octave-nkf: doc/interpreter/quad.txi annotate

annotate doc/interpreter/quad.txi @ 14138:72c96de7a403 stable

maint: update copyright notices for 2012

author	John W. Eaton <jwe@octave.org>
date	Mon, 02 Jan 2012 14:25:41 -0500
parents	16cca721117b
children	c3fd61c59e9c

rev	line source
14138 72c96de7a403 maint: update copyright notices for 2012 John W. Eaton <jwe@octave.org> parents: 12612 diff changeset	1 @c Copyright (C) 1996-2012 John W. Eaton
7018 fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	2 @c
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	3 @c This file is part of Octave.
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	4 @c
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	5 @c Octave is free software; you can redistribute it and/or modify it
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	6 @c under the terms of the GNU General Public License as published by the
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	7 @c Free Software Foundation; either version 3 of the License, or (at
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	8 @c your option) any later version.
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	9 @c
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	10 @c Octave is distributed in the hope that it will be useful, but WITHOUT
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	13 @c for more details.
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	14 @c
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	15 @c You should have received a copy of the GNU General Public License
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	16 @c along with Octave; see the file COPYING. If not, see
fd42779a8428 [project @ 2007-10-13 00:52:12 by jwe] jwe parents: 6939 diff changeset	17 @c <http://www.gnu.org/licenses/>.
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	18
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	19 @node Numerical Integration
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	20 @chapter Numerical Integration
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	21
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	22 Octave comes with several built-in functions for computing the integral
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	23 of a function numerically (termed quadrature). These functions all solve
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	24 1-dimensional integration problems.
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	25
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	26 @menu
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	27 * Functions of One Variable::
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	28 * Orthogonal Collocation::
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	29 * Functions of Multiple Variables::
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	30 @end menu
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	31
4167 aae05d51353c [project @ 2002-11-12 02:52:50 by jwe] jwe parents: 3368 diff changeset	32 @node Functions of One Variable
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	33 @section Functions of One Variable
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	34
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	35 Octave supports five different algorithms for computing the integral
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	36 @tex
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	37 $$
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	38 \int_a^b f(x) d x
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	39 $$
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	40 @end tex
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	41 of a function @math{f} over the interval from @math{a} to @math{b}.
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	42 These are
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	43
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	44 @table @code
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	45 @item quad
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	46 Numerical integration based on Gaussian quadrature.
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	47
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	48 @item quadv
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	49 Numerical integration using an adaptive vectorized Simpson's rule.
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	50
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	51 @item quadl
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	52 Numerical integration using an adaptive Lobatto rule.
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	53
7984 bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	54 @item quadgk
9067 8970b4b10e9f Cleanup documentation for quad.texi and diffeq.texi Rik <rdrider0-list@yahoo.com> parents: 8920 diff changeset	55 Numerical integration using an adaptive Gauss-Konrod rule.
7984 bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	56
11562 1811f4f8b3b5 Rename cquad to quadcc and add to documentation. Rik <octave@nomad.inbox5.com> parents: 11523 diff changeset	57 @item quadcc
1811f4f8b3b5 Rename cquad to quadcc and add to documentation. Rik <octave@nomad.inbox5.com> parents: 11523 diff changeset	58 Numerical integration using adaptive Clenshaw-Curtis rules.
1811f4f8b3b5 Rename cquad to quadcc and add to documentation. Rik <octave@nomad.inbox5.com> parents: 11523 diff changeset	59
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	60 @item trapz, cumtrapz
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	61 Numerical integration of data using the trapezoidal method.
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	62 @end table
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	63
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	64 @noindent
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	65 The best quadrature algorithm to use depends on the integrand. If you have
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	66 empirical data, rather than a function, the choice is @code{trapz} or
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	67 @code{cumtrapz}. If you are uncertain about the characteristics of the
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	68 integrand, @code{quadcc} will be the most robust as it can handle
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	69 discontinuities, singularities, oscillatory functions, and infinite intervals.
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	70 When the integrand is smooth @code{quadgk} may be the fastest of the
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	71 algorithms.
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	72
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	73 @multitable @columnfractions 0.05 0.15 .80
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	74 @headitem @tab Function @tab Characteristics
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	75 @item @tab quad @tab Low accuracy with nonsmooth integrands
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	76 @item @tab quadv @tab Medium accuracy with smooth integrands
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	77 @item @tab quadl @tab Medium accuracy with smooth integrands. Slower than quadgk.
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	78 @item @tab quadgk @tab Medium accuracy (@math{1e^{-6}}--@math{1e^{-9}}) with smooth integrands.
