Sample Problems
Math 464
The test (on Friday, October 31 (Halloween)) will be closed book. One notebook-size page of notes
will be allowed on the test. You should give any numerical answers in the form of a rational number or
a simple expression involving radicals. Scientific calculators in which no formulas or text has been stored
will be allowed.
1. In using Newton’s method to find a root of an equation f(x) = 0, with initial guess x0= 3, we
compute f(x0) = 1, and find that x1= 5. What is f0(x0)?
2. Let abe a fixed number and suppose that 0, < 1 is also a fixed number, Show that the equation
a=x−sin xhas a unique solution and it can be found by fixed point iteration of an appropriate
function with any starting value.
3. The following formula is used to compute a function f:
f(x) = √x−1−√x−2
Suggest a more accurate way to compute the same function.
4. Each of the following iterations may converge. If the iteration does converge it will be to a root of a
polynomial p(x). In all three cases the convergence will be to some root of the same polynomial.
xk+1 =x2
k−2xk+ 2 (1)
xk+1 =xk−x2
k−3xk+ 2
2xk−3(2)
xk+1 =−x2
k+ 4xk−2 (3)
a) What is p(x)? What are its roots?
b) One of the iterations will converge to either of the two roots of p(x) if the initial guess is
sufficiently close. Each of the others will converge to one root, but can’t possibly converge
to the other unless the initial guess is exactly equal to the root. Which is which? Give a
justification for your answer.
5. a) Find the P A =LU factorization of
122
4 8 16
−2−1−2
by using Gaussian elimination and pivoting. Lshould have 1’s on the main diagonal.
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