Mercurial > hg > octave-lyh
annotate doc/interpreter/quad.txi @ 17289:bc924baa2c4e
doc: Add new @qcode macro for code samples which are quoted.
Macro handles options ("on") or properties ("position") more elegantly
than @code{"text"}.
* doc/interpreter/macros.texi: Add new @qcode macro.
* doc/interpreter/tips.txi: Add documentation about @qcode macro.
* doc/interpreter/basics.txi, doc/interpreter/container.txi,
doc/interpreter/emacs.txi, doc/interpreter/errors.txi,
doc/interpreter/eval.txi, doc/interpreter/expr.txi,
doc/interpreter/external.txi, doc/interpreter/func.txi,
doc/interpreter/grammar.txi, doc/interpreter/image.txi,
doc/interpreter/install.txi, doc/interpreter/interp.txi,
doc/interpreter/io.txi, doc/interpreter/matrix.txi,
doc/interpreter/numbers.txi, doc/interpreter/oop.txi,
doc/interpreter/package.txi, doc/interpreter/plot.txi,
doc/interpreter/quad.txi, doc/interpreter/sparse.txi,
doc/interpreter/strings.txi, doc/interpreter/system.txi,
doc/interpreter/vectorize.txi, libinterp/corefcn/balance.cc,
libinterp/corefcn/bitfcns.cc, libinterp/corefcn/cellfun.cc,
libinterp/corefcn/conv2.cc, libinterp/corefcn/data.cc,
libinterp/corefcn/debug.cc, libinterp/corefcn/defaults.cc,
libinterp/corefcn/dirfns.cc, libinterp/corefcn/dlmread.cc,
libinterp/corefcn/error.cc, libinterp/corefcn/file-io.cc,
libinterp/corefcn/find.cc, libinterp/corefcn/gammainc.cc,
libinterp/corefcn/graphics.cc, libinterp/corefcn/help.cc,
libinterp/corefcn/hex2num.cc, libinterp/corefcn/input.cc,
libinterp/corefcn/load-path.cc, libinterp/corefcn/load-save.cc,
libinterp/corefcn/ls-oct-ascii.cc, libinterp/corefcn/lu.cc,
libinterp/corefcn/luinc.cc, libinterp/corefcn/matrix_type.cc,
libinterp/corefcn/oct-hist.cc, libinterp/corefcn/pager.cc,
libinterp/corefcn/pr-output.cc, libinterp/corefcn/pt-jit.cc,
libinterp/corefcn/qz.cc, libinterp/corefcn/rand.cc,
libinterp/corefcn/regexp.cc, libinterp/corefcn/schur.cc,
libinterp/corefcn/sighandlers.cc, libinterp/corefcn/sparse.cc,
libinterp/corefcn/spparms.cc, libinterp/corefcn/str2double.cc,
libinterp/corefcn/svd.cc, libinterp/corefcn/symtab.cc,
libinterp/corefcn/syscalls.cc, libinterp/corefcn/toplev.cc,
libinterp/corefcn/tril.cc, libinterp/corefcn/typecast.cc,
libinterp/corefcn/utils.cc, libinterp/corefcn/variables.cc,
libinterp/dldfcn/__init_fltk__.cc, libinterp/dldfcn/chol.cc,
libinterp/dldfcn/colamd.cc, libinterp/dldfcn/fftw.cc, libinterp/dldfcn/qr.cc,
libinterp/dldfcn/symbfact.cc, libinterp/octave-value/ov-base.cc,
libinterp/octave-value/ov-fcn-handle.cc,
libinterp/octave-value/ov-fcn-inline.cc, libinterp/octave-value/ov-java.cc,
libinterp/octave-value/ov-range.cc, libinterp/octave-value/ov-struct.cc,
libinterp/octave-value/ov-usr-fcn.cc, libinterp/parse-tree/oct-parse.in.yy,
libinterp/parse-tree/pt-binop.cc, libinterp/parse-tree/pt-eval.cc,
libinterp/parse-tree/pt-mat.cc, scripts/@ftp/ftp.m,
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scripts/optimization/pqpnonneg.m, scripts/pkg/pkg.m, scripts/plot/allchild.m,
scripts/plot/ancestor.m, scripts/plot/area.m, scripts/plot/axis.m,
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scripts/plot/caxis.m, scripts/plot/cla.m, scripts/plot/clabel.m,
