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Sample Problems
Math 465
One notebook sized page of notes (both sides) allowed. Bring calculators
to the exam. There are three pages of problems.
- 1.
- Find the Gaussian quadrature rule of precision 3 for , where w(x)=1+x4. Leave any irrationals in
simplified radical form (i.e. . etc.).
- 2.
- Let . Find the monic
orthogonal polynomials of degree 0,1,2 for this inner product. What
happens if you try to find the polynomial of degree 3?
- 3.
- Let N=2n and let denote the discrete Fourier
transform from to .
- (a)
- Give formulas which allow you to compute in terms of
(one step of the FFT).
- (b)
- Let
Compute .
- (c)
- Let the components of the vector y of part (b) be denoted by
y0,y1,y2,y3. Find the interpolating function of the form
, such that .
- 4.
- We use the same notation as in problem 3.
- (a)
- Express the matrix
in terms of . (This may not be exactly
the same notation in class.)
- (b)
- Let
where y is any vector in .Derive the following factorization:
-
- 5.
- Do one step of Newton's method for solving the system
starting at the point (x0,y0)=(1,0).
- 6.
- Using Householder transformations, find a symmetric tridiagonal
matrix which is orthogonally similar to
- 7.
- Let V be a real vector space with inner product <u,v>. Let
be an orthonormal set in V. Let
and let . Prove Bessel's inequality:
- 8.
- Let A be a real symmetric matrix with eigenvalues -5, 1,
3. Let x0 be a fixed vector in which is not orthogonal
to any of the eigenvectors of A.
- a)
- Let the vectors xk and the values be generated by the
following scheme:
What is ? Explain your answer.
- b)
- If we generate xk and the values by the
following scheme what is ? Explain your answer.
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David Ragozin
2/6/1998