Diferencia entre revisiones de «Trapezoidal rule to approximate integrals»
| Línea 7: | Línea 7: | ||
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: | 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> | + | :<math> \int_a^b f(u) \; du \sim h\frac12 f(u_0)+h\sum_{i=1}^{N-1}f(u_i)+h\frac12 f(u_N) </math> |
that can be written as | that can be written as | ||
| − | :<math> \int_a^b f(u) \; du \sim \sum_{i=0}^{N}w_if(u_i), </math> | + | :<math> \int_a^b f(u) \; du \sim h\sum_{i=0}^{N}w_if(u_i)=hw^t*f, </math> |
| − | where <math> w_i </math> are the components of the weight vector <math> w=(1/2,1,1,...,1,1,1/2) </math>. | + | where <math> w_i </math> are the components of the weight column vector <math> w=(1/2,1,1,...,1,1,1/2)^t </math> and <math> f </math> is the column vector <math> f=(f(u_0),f(u_1),...,f(u_N))^t </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> | '''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> | ||
Revisión del 00:55 26 nov 2014
In this article we approximate integrals by the trapezoidal rule.
- 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 h\frac12 f(u_0)+h\sum_{i=1}^{N-1}f(u_i)+h\frac12 f(u_N) [/math]
that can be written as
- [math] \int_a^b f(u) \; du \sim h\sum_{i=0}^{N}w_if(u_i)=hw^t*f, [/math]
where [math] w_i [/math] are the components of the weight column vector [math] w=(1/2,1,1,...,1,1,1/2)^t [/math] and [math] f [/math] is the column vector [math] f=(f(u_0),f(u_1),...,f(u_N))^t [/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
