Forward difference and central difference help - expected rates of convergence

Hi, if anyone could help me with this question on my coursework I'd be really grateful! I've already had help on the first part, now it's just the second part:

Within your file , test your difference quotient formulas with h = 10^−i for i = 1, 2, . . . , 10. Plot the errors versus h in a figure with double-log axes. (This part I've already done)

Can you observe the expected rate of convergence (asymptotic behaviour as h → 0)? What happens for very small h? Hint: It may be helpful to include in your plot the graphs of functions r(h) = h^α with α the expected rates of convergence.

The mark scheme says that I need two error plots. Which is the second one? Is it r(h) = h^α? Do I need to plot the expected error? How would I go about doing that? Thank you in advance for your help!

Here is my code:

f = @(x) sin(x); % function handle

for i = 1:10

h(i) = 10.^(-i);

[fd(i),cd] = FDCD(f,0.9,h(i));

end

% f'(0) = 1 for this question

errors = 1 - fd; % Approximation error for forward difference quotient

% Setting the figure

figure

loglog(errors, h);

grid on;

% Formatting the plot

(just title, axes labels, etc)

I know that all this is right, but i have no idea what the second plot is spoken about in the mark sceme and how to go about making it, if it is r(h) = h^a. Thank you!

[Sriram Tadavarty]

What is 'a' here in r(h) = h^a?

[Savannah Phillips]

sorry, it's just alpha! I don't have the alpha key and I was being a bit lazy- my apologies :-)

[darova]

What is this

[Savannah Phillips]

that's just a function used to find the forward and central difference (it's all correct)

Answer 1

[Ayush Gupta]

Try using with this code for section 2

x=[-10:0.01:20]; 
dt = 0.00001; 
values_y = [-10:0.01:20]; 
y1 = [-10:0.01:20]; 
values_error = [-10:0.01:20]; 
y2 = [-10:0.01:20]; 

for i =1:length(x) 

    values_y(i) = 1/(10^(x(i))); 

    y1(i) = log(values_y(i)); 

    values_error(i) = (1/((10^(x(i)+dt)) - values_y(i)))/dt; 

    y2(i) = log(values_error(i)); 

end 
hold on 

plot(x,y1) 

plot(x,y2) 

hold off