Question

write a program to compute an approximate value for the derivative of a function using the finite difference formula f ′ (x) ≈ f(x + h) − f(x) h . (2) Test your program using the function sin(x) for x = 1. Determine the error by comparing with the built-in function cos(x). Plot the magnitude of the error as a function of h, for h = 1 2 , 1 4 , 1 8 , . . . You should use a log scale for h and the magnitude of the error. Is there a minimum value for the magnitude of the error? b) (5 points) Check if the corresponding value of h to the minimum value for the magnitude of the error can be approximately equal to √ eps ? Why?

Answer 1

Hey,

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clc;
close all
clear all
f = @(x) sin(x); % function handler for The function sin(x)
x = 1;% The point at which the function computing the derivative
k = 1:50;% number of h values tested
h = 2.^(-k);% creating a vector of h of various values as mentioned in problem
df = (f(x+h) - f(x))./h;% finding the derivative
err = abs(df - cos(x));% finding the error
loglog(h,err) % plotting the figure in log x and log y graph
grid on % grid on command
xlabel('h');% x axis lable
ylabel('error');% y axis label
for i=1:50
if(err(i) == min(err))
H = h(i); % h corresponding to minimum error
end
end
ApproxH = sqrt(abs(3*(4/3-1)-1))*abs(x); % h approx sqrt(exs mach)
fprintf(' h corresponding to minumum error %e rule of thum h = %e ',H,ApproxH);

OUTPUT

h corresponding to minumum error 3.725290e-09
rule of thum h = 1.490116e-08

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