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Unformatted text preview: Numerical Linear Algebra Midterm Exam Takehome 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. Lowrank 25 Displacement 5. Sparse Primitives 25 Total 130 Points Name: Alias: to be used when posting anonymous grade list. 1 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 ≤ k ≤ n 1 γ k γ . 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. Solution: For A we have μ = 1. This can be seen by noting that the Schur complements are all trivially related to A . Consider A (0) and A (1) A (0) = 1 1 1 1 1 2 2 2 1 2 3 3 1 2 3 4 A (1) = 2 2 2 2 3 3 2 3 4  1 1 1 (1) 1 ( 1 1 1 ) = 1 1 1 1 2 2 1 2 3 Similarly we have A (2) = 1 1 1 2 and A (3) = ( 1 ) So all of the Schur complements are submatrices of A and therefore the maximum ele ments in the numerator and denominator of μ are equal. 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 are performed exactly, i.e., ignore finite precision effects. Give the number of 2 operations in the algorithm in the form Cn k + O ( n k 1 ), where C is a constant independent of n and k > 0. (ii) (5 points) Apply the algorithm to the matrix A given in first part of the problem to determine if it is symmetric positive definite. Solution: The check for symmetry is trivial and has O ( n 2 ) complexity. The simplest finite, but not necessarily numerically reliable, method is to compute a Cholesky factorization. If A = LL T with positive elements on the diagonal of the triangular matrix L then A is positive definite. This has complexity n 3 3 + O ( n 2 )....
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 Fall '06
 gallivan

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