I have a project on rounding errors and truncation error

Project 1 (Rounding vs. Truncation Errors)

1. Apply the backward difference schemes to approximate the derivative of the function, f(x)=e^x , at x = 1. 2. To investigate the effect of spacing of discrete points, h, a series of values of h are adopted, that is, h = 1/2n (n = 1, 2, …, 100). 3. Calculate and list the absolute error, eabs, for each choice of h (or n). Plot log10(eabs) vs. log10(h) and explain what you observe. Also derive the order of accuracy by calculating the slope in the truncation-error dominated part.

How can I solve this Project? I need to be calculate by hand and use MATLAB.

[Zoltán Csáti]

This is a very simple task. You will experience that by reducing the step-size, the rounding error will grow, while the truncation error will be smaller and smaller. Show what you have done so far.

[Ming-Yi Sun]

I don't know what is the next step! It's my progress:

clc

format long

hn=[];

for n=1:100

    h=1/(2^n);

    hn=[hn;h];

end

disp(hn)

Answer 1

[Arkayan Laha]

x=1.2; %value at which error has to be determined

f=@(x) exp(x); %defining the function

true_derivative=exp(x);%exact derivative result

for i=1:20

h(i,1)=0+.01*i;

n_d(i,1)=(f(x+h(i,1))-f(x))/h(i,1);%numerical derivative

e(i,1)=n_d(i,1)-true_derivative; %error

err(i,1)=abs(e(i,1)); %absolute error

end

plot(h,err);