Diferencia entre revisiones de «Trapezoidal rule to approximate integrals»

Revisión del 00:50 26 nov 2014

In this article we approximate integrals by the trapezoidal rule.

1. One dimensional integrals

Let [math] [a,b] [/math] be an interval and [math] f:[a,b]\to \mathbb{R} [/math] a real function. We want to approximate the integral [math] \int_a^bf(u) \; du [/math].

Consider a partition of the interval [math] [a,b] [/math] in [math] N [/math] equal subintervals of length [math] h=\frac{b-a}{N} [/math], given by [math] u_n=a+nh, [/math] where [math] n=0,1,...,N [/math]. The trapezoidal rule is as follows:

[math] \int_a^b f(u) \; du \sim \frac12 f(u_0)+\sum_{i=1}^{N-1}f(u_i)+\frac12 f(u_N) [/math]

that can be written as

[math] \int_a^b f(u) \; du \sim \sum_{i=0}^{N}w_if(u_i), [/math]

where [math] w_i [/math] are the components of the weight vector [math] w=(1/2,1,1,...,1,1,1/2) [/math].

Example: En Matlab/Octave vamos a aproximar la integral de [math] e^{-x^2} [/math] en [math] [-1,1] [/math] con [math] h=0.1 [/math]

N=200;        %Number of points
a=-1; b=1;    %Extremes of the interval
h=(b-a)/N;
u=a:h:b;      %coordinates of the partition
f=exp(-u.^2); %function
w=ones(1,N+1); %weights vector
w(1)=1/2; w(N+1)=1/2;
result=w*f'   % result