Exam I will be on Tuesday, October 10, 1995 and will cover chapters 1, 2 and 6. Bring a blue book.
The problems given below are representative of the level of difficulty of the problems that will be given on the exam. The exam problem types may be different and the length of the actual exam will likely be shorter than the one given below.
1. Suppose p* approximates p with a relative error of 10^(-3).
Find the interval in which p* must lie if p is a) .01, b) 600
2. Write a short algorithm that uses the quadratic formula to
find both real roots to the equation ax^2+bx+c=0.
Construct your algorithm to minimize round-off error.
(Hint, separately consider the cases where b>0 and b<0).
3. Find the rates of convergence (i.e. O(h), O(h^2), etc.) of the
following expressions as h -> 0.
h^3
a) tan(h) -h b) (e -1)/h
4. Suppose the Taylor polynomial for sin(x) about the origin of degree n
is used to approximate sin(x) for |x| <= .1 (radians).
How large must n be to make the absolute value of the error
less than 10^(-7)? What is the largest relative error of this
approximation in this interval?
5. Suppose f(x) has a root at x=p and suppose an algorithm, such as
bisection or Newton produces a sequence p(n) which
converges to p as n -> infinity. Suppose that f'(p) =3,
then which of the following stopping criteria will likely
produce an approximate root that is closest to p?
a) |p(n)-p(n-1)| < tol b) |f(p(n)) - f(p(n-1))| < tol
6. Suppose f(x) has a root in the interval [1,3]. How many iterations
of bisection would be required to obtain an approximation
to the root accurate to within 10^(-3)?
7. Suppose f(x) has a fixed point on the interval [0,1] and that
|f'| < 1/3 on this interval. Starting with a point x_0 in
the interval [0,1], how many iterations of the fixed point
algorithm would be required to obtain an approximation to the
fixed point accurate to within 0.01?
8. State Newton's algorithm for finding the root of a function
f(x). Newton's algorithm is equivalent to the fixed point algorithm
finding a fixed point, x=g(x), for some function g(x).
In terms of f(x), find the function g(x).
9. The function f(x)=x^5+x-1 has a root between x=0 and x=1.
Describe how you would use the fixed point algorithm
to find its root.
10. a) In general, if g(x)=x has a solution
on an interval where |g'| < 1, then how fast will the
sequence of approximate roots generated by the fixed
point algorithm converge to the fixed point of g(x)?
(linearly or quadratically).
b) Answer the same question as in part a) but assuming that
g'(p)=0 where p is the fixed point.
11. How fast (linearly or quadratically) do the following sequences
converge to zero?
-(2^n) -n -n!
a) 3 b) 3 c) 3
12. The function f(x)=x^3+x-1 has a root in the interval [0,1].
With x_0=2, x_1=1, x_2=0, set-up, but do not solve, the
necessary equations for finding the next approximation, x_3,
to the root of f(x) by using Mueller's method (i.e. using parabolas).
13. Derive the formula for the secant algorithm (i.e given x_0, x_1,
explain the formula for x_2 in the secant algorithm).
14. True/false
a) If A has an inverse, then the columns of A are linearly independent.
b) det(A+B)=det(A) + det(B)
c) If det(A)=0, then the matrix equation A(X)=B never
has a solution.
15. Be prepared to solve a 3X3 matrix equation with partial pivoting.