matlab4 - b-AT The value of κ as I’ve written it depends...

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Math 3353 Matlab Assignment 4 - Due 11/18/09 T. Hagstrom 1. Using random similarity transforms build 3 × 3 matrices, A , with the indicated eigenvalues. Then, starting with a random 3-vector, x (0) , compute x ( k ) = Ax ( k - 1) , k = 1 , . . . 25 . Plot the iterates (use plot3 ) and explain the relationship between the eigenvalues and the observed results. i. { 1 . 02 , - . 8 , . 2 } ii. n 1+ i 2 , 1 - i 2 , 1 3 o iii. ± 2+ i 3 , 2 - i 3 , - 1 2 ² 2. Consider the dynamic heat transfer problem associated with Problem 2 in Homework 2. In that problem, given a p × p lattice of temperature values, you set up and solved a system of p 2 equations AT = b where the vector b was determined by the boundary temperatures. Now suppose the temperatures evolve by the differential equation: dT j dt = - κ ³ T j - 1 4 ( T left + T right + T up + T down ) ´ where κ measures the heat conduction. In matrix form dT dt = κ
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Unformatted text preview: ( b-AT ) The value of κ as I’ve written it depends on the lattice spacing. You can take it to be κ = p 2 10 . 1 a. Let T s be the steady state solution as computed in Homework 2. That is AT s = b verify that if we write T ( t ) = T s + W ( t ) then dW dt =-κAW, W (0) = T (0)-T s b. Use eig to compute the eigenvalues of A for p = 10. What do you conclude about the convergence of the temperature to the steady state value? c. Suppose the sample is intially heated to a temperature of 80. Use expm to compute the temper-ature at times t = . 5 : . 5 : 5 and use surf to plot it. (Note you are solving for W so you need to add in T s to get the total temperature field. Use p = 10.) 2...
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matlab4 - b-AT The value of κ as I’ve written it depends...

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