scripts/plot/clf.m, scripts/plot/close.m, scripts/plot/colorbar.m,
scripts/plot/daspect.m, scripts/plot/ezmesh.m, scripts/plot/ezmeshc.m,
scripts/plot/ezsurf.m, scripts/plot/ezsurfc.m, scripts/plot/findall.m,
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scripts/plot/gco.m, scripts/plot/grid.m, scripts/plot/guihandles.m,
scripts/plot/hdl2struct.m, scripts/plot/hidden.m, scripts/plot/hold.m,
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scripts/plot/newplot.m, scripts/plot/orient.m, scripts/plot/pareto.m,
scripts/plot/patch.m, scripts/plot/pbaspect.m, scripts/plot/pcolor.m,
scripts/plot/plot.m, scripts/plot/print.m,
scripts/plot/private/__add_default_menu__.m, scripts/plot/quiver.m,
scripts/plot/quiver3.m, scripts/plot/refreshdata.m, scripts/plot/saveas.m,
scripts/plot/scatter.m, scripts/plot/scatter3.m, scripts/plot/shading.m,
scripts/plot/shrinkfaces.m, scripts/plot/slice.m, scripts/plot/stem.m,
scripts/plot/stem3.m, scripts/plot/struct2hdl.m, scripts/plot/subplot.m,
scripts/plot/surf.m, scripts/plot/surfc.m, scripts/plot/surfl.m,
scripts/plot/tetramesh.m, scripts/plot/uigetfile.m, scripts/plot/uimenu.m,
scripts/plot/uiputfile.m, scripts/plot/waterfall.m, scripts/plot/whitebg.m,
scripts/plot/xlim.m, scripts/plot/ylim.m, scripts/plot/zlim.m,
scripts/polynomial/conv.m, scripts/polynomial/polyout.m,
scripts/polynomial/splinefit.m, scripts/set/ismember.m, scripts/set/powerset.m,
scripts/set/setdiff.m, scripts/set/union.m, scripts/set/unique.m,
scripts/signal/detrend.m, scripts/signal/filter2.m, scripts/signal/freqz.m,
scripts/signal/periodogram.m, scripts/signal/spectral_adf.m,
scripts/signal/spectral_xdf.m, scripts/sparse/eigs.m, scripts/sparse/svds.m,
scripts/specfun/legendre.m, scripts/special-matrix/gallery.m,
scripts/statistics/base/mean.m, scripts/statistics/base/moment.m,
scripts/statistics/tests/cor_test.m,
scripts/statistics/tests/kolmogorov_smirnov_test.m,
scripts/statistics/tests/kolmogorov_smirnov_test_2.m,
scripts/statistics/tests/kruskal_wallis_test.m,
scripts/statistics/tests/prop_test_2.m, scripts/statistics/tests/sign_test.m,
scripts/statistics/tests/t_test.m, scripts/statistics/tests/t_test_2.m,
scripts/statistics/tests/t_test_regression.m,
scripts/statistics/tests/u_test.m, scripts/statistics/tests/var_test.m,
scripts/statistics/tests/welch_test.m,
scripts/statistics/tests/wilcoxon_test.m, scripts/statistics/tests/z_test.m,
scripts/statistics/tests/z_test_2.m, scripts/strings/base2dec.m,
scripts/strings/index.m, scripts/strings/isstrprop.m,
scripts/strings/mat2str.m, scripts/strings/regexptranslate.m,
scripts/strings/rindex.m, scripts/strings/str2num.m, scripts/strings/strcat.m,
scripts/strings/strjust.m, scripts/strings/strmatch.m,
scripts/strings/validatestring.m, scripts/testfun/demo.m,
scripts/testfun/example.m, scripts/testfun/test.m, scripts/time/addtodate.m,
scripts/time/asctime.m, scripts/time/datestr.m, scripts/time/datetick.m,
scripts/time/weekday.m, scripts/ui/errordlg.m, scripts/ui/helpdlg.m,
scripts/ui/inputdlg.m, scripts/ui/listdlg.m, scripts/ui/msgbox.m,
scripts/ui/questdlg.m, scripts/ui/warndlg.m: Use new @qcode macro.
