### Math. 455-01, Spring, 2005 Assignments

This list is updated as the assignments are made.  Unless otherwise stated, the assignments are from Faires and Burden, third edition.  Assignments should be written up in a careful and organized way; computations, not just the answer or result, should be shown.
R. Baker Kearfott

For Thursday, January 20:
p. 13ff -- 1, 2, 4, 10;
p. 20ff -- 1a, 1c, 1e, 1h, 2a, 9, 10
For Thursday, January 27:
p. 27ff --  3, 4, 7, 10, 11
For Thursday, February 10, 2005:
1. Complete the program for the method of bisection, as discussed in class.
2. Use your program to find solutions within the intervals you considered in problems 1 and 2 (a,, b, c, and d) of page 13 of the text.
3. Organize and explain your results carefully.

4. Here are some "m" files we discussed in class, that may be useful as patterns for you to follow.
test2.m
xsqm2.m
my_function.m
bisection.m
For Tuesday, March 15, 2005:
1. Consider problem 3(c), page 75 of the text.
1. Use Lagrange polynomials (as requested in the text) to write down the interpolating polynomial and compute its value at 0.25.
2. Use the Vandermonde system to write down the interpolating poilynomial and compute its value at 0.25.
3. Use divided differences to write down the interpolating polynomial and compute its value at 0.25.
4. Simplify the representations algebraically to show that the three polynomials you get above represent the same function.
5. Do problem 7(c) of the text.
2. Consider f(x) = sin(x).
1. Generate 6 equally spaced points in [-1,1] (that is, x1=-1, x2=-.6, x3=-.2, x4=.2, x5=.6, x6=1) and compute the coefficients of the interpolating polynomial of degree 5.  Compute a bound on the error for x in [-0.01,0.01].
2. Do the same, except with six equally spaced points in [-0.1,0.1].
3. Do the same, except with six equally spaced points in [-0.01,0.01].
4. Compute the Taylor polynomial for f of degree 5 centered at x0=0.  Exhibit the error term, and compute a bound on the error for x in [-0.01,0.01].
5. Compare the results for the previous parts of this problem.  (That is, provide a written discussion.)
In the above, you may use any technology at your disposal, including Matlab, Mathematica or other symbolic manipulation program, a calculator, and pencil and paper.
The following m-files were presented in class on Tuesday, March 3, and may be useful:
setup_and_solve_Vandermonde.m
interp_poly_val.m
generate_points_for_Runges_function.m
(You may also find the Matlab functions Vander and polyval useful.)
For Thursday, March 24, 2005:
pp. 118-119 of the text: #12;
p. 143: #6;
pp. 148--149: #2
For Thursday, April 14, 2005:
p. 251, #2 and #7
p. 259, 5a, 6a, 7a, and 8a
p. 277, 1a and 2a
For Tuesday, April 25, 2005:
Apply Euler's method and the second order Taylor method to the system y'' + y = 0, y(0) = 1, y'(0) = 0, for t between 0 and 1, with 5, 10, 20, 40, and 80 subintervals.  Perform an empirical analysis of the order (by exhibiting the errors in a table and computing ratios), as was done in class.
Final assignment (Due Monday, May 2, 2005, in MDD 212, between 07:30 and 10:00):
p. 238, 1d, 2d, and 5d.