Please submit to CANVAS a .zip le that includes the following Matlab functions:
int midpoint rule.m
int trapezoidal rule.m
int Simpson rule.m
test integration.m
Exercise 1 Consider the following integral of a function f(x) on a nite interval [a; b]
b
I(f) = f(x)dx: (1)
a
Write three Matlab/Octave functions implementing, respectively, the composite midpoint rule, the composite trapezoidal rule, and the composite Simpson rule to compute the numerical approxima-tion of I(f). Such functions should be of the form
function [I]=int
midpoint
rule(fun,a,b,n)
(composite midpoint rule)
function
[I]=int
trapezoidal
rule(fun,a,b,n)
(composite trapezoidal rule)
function
[I]=int
Simpson
rule(fun,a,b,n)
(composite Simpson rule)
Input:
fun: function handle representing f(x)
a,b: endpoints of the integration interval
n: number of evenly-spaced points in [a; b] (including endpoints)
xj = a + (j 1)h;
h =
b
a
;
j = 1; :::n:
n
1
Output:
I: numerical approximation of the integral (1).
Exercise 2 Use the functions you coded in Exercise 1 to compute the numerical approximation of the integral
1
1
3
2
x3
I = Z
cos
e
x
dx:
(2)
3
1 + x2
2
30
To this end, write a Matlab/Octave function
1
[em,et,es] = function test integration()
that returns the following items:
en, et, es: row vectors with components the absolute values of the integration errors
jIref Inj n = 2; 3; :::; 10000 (3)
obtained with the midpont (vector en), trapezoidal (vector et) and Simpson (vector es) rules. Here,
Iref = 1:6851344770476
is the reference value of the integral (2) while In is the numerical approximation obtained by using the composite midpoint, trapezoidal, and Simpson rules with n nodes.
The function test integration() should also return the plot of the integrand function ap-pearing in (2) in figure(1), and the plots of the errors en, et and es versus n in a log-log scale in figure(2) (one gure with three plots). (Hint: use the Matlab command loglog()).
2
