Numerical Analysis II Math 6070

## HW Guidelines

You are encouraged to discuss homework problems with each other and to learn from each other. Help received from others must be acknowledged by a note at the beginning of your assignment. Under no circumstances is it acceptable for you to copy each other's work. See the section on Academic Integrity .

You are strongly advised to do your homework assignments promptly. Problems will cover both numerical and theoretical aspects of the course. Solutions must be legible, organized and always include reasoning (whether the problem explicitly calls for it or not).

Numerical experimentation is an integral part of the course. Its goal is to reinforce and reflect theory, compare and contrast theory with practise, and make numerical and theoretical conjectures.

Here are some tips on what makes a good solution to a problem involving numerical computation. Not all of these may be applicable in very problem:

1. Your Matlab code should either be a script or an m-file, created using the built-in Matlab editor. See Chapter 2 of our text.
3. Describe the design of your experiment. If you are comparing Method A with Method B, consider comparing the performance (for example, in terms of efficiency and accuracy) of each of the methods against a standard benchmark. A benchmark could be the exact solution computed by analytical (rather than numerical) means, or it could be a different method (which could be a built-in Matlab function).
4. Describe the test data, and why you chose it. Do the same for any other design parameters.
5. Do NOT submit reams of output. This will almost certainly earn you negative credit.
6. Organize your output in a compact and readable fashion. Use format compact to suppress unnecessary line feeds. Calculate relative errors if appropriate; summarize results in a table.
7. When appropriate, plot your numerical data. Learn about Matlab's subplot command which allows you to display multiple plots in the same graphing window.
8. All output, numerical and graphical, must be annotated, clearly identifying the features of interest. Annotations by hand are okay, even preferable at times.
9. Discussion of your output should be alongside your output. Don't make the reader keep turning pages back and forth.
10. What do you observe? What does the output suggest? Can you explain your observations?
11. What conclusions can you draw from your experiments? Why? How does theory compare with practise? For example, how do the mathematically exact solutions compare with the numerical ones? Or, can you prove by analytical means the patterns you've observed or been led to conjecture by numerical means?
12. Does your experiment suggest directions for further exploration? New conjectures? Generalizations? Include any progress made on such investigations. You are encouraged to extend exercises in this manner, and carry out your own on-going investigations during the semester.

#### Some Matlab Tips

1. To save output to a file, you can use Matlab's diary command. To learn more, type help diary at the Matlab prompt.
2. Creating special matrices:
>> help ones
>> help zeros
>> help eye
>> help diag
Example: >> n = 5; a = 5*eye(n) - 2*diag(ones(n-1,1),1) + 3*diag(ones(n-1,1),-1);
3. Formats: >> help format
Useful formats are: format short, format long, format rat.
Always use format compact when printing to save paper. Don't confuse format compact with format short .
4. Entering a polynomial: >> help poly
Finding its roots: >> help roots
Finding its derivative: >> help polyder
Evaluating a polynomial: >> help polyval
Multiplying polynomials: >> help conv
Dividing polynomials: >> help deconv
5. Generating evenly spaced points for function evaluation: >> help linspace
6. Plotting: >> help plot
Multiple plots in the same window: >>help subplot