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comparison doc/interpreter/quad.txi @ 7984:bbaa5d7d0143

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Some documentation updates
author David Bateman <dbateman@free.fr>
date Mon, 28 Jul 2008 15:47:40 +0200
parents fd42779a8428
children 23c248d415b5
comparison
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7983:91d020444da7 7984:bbaa5d7d0143
48 Numerical integration based on Gaussian quadrature. 48 Numerical integration based on Gaussian quadrature.
49 49
50 @item quadl 50 @item quadl
51 Numerical integration using an adaptive Lobatto rule. 51 Numerical integration using an adaptive Lobatto rule.
52 52
53 @item quadgk
54 Numerical integration using an adaptive Guass-Konrod rule.
55
56 @item quadv
57 Numerical integration using an adaptive vectorized Simpson's rule.
58
53 @item trapz 59 @item trapz
54 Numerical integration using the trapezoidal method. 60 Numerical integration using the trapezoidal method.
55 @end table 61 @end table
56 62
57 @noindent 63 @noindent
115 Although @code{quad} returns a nonzero value for @var{ier}, the result 121 Although @code{quad} returns a nonzero value for @var{ier}, the result
116 is reasonably accurate (to see why, examine what happens to the result 122 is reasonably accurate (to see why, examine what happens to the result
117 if you move the lower bound to 0.1, then 0.01, then 0.001, etc.). 123 if you move the lower bound to 0.1, then 0.01, then 0.001, etc.).
118 124
119 @DOCSTRING(quadl) 125 @DOCSTRING(quadl)
126
127 @DOCSTRING(quadgk)
128
129 @DOCSTRING(quadv)
120 130
121 @DOCSTRING(trapz) 131 @DOCSTRING(trapz)
122 132
123 @DOCSTRING(cumtrapz) 133 @DOCSTRING(cumtrapz)
124 134
171 @end example 181 @end example
172 182
173 @node Functions of Multiple Variables 183 @node Functions of Multiple Variables
174 @section Functions of Multiple Variables 184 @section Functions of Multiple Variables
175 185
176 Octave does not have built-in functions for computing the integral 186 Octave does not have built-in functions for computing the integral of
177 of functions of multiple variables. It is however possible to compute 187 functions of multiple variables directly. It is however possible to
178 the integral of a function of multiple variables using the functions 188 compute the integral of a function of multiple variables using the
179 for one-dimensional integrals. 189 functions for one-dimensional integrals.
180 190
181 To illustrate how the integration can be performed, we will integrate 191 To illustrate how the integration can be performed, we will integrate
182 the function 192 the function
183 @iftex 193 @iftex
184 @tex 194 @tex
213 223
214 I = quadl("g", 0, 1) 224 I = quadl("g", 0, 1)
215 @result{} 0.30022 225 @result{} 0.30022
216 @end example 226 @end example
217 227
228 The above process can be simplified with the @code{dblquad} and
229 @code{triplequad} functions for integrals over two and three
230 variables. For example
231
232 @example
233 I = dblquad (@(x, y) sin(pi.*x.*y).*sqrt(x.*y), 0, 1, 0, 1)
234 @result{} 0.30022
235 @end example
236
237 @DOCSTRING(dblquad)
238
239 @DOCSTRING(triplequad)
240
218 The above mentioned approach works but is fairly slow, and that problem 241 The above mentioned approach works but is fairly slow, and that problem
219 increases exponentially with the dimensionality the problem. Another 242 increases exponentially with the dimensionality the problem. Another
220 possible solution is to use Orthogonal Collocation as described in the 243 possible solution is to use Orthogonal Collocation as described in the
221 previous section. The integral of a function @math{f(x,y)} for 244 previous section. The integral of a function @math{f(x,y)} for
222 @math{x} and @math{y} between 0 and 1 can be approximated using @math{n} 245 @math{x} and @math{y} between 0 and 1 can be approximated using @math{n}
248 of the approximation. If the integration needs to be performed between 271 of the approximation. If the integration needs to be performed between
249 @math{a} and @math{b} instead of 0 and 1, a change of variables is needed. 272 @math{a} and @math{b} instead of 0 and 1, a change of variables is needed.
250 273
251 274
252 275
253