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	79 @item @tab @tab Handles oscillatory functions and infinite bounds
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	80 @item @tab quadcc @tab Low to High accuracy with nonsmooth/smooth integrands
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	81 @item @tab @tab Handles oscillatory functions, singularities, and infinite bounds
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	82 @end multitable
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	83
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	84
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	85 Here is an example of using @code{quad} to integrate the function
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	86 @tex
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	87 $$
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	88 f(x) = x \sin (1/x) \sqrt {\|1 - x\|}
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	89 $$
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	90 from $x = 0$ to $x = 3$.
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	91 @end tex
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	92 @ifnottex
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	93
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	94 @example
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	95 @var{f}(@var{x}) = @var{x} * sin (1/@var{x}) * sqrt (abs (1 - @var{x}))
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	96 @end example
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	97
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	98 @noindent
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	99 from @var{x} = 0 to @var{x} = 3.
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	100 @end ifnottex
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	101
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	102 This is a fairly difficult integration (plot the function over the range
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	103 of integration to see why).
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	104
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	105 The first step is to define the function:
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	106
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	107 @example
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	108 @group
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	109 function y = f (x)
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	110 y = x .* sin (1./x) .* sqrt (abs (1 - x));
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	111 endfunction
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	112 @end group
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	113 @end example
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	114
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	115 Note the use of the `dot' forms of the operators. This is not necessary for
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	116 the @code{quad} integrator, but is required by the other integrators. In any
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	117 case, it makes it much easier to generate a set of points for plotting because
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	118 it is possible to call the function with a vector argument to produce a vector
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	119 result.
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	120
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	121 The second step is to call quad with the limits of integration:
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	122
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	123 @example
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	124 @group
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	125 [q, ier, nfun, err] = quad ("f", 0, 3)
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	126 @result{} 1.9819
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	127 @result{} 1
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	128 @result{} 5061
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	129 @result{} 1.1522e-07
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	130 @end group
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	131 @end example
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	132
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	133 Although @code{quad} returns a nonzero value for @var{ier}, the result
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	134 is reasonably accurate (to see why, examine what happens to the result
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	135 if you move the lower bound to 0.1, then 0.01, then 0.001, etc.).
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	136
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	137 The function "f" can be the string name of a function, a function handle, or
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	138 an inline function. These options make it quite easy to do integration
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	139 without having to fully define a function in an m-file. For example:
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	140
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	141 @example
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	142 @group
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	143 # Verify integral (x^3) = x^4/4
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	144 f = inline ("x.^3");
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	145 quadgk (f, 0, 1)
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	146 @result{} 0.25000
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	147
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	148 # Verify gamma function = (n-1)! for n = 4
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	149 f = @@(x) x.^3 .* exp (-x);
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	150 quadcc (f, 0, Inf)
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	151 @result{} 6.0000
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	152 @end group
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	153 @end example
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	154
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	155 @DOCSTRING(quad)
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	156
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	157 @DOCSTRING(quad_options)
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	158
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	159 @DOCSTRING(quadv)
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	160
6502 6ab0a8767780 [project @ 2007-04-06 03:32:06 by jwe] jwe parents: 4167 diff changeset	161 @DOCSTRING(quadl)
6ab0a8767780 [project @ 2007-04-06 03:32:06 by jwe] jwe parents: 4167 diff changeset	162
7984 bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	163 @DOCSTRING(quadgk)
bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	164
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	165 @DOCSTRING(quadcc)
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	166
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	167 Sometimes one does not have the function, but only the raw (x, y) points from
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	168 which to perform an integration. This can occur when collecting data in an
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	169 experiment. The @code{trapz} function can integrate these values as shown in
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	170 the following example where "data" has been collected on the cosine function
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	171 over the range [0, pi/2).
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	172
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	173 @example
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	174 @group
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	175 x = 0:0.1:pi/2; # Uniformly spaced points
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	176 y = cos (x);
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	177 trapz (x, y)
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	178 @result{} 0.99666
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	179 @end group
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	180 @end example
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	181
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	182 The answer is reasonably close to the exact value of 1. Ordinary quadrature
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	183 is sensitive to the characteristics of the integrand. Empirical integration
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	184 depends not just on the integrand, but also on the particular points chosen to
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	185 represent the function. Repeating the example above with the sine function
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	186 over the range [0, pi/2) produces far inferior results.
7984 bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	187
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	188 @example
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	189 @group
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	190 x = 0:0.1:pi/2; # Uniformly spaced points
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	191 y = sin (x);
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	192 trapz (x, y)
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	193 @result{} 0.92849
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	194 @end group
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	195 @end example
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	196
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	197 However, a slightly different choice of data points can change the result
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	198 significantly. The same integration, with the same number of points, but
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	199 spaced differently produces a more accurate answer.
