practice final w 13 numerical analysis methods

Practice Exercises for the Final Exam, Math 104B 1. True (T) or False (F). Suppose A is an n × n positive definite matrix: a) ( ) The search dire cti ons for the conj uga te gra die nt (CG) method are always the residuals. b) ( ) The CG met hod will con ve rge to the exac t solut ion of Ax = b in at most n 2 iterations. c) ( ) Two vectors u and v are said to be conjugate with respect to A if and only if u T Av > 0. d) ( ) The most expensive part in a CG iteration is computing the product of A and a vector. e) ( ) For A sparse, the CG method generally beats Jacobi, GS, and S.O.R. 2. The concentr ation c of a radioactive material decays according to the law c(t) = be −at where t represents time in seconds, a = 0.1 sec −1 , and b is the initial conc ent ration. a)Usi ng the Least Squares method and the data table (Table 1) below find b. b) Find the error in the least squares approximation. t i (sec) C i 1 0.91 2 0.80 3 0.76 4 0.65 Table 1: 3. Giv en a collection of data points {(x i , y i )} m i=1 find the best least squares approximation of the form y = ax 2 + bx 3 . 4. (a) Given a collection of data points {(x i , y i )} m i=1 find the best least squares approximation of the form y = ax + bx 2 . (b) Use this approximation to fit the data in Table 2. (c) Find the error in the least squares approximation. 5. a) Obtain the first 4 Legendre polynomials. b) Find the least squares polynomial approximations of degrees 1, 2, and 3 for the function f (x) = e x on [−1, 1]. c) What is the polynomial least squares approximation of degree 4 for f (x) = x 3 on [−1, 1]? Explain 1

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Page 1: 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)]

√ 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