Question

1. Approximate the derivative of each of the following functions using the forward, backward, and centered differ- ence formulas on the grid linspace (-5,5,100) (x+h)-f(z thforward, (r)-fr-h ckward th)-fle-h centered. For each part, make a single plot (with three curves) showing the absolute error at each grid point. (Note that the approximations are undefined at one or both endpoints.) Also state which approximations are exact (within roundoff error) (b) f:x→z? (d) f:Hsin(x) 2. Use the centered difference formula to approximate f,(z) for f-sin and z-1, using h-1, 1/10, 1/100. . . . , 1/1015·Plot the absolute error. Explain the behavior as h decreases. (Hint: Read the last subsection of §4.1 concerning roundoff error.) Why does this instability not arise with 0?

Answer 1

SOLUTION

1)

MATLAB CODE :

clc
clear all
x=linspace(-5,5,100);
h=1e-4;
fd1=[];
bd1=[];
cd1=[];
fd2=[];
bd2=[];
cd2=[];
fd3=[];
bd3=[];
cd3=[];
fd4=[];
bd4=[];
cd4=[];
f1=@(x) x;
f2=@(x) x.^2;
f3=@(x) x.^3;
f4=@(x) sin(x);
for i=1:length(x)-1
fd1(i)=(f1(x(i)+h)-f1(x(i)))/h;
fd2(i)=(f2(x(i)+h)-f2(x(i)))/h;
fd3(i)=(f3(x(i)+h)-f3(x(i)))/h;
fd4(i)=(f4(x(i)+h)-f4(x(i)))/h;

end
for i=2:length(x)
bd1(i-1)=(f1(x(i))-f1(x(i)-h))/h;
bd2(i-1)=(f2(x(i))-f2(x(i)-h))/h;
bd3(i-1)=(f3(x(i))-f3(x(i)-h))/h;
bd4(i-1)=(f4(x(i))-f4(x(i)-h))/h;

end
for i=2:length(x)-1
cd1(i-1)=(f1(x(i)+h)-f1(x(i)-h))/(2h);
cd2(i-1)=(f2(x(i)+h)-f2(x(i)-h))/(2h);
cd3(i-1)=(f3(x(i)+h)-f3(x(i)-h))/(2h);
cd4(i-1)=(f4(x(i)+h)-f4(x(i)-h))/(2h);

end
plot(x(1:end-1),fd1,x(2:end),bd1,x(2:end-1),cd1);
title('Plot of finite difference for part a');
legend('Forward difference','Backward difference','Central difference')
figure;
plot(x(1:end-1),fd2,x(2:end),bd2,x(2:end-1),cd2);
title('Plot of finite difference for part b');
legend('Forward difference','Backward difference','Central difference')
figure;
plot(x(1:end-1),fd3,x(2:end),bd3,x(2:end-1),cd3);
title('Plot of finite difference for part c');
legend('Forward difference','Backward difference','Central difference')

figure;
plot(x(1:end-1),fd4,x(2:end),bd4,x(2:end-1),cd4);
title('Plot of finite difference for partd');
legend('Forward difference','Backward difference','Central difference')

OUTPUT :

PLOTS ;

part A :

plot 1

plot 2

Plot of finite difference for part a 1.15 - 1.05 - 0.95 0.9 43 -4 -2

plot 3

part B:

Plot of finite difference for part a 1.15 - Forward difference backward difference 1.1 1.05 - 0.95 0.9 43 -4 -2

plot 1

plot 2

plot 3

part C

plot 1

plot 2

plot 3

Plot of finite difference for part c 80 70 10) 50 40 30 20 10 41 .4 -2

part D :

plot 1

plot 2

plot 3

Plot of finite difference for partd 0.5 05 -1 -6 -4

Plot of finite difference for partd Forward difference Backward difference andle 0.5 0.5 -1 -6 -4