Learning Matlab: Plotting Error in f'(1) Using Loglog Scale & Subplot

[bobsmiters]

I am taking a Math/Cmpt class and I am useless with this program that we need to use Matlab.

Q: The derivative of a function f(x) can be approximated at the point x=x(sub)0 by a forward divided difference

f'(x(sub)0) ≈ [f(x(sub)0 + h) - f(x(sub)0)] / h

or by a central divided difference

f'(x(sub)0) ≈ [f(x(sub)0 + h) - f(x(sub)0)] / 2h

For the function f(x)= sin x, plot the error as a function of h when these divided differences are used to approximate f'(1) = cos1 for h = 10^-1, 10^-2, 10^-3, ..., 10^-16. Use loglog scale and display your results in one window using subplot.

As far and the loglog and subplot is concerned I understand what to do... it is the rest of the question that is causing a lot of difficulties. if there are any pointers on where to begin it would be much appreciated.

 

[mda]

You question is still a bit broad, but I'll bite.

Firstly I don't think the central difference is defined correctly... it should be
f'(x_0) = [f(x_0+h/2) - f(x_0-h/2)] / h

You're being asked to compute the convergence of the error of forward and central differentials, by comparing to the best answer you have available.
Fortunately your function has a well known analytic differential, otherwise you would have to compare to your best numerical estimate (you will we see how this works when you do the analytical comparison).

That aside, it seems that the main part of your question is how to compute the numerical differentials. I'm not going to tell you the answer, but you need to set up a vector for h (perhaps using logspace) and calculate the formulas as written. Just remember to use ./h not /h

 

[tacman]

A: Learning a new program, especially one as complex as Matlab, can be challenging. However, with some practice and understanding of the basic concepts, you will be able to use it effectively.

For this specific problem, the first step is to understand the concept of divided differences and how they can be used to approximate derivatives. The forward divided difference uses the function values at two points, while the central divided difference uses the function values at three points.

Next, you need to understand the given function f(x)= sin x and how to calculate its derivative at a specific point. In this case, we are interested in approximating f'(1) = cos1. You can use the formula provided in the question to calculate the approximate value of the derivative at x=1 for different values of h.

Now, for the plotting part, you need to use the loglog scale, which is a logarithmic scale on both the x and y axes. This is useful when dealing with very small or very large values, such as in this case where h ranges from 10^-1 to 10^-16. Using the loglog scale will help visualize the data better.

Lastly, the question asks for the results to be displayed in one window using subplot. Subplot is a function in Matlab that allows you to create multiple plots in one figure. You can use this to plot the error as a function of h for both the forward and central divided differences in one window.

To begin, you can start by writing a code to calculate the approximate values of f'(1) using both the forward and central divided differences for different values of h. Then, use the loglog and subplot functions to plot the error as a function of h for both methods. You can also add a title and labels to your plot to make it more clear and informative.

Remember, practice makes perfect when it comes to learning a new program. Don't be discouraged if it takes some time to get the hang of it, just keep practicing and seeking help when needed. Good luck!