| author | Rik <rik@octave.org> |
|---|---|
| date | Mon, 19 Aug 2013 20:46:38 -0700 |
| parents | f2a8592b8fbd |
| children |
| rev | line source |
|---|---|
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1 @c Copyright (C) 1996-2012 John W. Eaton |
| 7018 | 2 @c |
| 3 @c This file is part of Octave. | |
| 4 @c | |
| 5 @c Octave is free software; you can redistribute it and/or modify it | |
| 6 @c under the terms of the GNU General Public License as published by the | |
| 7 @c Free Software Foundation; either version 3 of the License, or (at | |
| 8 @c your option) any later version. | |
| 9 @c | |
| 10 @c Octave is distributed in the hope that it will be useful, but WITHOUT | |
| 11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
| 12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
| 13 @c for more details. | |
| 14 @c | |
| 15 @c You should have received a copy of the GNU General Public License | |
| 16 @c along with Octave; see the file COPYING. If not, see | |
| 17 @c <http://www.gnu.org/licenses/>. | |
| 3294 | 18 |
| 6741 | 19 @node Numerical Integration |
| 20 @chapter Numerical Integration | |
| 21 | |
| 22 Octave comes with several built-in functions for computing the integral | |
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23 of a function numerically (termed quadrature). These functions all solve |
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24 1-dimensional integration problems. |
| 3294 | 25 |
| 26 @menu | |
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27 * Functions of One Variable:: |
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28 * Orthogonal Collocation:: |
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29 * Functions of Multiple Variables:: |
| 3294 | 30 @end menu |
| 31 | |
| 4167 | 32 @node Functions of One Variable |
| 3294 | 33 @section Functions of One Variable |
| 34 | |
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35 Octave supports five different algorithms for computing the integral |
| 6741 | 36 @tex |
| 37 $$ | |
| 38 \int_a^b f(x) d x | |
| 39 $$ | |
| 40 @end tex | |
| 41 of a function @math{f} over the interval from @math{a} to @math{b}. | |
| 42 These are | |
| 43 | |
| 44 @table @code | |
| 45 @item quad | |
| 46 Numerical integration based on Gaussian quadrature. | |
| 47 | |
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48 @item quadv |
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49 Numerical integration using an adaptive vectorized Simpson's rule. |
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50 |
| 6741 | 51 @item quadl |
| 52 Numerical integration using an adaptive Lobatto rule. | |
| 53 | |
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54 @item quadgk |
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55 Numerical integration using an adaptive Gauss-Konrod rule. |
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56 |
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57 @item quadcc |
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58 Numerical integration using adaptive Clenshaw-Curtis rules. |
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59 |
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60 @item trapz, cumtrapz |
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61 Numerical integration of data using the trapezoidal method. |
| 6741 | 62 @end table |
| 63 | |
| 64 @noindent | |
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65 The best quadrature algorithm to use depends on the integrand. If you have |
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66 empirical data, rather than a function, the choice is @code{trapz} or |
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67 @code{cumtrapz}. If you are uncertain about the characteristics of the |
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68 integrand, @code{quadcc} will be the most robust as it can handle |
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69 discontinuities, singularities, oscillatory functions, and infinite intervals. |
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70 When the integrand is smooth @code{quadgk} may be the fastest of the |
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71 algorithms. |
| 6741 | 72 |
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73 @multitable @columnfractions 0.05 0.15 .80 |
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74 @headitem @tab Function @tab Characteristics |
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75 @item @tab quad @tab Low accuracy with nonsmooth integrands |
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76 @item @tab quadv @tab Medium accuracy with smooth integrands |
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77 @item @tab quadl @tab Medium accuracy with smooth integrands. Slower than quadgk. |
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78 @item @tab quadgk @tab Medium accuracy (@math{1e^{-6}}--@math{1e^{-9}}) with smooth integrands. |
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79 @item @tab @tab Handles oscillatory functions and infinite bounds |
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80 @item @tab quadcc @tab Low to High accuracy with nonsmooth/smooth integrands |
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81 @item @tab @tab Handles oscillatory functions, singularities, and infinite bounds |
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82 @end multitable |
| 3294 | 83 |
| 84 | |
| 85 Here is an example of using @code{quad} to integrate the function | |
