MATH 465 FINAL EXAM

8:30 AM, WEDNESDAY, MARCH 15, 1995

Name: One notebook sized page of notes (both sides) allowed. Calculators are recom-

mended. Show all your work

1. Find the Gaussian quadrature rule of precision 3 for ∫_{−1}^{1} w(x)f(x)dx, where w(x) = 1 + x^{2}. Leave any irrationals in simplified radical form (i.e. \frac{\sqrt{5}}{3}. etc.).

2 8:30 AM, WEDNESDAY, MARCH 15, 1995

2. Let < f, g >= ∑_{j=0}^{2}(1+j^{2})f(j)g(j). Find the monic orthogonal polynomials of degree 0, 1, 2 for this inner product.

3. Let N = 2n and let F_{k} denote the discrete Fourier transform from C^{k} to

C^{k}.

(a) Give formulas which allow you to compute F_{N} in terms of F_{n} (one step of the FFT).

(b) Let

y = [\begin{matrix}1 \\ 0 \\ 1 \\ 3 \end{matrix}] .

Compute F_{4}(y).

(c) Let the components of the vector y of part (b) be denoted by y_{0}, y_{1}, y_{2}, y_{3}. Find the interpolating function of the form p(x) = ∑_{k=0}^{3} a_{k} exp(ikx), such that p(\frac{jπ}{2} ) = y_{j}.

4 8:30 AM, WEDNESDAY, MARCH 15, 1995

4. Do one step of Newton’s method for solving the system

x^{2} + y^{2} = 2, x + y = 0,

starting at the point (x_{0}, y_{0}) = (1, 0).

FINAL EXAM 5

5. Using Householder transformations, find a symmetric tridiagonal matrix

which is orthogonally similar to

[\begin{matrix}1 & 4 & 3 \\ 4 & 0 & 2 \\ 3 & 2 & 0 \end{matrix}]

6. Let A be a real symmetric matrix with eigenvalues −6, 1, 2. Let x^{0} be a

fixed vector in R^{3} which is not orthogonal to any of the eigenvectors of A.

a) Let the vectors x^{k} and the values μ_{k} be generated by the following scheme:

y^{k} = Ax^{k−1},
μ_{k} = (y^{k})^{T} x^{k−1},
x^{k} = \frac{y^{k}}{∥y^{k}∥_{2}}

What is lim_{k→∞} μ_{k}? Explain your answer.

b) If we generate x^{k} and the values ν_{k} by the following scheme what is lim_{k→∞} ν_{k}? Explain your answer.

(A − 3I)y^{k} = x^{k−1},
ν_{k} = (y^{k})^{T} x^{k−1},
x^{k} = \frac{y^{k}}{∥y^{k}∥_{2}}