637
Notes Math to the Student
Mackey
1. You should be thoroughly familiar with the following topics from undergraduate linear algebra: vectors, vector spaces, subspaces linear independence, basis, dimension matrices and linear transformations rowspace, columnspace(range), nullspace(kernel), left nullspace rank and invertibility of a matrix linear systems, LU decomposition of a matrix, and its relation to Gaussian elimination dot product of vectors, orthogonality eigenvalues, eigenvectors and similarity transformations Please review these topics if you are rusty. 2. Numerical experimentation is an integral part of the course. Its goal is to reinforce and reect 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. (a) Describe the design of your experiment. If you are comparing method A with method B, consider comparing the performance (for example, in terms of eciency 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 dierent method (which could be a built-in function). matlab (b) Describe the test data, and why you chose it. Do the same for any other design parameters. (c) 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. (d) Annotate your output (by hand is ok, and even preferable at times!). Comment your code. (e) Do not submit reams of numerical output. This will almost certainly earn you negative credit. When appropriate, plot your numerical output. Learn about matlabs subplot command. (f) What do you observe? What does the output suggest? Can you explain your observations? (g) What conclusions can you draw from your experiments? Why? How does theory c...