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Unformatted text preview: 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 )  < x < 1 , < 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 secondorder accuracy of the numerical method....
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 Winter '08
 LinZhi
 Numerical Analysis, Matrices, finite difference, Gauss–Seidel method, Jacobi method, Positivedefinite matrix, Iterative method

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