Final6643Fall2008

Final6643Fall2008 - the Rayleigh quotient defined by A for...

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Math 6643 Final Exam Fall 2008 Name: 80 minutes, 6 questions for 60 points total. No calculators or notes permitted. Only the  textbook for the course and a pen is permitted. No credit is given for answers without the  reasoning that leads to them: show all work. If you need more paper, raise your hand. 1. (10 points) Let A = [ 2 1 0 1 ] . Consider the Simultaneous Iteration algorithm starting with the vector v = [ 1 1 ] (in place of Q (0) in Algorithm 28.3). What vector does it converge to and why?
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2. (10 points) Let A be a symmetric positive definite matrix. Is the Cholesky factoriza- tion of A the same as the LU factorization of A? If so, prove it. If not, find a counterexample.
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3. (10 points) Let A = [ 2 1 0 1 ] . What are the largest and smallest possible values of
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Unformatted text preview: the Rayleigh quotient defined by A for vectors in R 2 ? Explain your answer. 4. Let b be a nonzero vector in R m and A a symmetric real m-by-m matrix. a. (5 points) When is the dimension of the Krylov subspace K n = <b, Ab, A 2 b, . .., A n-1 b> less than n? b. (5 points) What is the most probable value for the dimension of K n if b is a ran-dom vector, and why? 5. (10 points) Both Arnoldi and GMRES relate to classes of polynomials with degree n. Why do the polynomials for Arnoldi have the coefficient of the degree-n term equal to 1, while those for GMRES have the constant term equal to 1? 6. (10 points) Why is the residual guaranteed not to increase from one iteration to the next for the Krylov subspace methods we discussed for solving Ax = b?...
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Final6643Fall2008 - the Rayleigh quotient defined by A for...

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