AMath 584, Autumn Quarter, 2011 Applied Linear Algebra and Introductory Numerical Analysis

# Final take-home¶

Due to dropbox by 6pm PDT, December 14, 2011

## Corrections and clarifications¶

Please submit your Matlab/Python codes for each problem as separate m-files or .py files. If something does not look right, it is good to be able to check the codes. It would also be helpful to put the main interesting parts of the code in a verbatim environment in your discussion of results.

Problem 1(a)

The solve_nopivot should be written without using the backslash operator in Matlab, following the algorithms as described in the book/lecture. You can use vectorized operations in both routines to multiply vectors together or subtracting a multiple of one row from another, rather than writing out these loops.

Problem 1(a,b)

For these problems the form of the functions (input and output) are specified. Please conform to these: it should be possible to run the function on other data than what you tested it on.

Regarding checking inputs: in solve_nopivot you can assume the L and U matrices passed in have the same size and the proper form. In a real code it would be best to check these things of course!

Problem 1(b)

Compute and print relative errors instead of absolute.

Machine epsilon is 1.1e-16, not 1.1e-15 as stated in the problem.

Problem 2(a)

Change to: Show that if |\alpha| < 1/3 then ...

The sum in the displayed equation should be summed only over $j neq k$.

Also note that it should say “the diagonal term in each column is larger than the sum of all the off-diagonals”, which really means “the sum of the absolute values of the non-diagonal elements of the same column”

Problem 2(b)

Change to \alpha = 0.25.

Problem 3

For a circulant matrix, $r_0 = c_0$ and $r_i = c_{m-i}$.

Extra Credit (e)

The last formula should involve \bar g rather than \bar f on both sides of the equation.

Extra Credit (f)

The definition of f(s) should say “if s < / > 0.5” rather than involving x. You can also define $f(0.5) = 2$.

Please use latex to write up your solutions, by modifying this latex file and inserting your name, NetID, and solutions:

The figure files...

To run latex you will also need to have this file containing some useful macros that are used in the assignment and that you might also want to use:

See Homework 1 for more about latex.