4. What is machine epsilon in the default Octave real variable precision. 5. What is machine epsilon in the Octave real variable single precision.

clc, clear, close all

% WRITING THE PROJECT DATE

printf("PROJECT 1 - Round-off vs Truncation Error ")

printf(" ")

display(date())

printf("======================================== ")

% f(x)= f

f=@(x) 12.5+3.5.*x.*cos(2.45.*x);

% df(x) is the first order derivative

df=@(x) -8.575*x*sin(2.45*x)+3.5*cos(2.45*x);

x = 2.75;

for i = 1:20

del (i,1)= 10^(-i);

%Backward finite difference

backward(i,1) = (f(x)-f(x - del(i,1)))/del(i,1);

%Forward finite difference

forward(i,1) = (f(x + del(i,1)) - f(x))/del(i,1);

%Central Finite Difference

central (i,1)= (f(x+del(i,1)) - f(x - del(i,1)))/(2*del(i,1));

%Calculate the errors compared to exact derivative

backward_error(i,1) = abs(backward(i,1) - df(x));

forward_error(i,1) = abs(forward(i,1) - df(x));

central_error(i,1) = abs(central(i,1) - df(x));

end

%plot the error vs del x

figure

hold on

plot(log10(del),log10(forward_error))

plot(log10(del),log10(backward_error))

plot(log10(del),log10(central_error))

xlabel('\Delta'), ylabel('Error')

title('Error on Forward, Backward, and Central')

legend('Forward','Backward','Central')

grid on, hold off

%assemble the results into a table (actually a matrix)

printf("===========Double precision results============== ")

printf(" del x forward backward central ")

Table = [del, forward_error, backward_error, central_error];

%print to screen

disp(Table)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%repeat everything in single precison

%select x value to evaluate derivative

x = single(x);

%evaluate the derivative

for i = single(1:20)

%Calculate delta x

del(i,1) = single(10^(-i));

%backward finite difference

backward(i,1) = single((f(x)-f(x - del(i,1)))/del(i,1));

%Forward finite difference

forward(i,1) = single((f(x + del(i,1)) - f(x))/del(i,1));

%Central Finite Difference

central (i,1)= single((f(x + del(i,1)) - f(x - del(i,1)))/(2*del(i,1)));

%Calculate the errors compared to exact derivative

backward_error(i,1) = single(abs(backward(i,1) - df(x)));

forward_error(i,1) = single(abs(forward(i,1) - df(x)));

central_error(i,1) = single(abs(central(i,1) - df(x)));

end

%Plot the error vs delta x for each method on the log scale

figure

hold on

plot(log10(del),log10(forward_error))

plot(log10(del),log10(backward_error))

plot(log10(del),log10(central_error))

xlabel('\Delta'), ylabel('Error')

title(' Error on Forward, Backward, and Central')

legend('Forward','Backward','Central')

grid on, hold off

printf("================================================= ")

%assemble the results into a table (actually a matrix)

printf("===========Single precision results============== ")

printf(" del x forward backward central ")

Table = [del, forward_error, backward_error, central_error];

%print to screen

disp(Table)