annotate scripts/general/quadv.m @ 14363:f3d52523cde1

Use Octave coding conventions in all m-file %!test blocks * wavread.m, acosd.m, acot.m, acotd.m, acoth.m, acsc.m, acscd.m, acsch.m, asec.m, asecd.m, asech.m, asind.m, atand.m, cosd.m, cot.m, cotd.m, coth.m, csc.m, cscd.m, csch.m, sec.m, secd.m, sech.m, sind.m, tand.m, accumarray.m, accumdim.m, bitcmp.m, bitget.m, bitset.m, blkdiag.m, cart2pol.m, cart2sph.m, celldisp.m, chop.m, circshift.m, colon.m, common_size.m, cplxpair.m, cumtrapz.m, curl.m, dblquad.m, deal.m, divergence.m, flipdim.m, fliplr.m, flipud.m, genvarname.m, gradient.m, idivide.m, int2str.m, interp1.m, interp1q.m, interp2.m, interp3.m, interpft.m, interpn.m, isa.m, isdir.m, isequal.m, isequalwithequalnans.m, issquare.m, logspace.m, nargchk.m, narginchk.m, nargoutchk.m, nextpow2.m, nthargout.m, num2str.m, pol2cart.m, polyarea.m, postpad.m, prepad.m, profile.m, profshow.m, quadgk.m, quadv.m, randi.m, rat.m, repmat.m, rot90.m, rotdim.m, shift.m, shiftdim.m, sph2cart.m, structfun.m, trapz.m, triplequad.m, convhull.m, dsearch.m, dsearchn.m, griddata3.m, griddatan.m, rectint.m, tsearchn.m, __makeinfo__.m, doc.m, get_first_help_sentence.m, help.m, type.m, unimplemented.m, which.m, imread.m, imwrite.m, dlmwrite.m, fileread.m, is_valid_file_id.m, strread.m, textread.m, textscan.m, commutation_matrix.m, cond.m, condest.m, cross.m, duplication_matrix.m, expm.m, housh.m, isdefinite.m, ishermitian.m, issymmetric.m, logm.m, normest.m, null.m, onenormest.m, orth.m, planerot.m, qzhess.m, rank.m, rref.m, trace.m, vech.m, ans.m, bincoeff.m, bug_report.m, bzip2.m, comma.m, compare_versions.m, computer.m, edit.m, fileparts.m, fullfile.m, getfield.m, gzip.m, info.m, inputname.m, isappdata.m, isdeployed.m, ismac.m, ispc.m, isunix.m, list_primes.m, ls.m, mexext.m, namelengthmax.m, news.m, orderfields.m, paren.m, recycle.m, rmappdata.m, semicolon.m, setappdata.m, setfield.m, substruct.m, symvar.m, ver.m, version.m, warning_ids.m, xor.m, fminbnd.m, fsolve.m, fzero.m, lsqnonneg.m, optimset.m, pqpnonneg.m, sqp.m, matlabroot.m, __gnuplot_drawnow__.m, __plt_get_axis_arg__.m, ancestor.m, cla.m, clf.m, close.m, colorbar.m, colstyle.m, comet3.m, contourc.m, figure.m, gca.m, gcbf.m, gcbo.m, gcf.m, ginput.m, graphics_toolkit.m, gtext.m, hggroup.m, hist.m, hold.m, isfigure.m, ishghandle.m, ishold.m, isocolors.m, isonormals.m, isosurface.m, isprop.m, legend.m, line.m, loglog.m, loglogerr.m, meshgrid.m, ndgrid.m, newplot.m, orient.m, patch.m, plot3.m, plotyy.m, __print_parse_opts__.m, quiver3.m, refreshdata.m, ribbon.m, semilogx.m, semilogxerr.m, semilogy.m, stem.m, stem3.m, subplot.m, title.m, uigetfile.m, view.m, whitebg.m, compan.m, conv.m, deconv.m, mkpp.m, mpoles.m, pchip.m, poly.m, polyaffine.m, polyder.m, polyfit.m, polygcd.m, polyint.m, polyout.m, polyval.m, polyvalm.m, ppder.m, ppint.m, ppjumps.m, ppval.m, residue.m, roots.m, spline.m, intersect.m, ismember.m, powerset.m, setdiff.m, setxor.m, union.m, unique.m, autoreg_matrix.m, bartlett.m, blackman.m, detrend.m, fftconv.m, fftfilt.m, fftshift.m, freqz.m, hamming.m, hanning.m, ifftshift.m, sinc.m, sinetone.m, sinewave.m, unwrap.m, bicg.m, bicgstab.m, gmres.m, gplot.m, nonzeros.m, pcg.m, pcr.m, spaugment.m, spconvert.m, spdiags.m, speye.m, spfun.m, spones.m, sprand.m, sprandsym.m, spstats.m, spy.m, svds.m, treelayout.m, bessel.m, beta.m, betaln.m, factor.m, factorial.m, isprime.m, lcm.m, legendre.m, nchoosek.m, nthroot.m, perms.m, pow2.m, primes.m, reallog.m, realpow.m, realsqrt.m, hadamard.m, hankel.m, hilb.m, invhilb.m, magic.m, rosser.m, vander.m, __finish__.m, center.m, cloglog.m, corr.m, cov.m, gls.m, histc.m, iqr.m, kendall.m, kurtosis.m, logit.m, mahalanobis.m, mean.m, meansq.m, median.m, mode.m, moment.m, ols.m, ppplot.m, prctile.m, probit.m, quantile.m, range.m, ranks.m, run_count.m, runlength.m, skewness.m, spearman.m, statistics.m, std.m, table.m, var.m, zscore.m, betacdf.m, betainv.m, betapdf.m, betarnd.m, binocdf.m, binoinv.m, binopdf.m, binornd.m, cauchy_cdf.m, cauchy_inv.m, cauchy_pdf.m, cauchy_rnd.m, chi2cdf.m, chi2inv.m, chi2pdf.m, chi2rnd.m, discrete_cdf.m, discrete_inv.m, discrete_pdf.m, discrete_rnd.m, empirical_cdf.m, empirical_inv.m, empirical_pdf.m, empirical_rnd.m, expcdf.m, expinv.m, exppdf.m, exprnd.m, fcdf.m, finv.m, fpdf.m, frnd.m, gamcdf.m, gaminv.m, gampdf.m, gamrnd.m, geocdf.m, geoinv.m, geopdf.m, geornd.m, hygecdf.m, hygeinv.m, hygepdf.m, hygernd.m, kolmogorov_smirnov_cdf.m, laplace_cdf.m, laplace_inv.m, laplace_pdf.m, laplace_rnd.m, logistic_cdf.m, logistic_inv.m, logistic_pdf.m, logistic_rnd.m, logncdf.m, logninv.m, lognpdf.m, lognrnd.m, nbincdf.m, nbininv.m, nbinpdf.m, nbinrnd.m, normcdf.m, norminv.m, normpdf.m, normrnd.m, poisscdf.m, poissinv.m, poisspdf.m, poissrnd.m, stdnormal_cdf.m, stdnormal_inv.m, stdnormal_pdf.m, stdnormal_rnd.m, tcdf.m, tinv.m, tpdf.m, trnd.m, unidcdf.m, unidinv.m, unidpdf.m, unidrnd.m, unifcdf.m, unifinv.m, unifpdf.m, unifrnd.m, wblcdf.m, wblinv.m, wblpdf.m, wblrnd.m, kolmogorov_smirnov_test.m, kruskal_wallis_test.m, base2dec.m, bin2dec.m, blanks.m, cstrcat.m, deblank.m, dec2base.m, dec2bin.m, dec2hex.m, findstr.m, hex2dec.m, index.m, isletter.m, mat2str.m, rindex.m, str2num.m, strcat.m, strjust.m, strmatch.m, strsplit.m, strtok.m, strtrim.m, strtrunc.m, substr.m, validatestring.m, demo.m, example.m, fail.m, speed.m, addtodate.m, asctime.m, clock.m, ctime.m, date.m, datenum.m, datetick.m, datevec.m, eomday.m, etime.m, is_leap_year.m, now.m: Use Octave coding conventions in all m-file %!test blocks
author Rik <octave@nomad.inbox5.com>
date Mon, 13 Feb 2012 07:29:44 -0800
parents 72c96de7a403
children 5d3a684236b0
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1 ## Copyright (C) 2008-2012 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 (isnan (q)(:) | isinf (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