| 86 @tex | |
| 87 $$ | |
| 88 f(x) = x \sin (1/x) \sqrt {|1 - x|} | |
| 89 $$ | |
| 90 from $x = 0$ to $x = 3$. | |
| 91 @end tex | |
| 6741 | 92 @ifnottex |
| 3294 | 93 |
| 94 @example | |
| 95 @var{f}(@var{x}) = @var{x} * sin (1/@var{x}) * sqrt (abs (1 - @var{x})) | |
| 96 @end example | |
| 97 | |
| 98 @noindent | |
| 99 from @var{x} = 0 to @var{x} = 3. | |
| 6741 | 100 @end ifnottex |
| 3294 | 101 |
| 102 This is a fairly difficult integration (plot the function over the range | |
| 103 of integration to see why). | |
| 104 | |
| 105 The first step is to define the function: | |
| 106 | |
| 107 @example | |
| 108 @group | |
| 109 function y = f (x) | |
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110 y = x .* sin (1./x) .* sqrt (abs (1 - x)); |
| 3294 | 111 endfunction |
| 112 @end group | |
| 113 @end example | |
| 114 | |
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115 Note the use of the `dot' forms of the operators. This is not necessary for |
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116 the @code{quad} integrator, but is required by the other integrators. In any |
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117 case, it makes it much easier to generate a set of points for plotting because |
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118 it is possible to call the function with a vector argument to produce a vector |
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119 result. |
| 3294 | 120 |
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121 The second step is to call quad with the limits of integration: |
| 3294 | 122 |
| 123 @example | |
| 124 @group | |
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125 [q, ier, nfun, err] = quad ("f", 0, 3) |
| 3294 | 126 @result{} 1.9819 |
| 127 @result{} 1 | |
| 128 @result{} 5061 | |
| 129 @result{} 1.1522e-07 | |
| 130 @end group | |
| 131 @end example | |
| 132 | |
| 133 Although @code{quad} returns a nonzero value for @var{ier}, the result | |
| 134 is reasonably accurate (to see why, examine what happens to the result | |
| 135 if you move the lower bound to 0.1, then 0.01, then 0.001, etc.). | |
| 136 | |
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137 The function @qcode{"f"} can be the string name of a function, a function |
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138 handle, or an inline function. These options make it quite easy to do |
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139 integration without having to fully define a function in an m-file. For |
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140 example: |
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141 |
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142 @example |
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143 @group |
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144 # Verify integral (x^3) = x^4/4 |
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145 f = inline ("x.^3"); |
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146 quadgk (f, 0, 1) |
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147 @result{} 0.25000 |
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148 |
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149 # Verify gamma function = (n-1)! for n = 4 |
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150 f = @@(x) x.^3 .* exp (-x); |
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151 quadcc (f, 0, Inf) |
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152 @result{} 6.0000 |
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153 @end group |
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154 @end example |
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155 |
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156 @DOCSTRING(quad) |
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157 |
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158 @DOCSTRING(quad_options) |
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159 |
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160 @DOCSTRING(quadv) |
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161 |
| 6502 | 162 @DOCSTRING(quadl) |
| 163 | |
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164 @DOCSTRING(quadgk) |
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165 |
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166 @DOCSTRING(quadcc) |
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167 |
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168 Sometimes one does not have the function, but only the raw (x, y) points from |
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169 which to perform an integration. This can occur when collecting data in an |
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170 experiment. The @code{trapz} function can integrate these values as shown in |
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171 the following example where "data" has been collected on the cosine function |
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172 over the range [0, pi/2). |
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173 |
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174 @example |
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175 @group |
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176 x = 0:0.1:pi/2; # Uniformly spaced points |
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177 y = cos (x); |
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178 trapz (x, y) |