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	200
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	201 @example
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	202 @group
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	203 x = linspace (0, pi/2, 16); # Uniformly spaced, but including endpoint
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	204 y = sin (x);
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	205 trapz (x, y)
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	206 @result{} 0.99909
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	207 @end group
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	208 @end example
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	209
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	210 In general there may be no way of knowing the best distribution of points ahead
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	211 of time. Or the points may come from an experiment where there is no freedom to
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	212 select the best distribution. In any case, one must remain aware of this
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	213 issue when using @code{trapz}.
11562 1811f4f8b3b5 Rename cquad to quadcc and add to documentation. Rik <octave@nomad.inbox5.com> parents: 11523 diff changeset	214
6502 6ab0a8767780 [project @ 2007-04-06 03:32:06 by jwe] jwe parents: 4167 diff changeset	215 @DOCSTRING(trapz)
6ab0a8767780 [project @ 2007-04-06 03:32:06 by jwe] jwe parents: 4167 diff changeset	216
6ab0a8767780 [project @ 2007-04-06 03:32:06 by jwe] jwe parents: 4167 diff changeset	217 @DOCSTRING(cumtrapz)
6ab0a8767780 [project @ 2007-04-06 03:32:06 by jwe] jwe parents: 4167 diff changeset	218
4167 aae05d51353c [project @ 2002-11-12 02:52:50 by jwe] jwe parents: 3368 diff changeset	219 @node Orthogonal Collocation
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	220 @section Orthogonal Collocation
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	221
3368 a4cd1e9d9962 [project @ 1999-11-20 17:22:48 by jwe] jwe parents: 3294 diff changeset	222 @DOCSTRING(colloc)
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	223
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	224 Here is an example of using @code{colloc} to generate weight matrices
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	225 for solving the second order differential equation
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	226 @tex
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	227 $u^\prime - \alpha u^{\prime\prime} = 0$ with the boundary conditions
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	228 $u(0) = 0$ and $u(1) = 1$.
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	229 @end tex
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	230 @ifnottex
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	231 @var{u}' - @var{alpha} * @var{u}'' = 0 with the boundary conditions
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	232 @var{u}(0) = 0 and @var{u}(1) = 1.
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	233 @end ifnottex
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	234
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	235 First, we can generate the weight matrices for @var{n} points (including
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	236 the endpoints of the interval), and incorporate the boundary conditions
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	237 in the right hand side (for a specific value of
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	238 @tex
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	239 $\alpha$).
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	240 @end tex
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	241 @ifnottex
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	242 @var{alpha}).
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	243 @end ifnottex
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	244
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	245 @example
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	246 @group
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	247 n = 7;
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	248 alpha = 0.1;
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	249 [r, a, b] = colloc (n-2, "left", "right");
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	250 at = a(2:n-1,2:n-1);
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	251 bt = b(2:n-1,2:n-1);
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	252 rhs = alpha * b(2:n-1,n) - a(2:n-1,n);
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	253 @end group
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	254 @end example
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	255
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	256 Then the solution at the roots @var{r} is
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	257
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	258 @example
9067 8970b4b10e9f Cleanup documentation for quad.texi and diffeq.texi Rik <rdrider0-list@yahoo.com> parents: 8920 diff changeset	259 @group
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	260 u = [ 0; (at - alpha * bt) \ rhs; 1]
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	261 @result{} [ 0.00; 0.004; 0.01 0.00; 0.12; 0.62; 1.00 ]
9067 8970b4b10e9f Cleanup documentation for quad.texi and diffeq.texi Rik <rdrider0-list@yahoo.com> parents: 8920 diff changeset	262 @end group
3294 bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe] jwe parents: diff changeset	263 @end example
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	264
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	265 @node Functions of Multiple Variables
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	266 @section Functions of Multiple Variables
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	267
7984 bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	268 Octave does not have built-in functions for computing the integral of
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	269 functions of multiple variables directly. It is possible, however, to
7984 bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	270 compute the integral of a function of multiple variables using the
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	271 existing functions for one-dimensional integrals.
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	272
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	273 To illustrate how the integration can be performed, we will integrate
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	274 the function
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	275 @tex
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	276 $$
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	277 f(x, y) = \sin(\pi x y)\sqrt{x y}
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	278 $$
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	279 @end tex
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	280 @ifnottex
10828 322f43e0e170 Grammarcheck .txi documentation files. Rik <octave@nomad.inbox5.com> parents: 9245 diff changeset	281
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	282 @example
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	283 f(x, y) = sin(pixy)sqrt(xy)
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	284 @end example
10828 322f43e0e170 Grammarcheck .txi documentation files. Rik <octave@nomad.inbox5.com> parents: 9245 diff changeset	285
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	286 @end ifnottex
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	287 for @math{x} and @math{y} between 0 and 1.
