hw2 - Homework#2 1(a Find the PLU factorization of the...

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Homework #2 1. (a) Find the PLU factorization of the matrix A = - 2 0 1 - 1 - 1 2 0 1 4 - 1 - 2 - 4 0 0 2 0 using Gaussian elimination with row pivoting. (b) Use these results to compute the determinant of A . (c) Use the PLU factorization and forward and back substitution to solve the linear system of equations A~x = ~ b , with ~ b = [1 , 0 , 1 , 0] t . (d) Use the PLU factorization and forward and back substitution to solve the linear system of equations A~x = ~ b , with ~ b = [0 , - 1 , 0 , - 1] t . 2. Consider the Babylonian method solving the fixed point problem x = 1 2 x + c x to approximate 2. (a) Perform fixed point iterations with initial guess x 0 = 1 and stop according to the “converged” stopping condition | x k - x k - 1 | ≤ 10 - 4 . (b) Find the relative error of your final approximation. 3. Consider the nonlinear equation x 2 - 3 = 0 and the naive attempt at a fixed point problem x 2 + x - 3 = x obtained by adding x on both sides. Starting with initial guess x 0 = 2, write down the results of fixed point iterations with the “looking bad”
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