
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138
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139 ## Force at least one adpative step.
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140 ## Not vectorizing q-q0 in the norm provides a more rigid criterion for
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141 ## matrix-valued functions.
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142 if (nfun == 5 || norm (q - q0, Inf) > tol)
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143 [q1, nfun, hmin] = simpsonstp (f, a, c, d, fa, fc, fd, q1, nfun, hmin,
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144 tol, trace, varargin{:});
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145 [q2, nfun, hmin] = simpsonstp (f, c, b, e, fc, fb, fe, q2, nfun, hmin,
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146 tol, trace, varargin{:});
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147 q = q1 + q2;
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148 endif
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149 endif
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150 endfunction
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151
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152
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153 %!assert (quadv (@sin, 0, 2 * pi), 0, 1e-5)
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154 %!assert (quadv (@sin, 0, pi), 2, 1e-5)
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155
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156 ## Handles weak singularities at the edge
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157 %!assert (quadv (@(x) 1 ./ sqrt (x), 0, 1), 2, 1e-5)
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158
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159 ## Handles vector-valued functions
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160 %!assert (quadv (@(x) [(sin (x)), (sin (2 * x))], 0, pi), [2, 0], 1e-5)
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161
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162 ## Handles matrix-valued functions
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163 %!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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164