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hw3 - Homework 3 Foundations of Computational Math 1 Fall...

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Homework 3 Foundations of Computational Math 1 Fall 2010 The solutions will be posted on Wednesday, 9/22/09 Problem 3.1 Suppose A R n × n is a nonsymmetric nonsingular diagonally dominant matrix with the following nonzero pattern (shown for n = 6) * * * * * * * * 0 0 0 0 * 0 * 0 0 0 * 0 0 * 0 0 * 0 0 0 * 0 * 0 0 0 0 * It is known that a diagonally dominant (row or column dominant) matrix has an LU factor- ization and it can be computed stably without pivoting. 3.1.a . Describe an algorithm that solves Ax = b as efficiently as possible. 3.1.b . Given that the number of operations in the algorithm is of the form Cn k + O ( n k - 1 ), where C is a constant independent of n and k > 0, what are C and k ? Problem 3.2 It is known that if partial or complete pivoting is used to compute PA = LU or PAQ = LU of a nonsingular matrix then the elements of L are less than 1 in magnitude, i.e., | λ ij | ≤ 1. Now suppose A R n × n is a symmetric positive definite matrix, i.e., A = A T and x = 0 x T Ax > 0. It is known that A has a factorization A = LL T
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