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179 @result{} 0.99666 |
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180 @end group |
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181 @end example |
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182 |
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183 The answer is reasonably close to the exact value of 1. Ordinary quadrature |
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184 is sensitive to the characteristics of the integrand. Empirical integration |
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185 depends not just on the integrand, but also on the particular points chosen to |
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186 represent the function. Repeating the example above with the sine function |
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187 over the range [0, pi/2) produces far inferior results. |
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188 |
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189 @example |
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190 @group |
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191 x = 0:0.1:pi/2; # Uniformly spaced points |
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192 y = sin (x); |
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193 trapz (x, y) |
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194 @result{} 0.92849 |
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195 @end group |
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196 @end example |
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197 |
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198 However, a slightly different choice of data points can change the result |
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199 significantly. The same integration, with the same number of points, but |
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200 spaced differently produces a more accurate answer. |
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201 |
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202 @example |
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203 @group |
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204 x = linspace (0, pi/2, 16); # Uniformly spaced, but including endpoint |
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205 y = sin (x); |
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206 trapz (x, y) |
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207 @result{} 0.99909 |
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208 @end group |
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209 @end example |
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210 |
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211 In general there may be no way of knowing the best distribution of points ahead |
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212 of time. Or the points may come from an experiment where there is no freedom to |
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213 select the best distribution. In any case, one must remain aware of this |
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214 issue when using @code{trapz}. |
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215 |
| 6502 | 216 @DOCSTRING(trapz) |
| 217 | |
| 218 @DOCSTRING(cumtrapz) | |
| 219 | |
| 4167 | 220 @node Orthogonal Collocation |
| 3294 | 221 @section Orthogonal Collocation |
| 222 | |
| 3368 | 223 @DOCSTRING(colloc) |
| 3294 | 224 |
| 225 Here is an example of using @code{colloc} to generate weight matrices | |
| 226 for solving the second order differential equation | |
| 227 @tex | |
| 228 $u^\prime - \alpha u^{\prime\prime} = 0$ with the boundary conditions | |
| 229 $u(0) = 0$ and $u(1) = 1$. | |
| 230 @end tex | |
| 6741 | 231 @ifnottex |
| 3294 | 232 @var{u}' - @var{alpha} * @var{u}'' = 0 with the boundary conditions |
| 233 @var{u}(0) = 0 and @var{u}(1) = 1. | |
| 6741 | 234 @end ifnottex |
| 3294 | 235 |
| 236 First, we can generate the weight matrices for @var{n} points (including | |
| 237 the endpoints of the interval), and incorporate the boundary conditions | |
| 238 in the right hand side (for a specific value of | |
| 239 @tex | |
| 240 $\alpha$). | |
| 241 @end tex | |
| 6741 | 242 @ifnottex |
| 3294 | 243 @var{alpha}). |
| 6741 | 244 @end ifnottex |
| 3294 | 245 |
| 246 @example | |
| 247 @group | |
| 248 n = 7; | |
| 249 alpha = 0.1; | |
| 250 [r, a, b] = colloc (n-2, "left", "right"); | |
| 251 at = a(2:n-1,2:n-1); | |
| 252 bt = b(2:n-1,2:n-1); | |
| 253 rhs = alpha * b(2:n-1,n) - a(2:n-1,n); | |
| 254 @end group | |
| 255 @end example | |
| 256 | |
| 257 Then the solution at the roots @var{r} is | |
| 258 | |
| 259 @example | |
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260 @group |
| 3294 | 261 u = [ 0; (at - alpha * bt) \ rhs; 1] |
| 262 @result{} [ 0.00; 0.004; 0.01 0.00; 0.12; 0.62; 1.00 ] | |
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263 @end group |
| 3294 | 264 @end example |
| 6741 | 265 |
| 266 @node Functions of Multiple Variables | |
| 267 @section Functions of Multiple Variables | |
| 268 | |
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269 Octave does not have built-in functions for computing the integral of |
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270 functions of multiple variables directly. It is possible, however, to |
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271 compute the integral of a function of multiple variables using the |
