Review Problems 1             Due 2/26/01

This problem consists of five examples of the types of questions that 
will be on the first hour exam.

Problem 1. Normal Equations:

	In this problem we are going to fit a parabola to the four data points given below:

Data:  (t,b):  (1,1); (2,4); (3,9); (4,15)

Model:  b = x(1) + x(2) * t + x(3) * t^2

a. Write down the least squares problem in the matrix form given in class, explicitly 
presenting the matrix A and the vector b.

b. Convert this problem to the corresponding normal equation problem (Hint: this 
new problem should be 3x3 - square).

c. (only if you have time left) solve this problem by gaussian elimination.

Problem 2. Orthogonal Transformations:

	a. A matrix P is orthogonal if P'P is the identity matrix.  Prove that orthogonal 
transformations preserve the 2-norm.

	b. How can this property of orthogonal transformations be useful in solving the 
linear least squares minimization problem min||Ax - b||?  A long answer is not required.

Problem 3.  Finite Differences:

The finite difference formula for a second derivative is given by:

f"(x) ~ [f(x - h) + f(x + h) - 2*f(x)] / h^2

a.  What is the error in this formula (include both algorithm error and machine error)?

b.  Given a machine precision of eps, determine the optimum value of h and the 
minimum possible error.

Problem 4. Gaussian Elimination:

Given the 2x2 matrix A shown below, compute the upper bound on the error of the 
solution x of Ax=b given that we have some error in the right hand side vector b.

A = [1, 2; 3, 7]

Hint: the formula for inv(A) is given by:

A = [a, b; c, d]     inv(A) = [d, -b; -c, a] / (ad - bc)

Problem 5. Statistics and Probability:

Suppose we have a set of N measurements [x(1), x(2), ... x(N)] for which we 
wish to calculate the variance.  Prove that an unbiased estimate of the population
variance is given by:

variance = (sum(x.^2) - sum(x)^2/N)/(N-1)

In particular, I want you to justify the N-1 factor.