Assignment #6
Due Monday 11/1 (CHANGE!)
Note: You may always use Matlab
to do
computations on the homework. However, when explanations,
arguments, "show that ..." are called for, you need to include your
written answer in English. Please try to minimize the
amount of paper your solution uses. You can copy a few numbers by
hand, sometimes, instead of a long printout. Suggestion:
Print out programs and write in the
white space.
From text (Leader):
Exercise I. Follow the method of Example 2.4.1 to
decompose A=[1 -1 1; -1
5 1; 1 1 6] into
A=L LT by hand.
Exercise II. Use MATLAB's chol
to compute the Cholesky factors of A
above. Next, use MATLAB's eig
to factor A above as A=S D ST.
Check that ST
= S-1.
Exercise III. Show that any matrix S such that
ST = S-1
has orthogonal columns.
Problems 2.4: # 1
Now read the handout on "Tridiagonal and Banded Systems" and do the
following:
Exercise IV. Write the routine tri on pages 275-276 of the
handout as a MATLAB program. Test it on the system
Ax=b where A = [2 -1 0 0; -1
2 -1 0; 0 -1 2 -1; 0
0 -1 2] and b = [5 3 1
3]. (Make sure you know the correct solution to this
system!) Test it on a 50 variable tridiagonal system of your own
choice and compare times to MATLAB's "A\b".
Exercise V. Give and example of a tridiagonal system which cannot
be solved without pivoting.
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www.math.uaf.edu/~bueler/