Question

Write a Matlab script-file probl1.m to execute the requested commands (as much as possible) in the exercises below. Increase N a number of times according to N = 4, 8, 16, 32, 64, 128, . . . (1) Determine for each N the (exact) error. (2) Determine for N ≥ 16 also the convergence ratio q(h/2).

This script should be based on a function-file trap.m (trapezoidal integration) as follows:

function [totarea] = trap(N)

format long;

a = 0; b = 1/2; h = (b-a)/N;

x = a:h:b; totarea = 0;

for i = 1:N

xl = x(i);

xr = x(i+1);

fxl = myfunct(xl);

fxr = myfunct(xr);

locarea = (h/2)*(fxl+fxr);

totarea = totarea + locarea;

end

end

You can refer to the integral as myfunct(). The interval is [0,1/2].

Answer 1

%Matlab code for Trapizoidal method
clear all
close all

%Matlab code for Trapizoidal integral for varing N
n=[4 8 16 32 64 128 256];
%exact integral for given myfunct(x) is 0.75;
for i=1:length(n)
     N=n(i);
    [totarea] = trap(N);
    err(i)=abs(totarea-0.75);
    fprintf('For N=%d integral using Trapizoidal is %f with error %e\n',N,totarea,err(i))
    if i>=3
        fprintf('\tConvergence ratio =%f\n',err(i)/err(i-1))
    end
end
  

%function for integrand
function f=myfunct(x)
    f=3*x^2+5*x;
end

%function for Trapizoidal integral
function [totarea] = trap(N)

    format long;

    a = 0; b = 1/2; h = (b-a)/N;

    x = a:h:b; totarea = 0;

    for i = 1:N

        xl = x(i);

        xr = x(i+1);

        fxl = myfunct(xl);

        fxr = myfunct(xr);

        locarea = (h/2)*(fxl+fxr);

        totarea = totarea + locarea;

    end

end

    %%%%%%%%%%%%%% End of code %%%%%%%%%%%%%%%