Exam6643Fall2009

Exam6643Fall2009 - Fall 2009 Name 3(10 points Calculate(by...

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Math 6643 Exam Fall 2009 Name: Put your name on all sheets now. 80 minutes, 6 questions for 60 points total. No calculators or electronic devices permitted. Only the textbook for the course, notes, 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 space, continue onto the back with a note that you have done so. 1. (10 points) Let . Consider the Simultaneous Iteration algorithm starting with a random orthogonal matrix = 1 0 0 2 A . ) ( 0 Q a. (5 points) What matrix Q does the matrix ) ( k Q converge to as k and why? b. (5 points) Estimate how quickly the norm of Q Q k ) ( tends to zero as a function of k. I.e. find f(k) with f(k) 0 as k so )). ( ( ) ( k f O Q Q k =
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Math 6643 Exam Fall 2009 Name: 2. (10 points) Determine the LU factorization of B with partial pivoting . . = 4 3 2 1 B
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Math 6643 Exam
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Unformatted text preview: Fall 2009 Name: 3. (10 points) Calculate (by hand) the condition number of the matrix Hint: you may wish to consider the matrix . ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 1 1 1 C . C C T Math 6643 Exam Fall 2009 Name: 4.(10 points) Let A be a symmetric positive definite matrix. Is the Cholesky factorization of A the same as the LU factorization of A? If so, prove it. If not, find a counterexample. Math 6643 Exam Fall 2009 Name: 5. (10 points) Let . ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 1 1 1 1 1 b D , Find an orthogonal projector onto the Krylov space Db b K , = 2 . Math 6643 Exam Fall 2009 Name: 6. (10 points) Consider the algorithm , where x and c are complex scalars. For which c is the algorithm backward stable? c x fl x f ⊕ = ) ( ) ( ~...
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This note was uploaded on 12/25/2011 for the course MATH 6643 taught by Professor Staff during the Fall '08 term at Georgia Tech.

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Exam6643Fall2009 - Fall 2009 Name 3(10 points Calculate(by...

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