midtermf11 - Numerical Linear Algebra Midterm Exam...

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Numerical Linear Algebra Midterm Exam Take-home Exam Open Notes, Textbook, Homework Solutions Only Calculators Allowed No collaboration with anyone Due beginning of Class Wednesday, October 26, 2011 Question Points Points Possible Awarded 1. Structured Schur 25 complements 2. LU 30 Factorizations 3. Kronecker Product 25 Factorizations 4. Low-rank 25 Displacement 5. Sparse Primitives 25 Total 130 Points Name: Alias: to be used when posting anonymous grade list. 1
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Problem 1 1.a (10 points) Suppose A R n × n . Define A (0) = A and denote by A ( k ) the Schur complement of A ( k - 1) with respect to e T 1 A ( k - 1) e 1 for k = 1 , . . . , n - 1. For each 0 k n - 1, let γ k be the maximum magnitude of all the elements in A ( k ) . Define a growth factor μ for the series of Schur complements as μ = max 0 k n - 1 γ k γ 0 . Determine μ for the symmetric matrix A = 1 1 1 1 1 2 2 2 1 2 3 3 1 2 3 4 . Justify your answer. 2
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1.b (i) (10 points) Propose an algorithm that determines whether or not a given n × n matrix is symmetric positive definite. You may assume that computations
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