#2A TEST (April 12, 2011)
PGE 310
Name:
Signature:
UT EID:
NOTES:
1.
You have 1 hour and 30 minutes. Points will be deducted for
tests turned in after 8:30
2. Write your name on all pages of the test
3. SHOW ALL WORK. Partial (or any credit) cannot be given
for problems without work shown
1. Qualitative Questions (20 points)
(a) Using the Taylor series (provided on your equation sheet), derive Newtons
method for solving for the root of a nonlinear equation. Show all work.
(b) Briefly explain how false position and secant method are different. Be
specific.
(c) Explain when I might choose to use LU decomposition instead of Gauss
Elimination. Give an example if it helps explain your point.
(d) Explain how Gauss Seidel and Jacobi methods for solving systems of
equations differ. Why is Gauss Seidel often much faster?
(d) Explain how I would derive equations to curve fit data to an equation of the
form y=a0 +a1*x + a2*x2 + a3*x3. You may use simple equations to explain
your answer, but you dont need to actually do any calculations, derivations,
or calculus.
2. (a) Use the first three iterations of the bisection method to estimate the root of
the following nonlinear equation. Use initial guesses of xL=0 and xU=1.0 and
calculate approximate the relative error. Show all work and fill in the table at
the bottom of the page (15 points)
x3 3x + 1 = 0
iteration
First
Second
Third
xr
App relative error
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3. (a) Use the first iteration of the Netwon-Raphson method and an initial guess of
x=1; y=0 to solve the following system of equations. SHOW ALL WORK (20
points)
12
1
x + xy =
2
2
14
1
x + 3 yx 3 =
2
2
(b) Is the solution after the first iteration better than the initial guess? Use
numbers/calculations to back up your answer. Your answer can be yes, no, or
inconclusive but you must EXPLAIN.
4. The relationship between viscosity, , and shear rate, , that describes a certain
surfactant used in enhanced oil recovery is given by the equation below. Given
the provided data, use curve fitting to estimate the unknown parameters k and n.
(20 points)
= k n 1
= [1 15 75 200]
= [1000 600 400 350]
5. Write a MATLAB function file called backsubstitution.m that solves for the x,
when sent an upper triangular matrix (U) and right-hand-side vector (b). Recall
that the system of equations Ux=b looks something like this:
u11 u12
u22
u13
u23
u33
... u1n x1 b1
... u2 n x2 b2
... u3n x3 = b3
O
M M
unn xn bn