MA 5310, Spring 2013 (Take Home) Exam 3
Due by noon on May 10, 2013
Advice:
1. Read the whole exam start to finish right now. Ask questions now, not later. Only then
start solving the exam.
2. You are not to communicate with anyone besides Derrick or me on any issue regarding
this exam.
3. Give details so that I can give you partial credit if your solution does not merit full credit.
4. Write clearly and large enough so I can read what you write. If I cannot read your
writing, you get no points.
5. Matlab and C++ are allowed. The textbook, similar books, and the class notes are also
allowed. Use the 2013sw.pdf version of the class notes that is available through the notes
web page for this class.
6. Acing this exam implies passing the math qualifier in computational math easily.
Problems (each one is worth 25 points):
1. Consider the polynomial f (x) = x 2 − 2x + 1 , which has a root at x = 1 .
a. Why are the Bisection and False Position methods not applicable?
b. Find a simplified form for xk+1 in Newton’s method.
c. What is the order of convergence for Newton’s method? Prove it for this specific
function.
2. Show that the backward time, central difference finite difference scheme,
n+1
uin+1 − uin
u n+1 − ui−1
+ a i+1
=0,
Δt
2Δx
is consistent with the partial differential equation
ut + au x = 0
and is unconditionally stable, where uin ≈ u(t n , xi ), t n = nΔt, and xi = iΔx . State
conditions required for a and u that are necessary in your proof.
3. Consider the function f (x1 , x2 ) = x12 x2 + x1e 4 x2 . Mimic what is on slides 184-185 of the
class notes.
a. Construct a table of original code statements and added automatic differentiation
statements for forward accumulation. Provide solutions for seeding each of x1
and x2 (separately).
b. Produce an acyclic directed graph.
4. We are given an order N real Toeplitz matrix M = [A, B,C] , i.e., M is a tridiagonal
matrix with real constants A, B, and C. We can write part of the forward substitution
algorithm for directly solving Mx = b , where x, and b are real N-vectors, as follows:
c0 = C / B and ci = C / (B − ci−1 ), i = 1,2,, N .
Suppose M is diagonally dominant such that A < 0, C < 0 , and B > − ( A + C ) .
a. What is the limit of ci as i, N → ∞ ?
b. If the ci ’s are computed on a computer in floating point with d digits of
precision, how many iterations are needed before ci − ci−1 = 0 ?