From crose Thu Sep 26 21:07:55 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA16331; Thu, 26 Sep 96 20:58:33 EDT Date: Thu, 26 Sep 96 20:58:33 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9609270058.AA16331@MOGLI.rutgers.edu> To: ece501 Subject: extra problem set, due tuesday Status: RO Hi Folks, These problems are worth points if you're on a grade border at the end of it all. Please submit them to me ELECTRONICALLY via email by Saturday 9/28/96 at 11:59PM. 1) You are given a tri-diagonal matrix. Its main diagonal elements are a, its upper diagonal elements are b and the lower diagonal elements are c. b,c = 0.2 and a=0.1 for example: a b 0 0 0 0 0 0 c c a b 0 0 0 0 0 0 0 c a b 0 0 0 0 0 0 0 c a b 0 0 0 0 0 0 0 c a b 0 0 0 = A 0 0 0 0 c a b 0 0 0 0 0 0 0 c a b 0 0 0 0 0 0 0 c a b b 0 0 0 0 0 0 c a You are given the iterative mapping x(n+1) = Ax(n). Is the mapping a contraction? 2) a) A demon(ess) is put in a room with N chairs and seated initially in a single chair. Being demonic it can't just sit in one chair at a time. So with the first tick of the celestial clock, it splits itself into three equal weight demons. One remains where the original demon was seated and the other two move to chairs to the left and right of the first chair. If there's no chair to the left, the left-going demon stays where it was. Likewise for the right-going demon. But demons, even smaller ones, are NEVER happy or satisfied so each smaller demon does the same splitting routine with the next tick of the celestial clock. This process is followed by each "sub-demon" for each tick of the celestial clock for all eternity. If the original demon has weight one unit. Please derive a matrix difference equation for the total weight of demons in each chair. That is, write your equation in terms of an N-vector x(n) where x_i(n) is the weight of demons sitting in chair i at tick n and x(n+1). Identify (if any) the fixed points of this mapping. You can assume x is any real N-vector for this question. Is this mapping a contraction? b) Now the chairs are arranged in a circle and the demon does not split itself but always moves one chair to the right. Write down a difference equation in matrix form for this system. What is the domain space of this mapping? What is the range space of the mapping given the domain space? Identify (if any) the fixed points of this mapping. Is this mapping a contraction? 3) What is A^N for the matrix of problem 2b? 4) What is e^(At) for the matrix of problem 2b with N=2? Happy solving!