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272 existing functions for one-dimensional integrals. |
| 6741 | 273 |
| 274 To illustrate how the integration can be performed, we will integrate | |
| 275 the function | |
| 276 @tex | |
| 277 $$ | |
| 278 f(x, y) = \sin(\pi x y)\sqrt{x y} | |
| 279 $$ | |
| 280 @end tex | |
| 281 @ifnottex | |
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282 |
| 6741 | 283 @example |
| 284 f(x, y) = sin(pi*x*y)*sqrt(x*y) | |
| 285 @end example | |
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286 |
| 6741 | 287 @end ifnottex |
| 288 for @math{x} and @math{y} between 0 and 1. | |
| 289 | |
| 290 The first approach creates a function that integrates @math{f} with | |
| 291 respect to @math{x}, and then integrates that function with respect to | |
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292 @math{y}. Because @code{quad} is written in Fortran it cannot be called |
| 6741 | 293 recursively. This means that @code{quad} cannot integrate a function |
| 294 that calls @code{quad}, and hence cannot be used to perform the double | |
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295 integration. Any of the other integrators, however, can be used which is |
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296 what the following code demonstrates. |
| 6741 | 297 |
| 298 @example | |
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299 @group |
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300 function q = g(y) |
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301 q = ones (size (y)); |
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302 for i = 1:length (y) |
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303 f = @@(x) sin (pi*x.*y(i)) .* sqrt (x.*y(i)); |
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304 q(i) = quadgk (f, 0, 1); |
| 6741 | 305 endfor |
| 306 endfunction | |
| 307 | |
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308 I = quadgk ("g", 0, 1) |
| 6741 | 309 @result{} 0.30022 |
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310 @end group |
| 6741 | 311 @end example |
| 312 | |
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313 The above process can be simplified with the @code{dblquad} and |
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314 @code{triplequad} functions for integrals over two and three |
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315 variables. For example: |
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316 |
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317 @example |
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318 @group |
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319 I = dblquad (@@(x, y) sin (pi*x.*y) .* sqrt (x.*y), 0, 1, 0, 1) |
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320 @result{} 0.30022 |
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321 @end group |
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322 @end example |
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323 |
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324 @DOCSTRING(dblquad) |
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325 |
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326 @DOCSTRING(triplequad) |
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327 |
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328 The above mentioned approach works, but is fairly slow, and that problem |
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329 increases exponentially with the dimensionality of the integral. Another |
| 6741 | 330 possible solution is to use Orthogonal Collocation as described in the |
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331 previous section (@pxref{Orthogonal Collocation}). The integral of a function |
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332 @math{f(x,y)} for @math{x} and @math{y} between 0 and 1 can be approximated |
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333 using @math{n} points by |
| 6741 | 334 @tex |
| 335 $$ | |
| 336 \int_0^1 \int_0^1 f(x,y) d x d y \approx \sum_{i=1}^n \sum_{j=1}^n q_i q_j f(r_i, r_j), | |
| 337 $$ | |
| 338 @end tex | |
| 339 @ifnottex | |
| 340 the sum over @code{i=1:n} and @code{j=1:n} of @code{q(i)*q(j)*f(r(i),r(j))}, | |
| 341 @end ifnottex | |
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342 where @math{q} and @math{r} is as returned by @code{colloc (n)}. The |
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343 generalization to more than two variables is straight forward. The |
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344 following code computes the studied integral using @math{n=8} points. |
| 6741 | 345 |
| 346 @example | |
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347 @group |
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348 f = @@(x,y) sin (pi*x*y') .* sqrt (x*y'); |
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349 n = 8; |
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350 [t, ~, ~, q] = colloc (n); |
| 6741 | 351 I = q'*f(t,t)*q; |
| 352 @result{} 0.30022 | |
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353 @end group |
| 6741 | 354 @end example |
| 355 | |
| 356 @noindent | |
| 357 It should be noted that the number of points determines the quality | |
| 358 of the approximation. If the integration needs to be performed between | |
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359 @math{a} and @math{b}, instead of 0 and 1, then a change of variables is needed. |
| 6741 | 360 |