Forward difference and central difference help - expected rates of convergence

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Savannah Phillips
Savannah Phillips 2020-3-17
回答： Ayush Gupta 2020-6-4
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!
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Ayush Gupta
Ayush Gupta 2020-6-4
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

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