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	288
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	289 The first approach creates a function that integrates @math{f} with
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	290 respect to @math{x}, and then integrates that function with respect to
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	291 @math{y}. Because @code{quad} is written in Fortran it cannot be called
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	292 recursively. This means that @code{quad} cannot integrate a function
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	293 that calls @code{quad}, and hence cannot be used to perform the double
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	294 integration. Any of the other integrators, however, can be used which is
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	295 what the following code demonstrates.
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	296
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	297 @example
9067 8970b4b10e9f Cleanup documentation for quad.texi and diffeq.texi Rik <rdrider0-list@yahoo.com> parents: 8920 diff changeset	298 @group
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	299 function q = g(y)
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	300 q = ones (size (y));
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	301 for i = 1:length (y)
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	302 f = @@(x) sin (pix.y(i)) .* sqrt (x.*y(i));
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	303 q(i) = quadgk (f, 0, 1);
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	304 endfor
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	305 endfunction
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	306
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	307 I = quadgk ("g", 0, 1)
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	308 @result{} 0.30022
9067 8970b4b10e9f Cleanup documentation for quad.texi and diffeq.texi Rik <rdrider0-list@yahoo.com> parents: 8920 diff changeset	309 @end group
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	310 @end example
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	311
7984 bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	312 The above process can be simplified with the @code{dblquad} and
bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	313 @code{triplequad} functions for integrals over two and three
10828 322f43e0e170 Grammarcheck .txi documentation files. Rik <octave@nomad.inbox5.com> parents: 9245 diff changeset	314 variables. For example:
7984 bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	315
bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	316 @example
9067 8970b4b10e9f Cleanup documentation for quad.texi and diffeq.texi Rik <rdrider0-list@yahoo.com> parents: 8920 diff changeset	317 @group
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	318 I = dblquad (@@(x, y) sin (pix.y) .* sqrt (x.*y), 0, 1, 0, 1)
7984 bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	319 @result{} 0.30022
9067 8970b4b10e9f Cleanup documentation for quad.texi and diffeq.texi Rik <rdrider0-list@yahoo.com> parents: 8920 diff changeset	320 @end group
7984 bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	321 @end example
bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	322
bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	323 @DOCSTRING(dblquad)
bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	324
bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	325 @DOCSTRING(triplequad)
bbaa5d7d0143 Some documentation updates David Bateman <dbateman@free.fr> parents: 7018 diff changeset	326
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	327 The above mentioned approach works, but is fairly slow, and that problem
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	328 increases exponentially with the dimensionality of the integral. Another
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	329 possible solution is to use Orthogonal Collocation as described in the
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	330 previous section (@pxref{Orthogonal Collocation}). The integral of a function
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	331 @math{f(x,y)} for @math{x} and @math{y} between 0 and 1 can be approximated
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	332 using @math{n} points by
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	333 @tex
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	334 $$
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	335 \int_0^1 \int_0^1 f(x,y) d x d y \approx \sum_{i=1}^n \sum_{j=1}^n q_i q_j f(r_i, r_j),
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	336 $$
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	337 @end tex
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	338 @ifnottex
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	339 the sum over @code{i=1:n} and @code{j=1:n} of @code{q(i)q(j)f(r(i),r(j))},
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	340 @end ifnottex
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	341 where @math{q} and @math{r} is as returned by @code{colloc(n)}. The
9067 8970b4b10e9f Cleanup documentation for quad.texi and diffeq.texi Rik <rdrider0-list@yahoo.com> parents: 8920 diff changeset	342 generalization to more than two variables is straight forward. The
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	343 following code computes the studied integral using @math{n=8} points.
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	344
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	345 @example
9067 8970b4b10e9f Cleanup documentation for quad.texi and diffeq.texi Rik <rdrider0-list@yahoo.com> parents: 8920 diff changeset	346 @group
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	347 f = @@(x,y) sin (pixy') .* sqrt (x*y');
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	348 n = 8;
16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	349 [t, ~, ~, q] = colloc (n);
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	350 I = q'f(t,t)q;
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	351 @result{} 0.30022
9067 8970b4b10e9f Cleanup documentation for quad.texi and diffeq.texi Rik <rdrider0-list@yahoo.com> parents: 8920 diff changeset	352 @end group
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	353 @end example
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	354
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	355 @noindent
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	356 It should be noted that the number of points determines the quality
00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	357 of the approximation. If the integration needs to be performed between
12612 16cca721117b doc: Update all documentation for chapter on Numerical Integration Rik <octave@nomad.inbox5.com> parents: 11562 diff changeset	358 @math{a} and @math{b}, instead of 0 and 1, then a change of variables is needed.
6741 00116015904d [project @ 2007-06-18 16:07:14 by jwe] jwe parents: 6502 diff changeset	359

Mercurial > hg > octave-nkf

annotate doc/interpreter/quad.txi @ 14138:72c96de7a403 stable