Download - Practice Final w 13 Numerical Analysis methods

7/27/2019 Practice Final w 13 Numerical Analysis methods

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10. (a) Use the zeros of T̃ 3 (the monic Chebyshev polynomial of degree 3) to con-struct and interpolating polynomial of degree 2 for f (x) = ex on [−1, 1].(b) Find a bound for the maximum error in the approximation of f by thisinterpolating polynomial.

11. Show that any polynomial P (x) of degree less or equal to n can be written on[−1, 1] as a linear combination of Chebyshev polynomials:

P (x) =n

k=0

akT k(x) x [−1, 1], (3)

for some constants a0, a1, . . . , a n .

12. Prove that

2π 1

− 1[T n (x)]

2

√ 1 −x2 dx = 1 (4)

13. Use Gerschgorin Theorem to determine bounds for the eigenvalues and thespectral radius of the following matrix:

A =−4 0 1 30 −4 2 11 2 −2 03 1 0 −4

.

14. Find the rst 3 iterations obtained by the Power Method applied to the matrix

A =4 2 10 3 21 1 4

,

using x (0) = [1 2 1]T .

15. Determine a shift that can be used in the Power Method to compute λ1 whenthe eigenvalues of A satisfy: λ1 = −λ2 > |λ3| ≥. . . ≥ |λn |.

16. Explain why Google’s PageRank algorithm is a huge eigenvalue problem.

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