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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)
- 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)
- Use Gerschgorin's theorem to locate the eigenvalues of A.
- b)
- Use
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)
- Compute the characteristic polynomial of A by Krylov's method.
- b)
- Compute the characteristic polynomial of A by Hyman's method.
- 4.
- Let
- 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
.
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.
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David Ragozin
2000-02-07