hw7 - Math 471 Fall 2007 Homework 7 Assigned Friday Due...

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Math 471, Fall 2007 Homework 7 Assigned: Friday, October 26, 2007. Due: Friday, November 2, 2007 . 1. (Positive Definite Matrices, page 221, #9) . Let A be an n × n symmetric positive definite matrix. (a) Show that a ii > 0 for each i = 1 , 2 , 3 , · · · , n . (b) Show that a 2 ij < a ii a jj for each i 6 = j . Some hints: Establish the 2 × 2 case. Then reduce the general problem to this special case using the definition of positive definiteness. 2. (Poisson BVP, page 741, #7) Consider the Poisson problem 2 u ∂x 2 + 2 u ∂y 2 = 0 on R = { ( x, y ) | 0 < x < 1 , 0 < y < 1 } u ( x, 0) = 0 , u ( x, 1) = 1 (1 + x 2 ) + 1 , u (0 , y ) = y 1 + y 2 , u (1 , y ) = y 4 + y 2 , whose exact solution is u ( x, y ) = y (1 + x ) 2 + y 2 . (a) Taking N = M = 4, set up and solve the corresponding system of finite differ- ence equations. (b) Numerically verify the second-order accuracy of the numerical method. 3. (Iterative Methods: Analysis). Consider a linear system with matrix A = 2 1 1 4 . (a) Write down the iteration matrices T jac and T gs for Jacobi’s Method and Gauss– Seidel.
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