Math 465 Sample Problems

Date:

One notebook sized page of notes will be allowed on the test. Calculators in which no text, formulas or programs are stored should be brought to the test.

1.

Let

$$A= \begin{bmatrix}1&1&0\\ 1&2&-1\\ 0&-1&3\end{bmatrix}$$

a)

Compute the Sturm sequence of A.

b)

Use the Sturm sequence to determine how many eigenvalues of A are in [0,4].

2.

Let

$$A= \begin{bmatrix}1&3&4\\ 3&1&-1\\ 4&-1&2\end{bmatrix}$$

a)

Use Gerschgorin's theorem to locate the eigenvalues of A.

b)

Use $\Vert A \Vert _{\infty}$ to give an upper bound for the eigenvalues of A.

c)

Does A have any complex eigenvalues? Give reasons.

d)

Compute the Rayleigh quotient of A in the direction of the vector [1,1,1]^T.

3.

Let

$$A= \begin{bmatrix}0&4&1\\ 4&0&1\\ 0&1&1\end{bmatrix}$$

a)

Compute the characteristic polynomial of A by Krylov's method.

b)

Compute the characteristic polynomial of A by Hyman's method.

4.

Let

$$A=\begin{bmatrix}2&12&-5\\ 12&1&2\\ -5&2&1\end{bmatrix}$$

a)

Either elimination or Householder transformations can be used to reduce A to Hessenberg form. Which would you use to preserve the symmetry of A?

b)

Only one Householder transformation Q is needed to reduce A to Hessenberg form. Q is of the form $${Q=I-\frac{2}{\vert u\vert ^2}uu^T}$$. Find u.

c)

Use the u and/or Q of part b) to reduce A to Hessenberg form.

5.

Let q(x)=x⁴+x³-2x+1.

a)

Describe Bairstow's method for finding a real quadratic factor of a real polynomial.

b)

Beginning with (u₀,v₀)=(1,0), compute the first step of Bairstow's method applied to q.

c)

What is the companion matrix A of q?

d)

Apply Gerschgorin's circle theorem to the companion matrix to find an upper bound for all of the zeros of q.