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Unformatted text preview: AIMS Exercise Set # 4 Peter J. Olver 1. Find the explicit formula for the solution to the following linear iterative system: u ( k +1) = u ( k ) 2 v ( k ) , v ( k +1) = 2 u ( k ) + v ( k ) , u (0) = 1 , v (0) = 0. 2. Determine whether or not the following matrices are convergent: ( a ) 2 3 3 2 , ( b ) 1 5 5 3 2 1 2 1 1 5 4 . 3. ( a ) Find the spectral radius of the matrix T = 1 1 1 7 6 . ( b ) Predict the long term behavior of the iterative system u ( k +1) = T u ( k ) + b , where b =  1 2 , in as much detail as you can. 4. Consider the linear system A x = b , where A = 4 1 2 1 4 1 1 1 4 , b = 4 4 . ( a ) First, solve the equation directly by Gaussian Elimination. ( b ) Using the initial approximation x (0) = , carry out three iterations of the Jacobi algorithm to compute x (1) , x (2) and x (3) . How close are you to the exact solution? ( c ) Write the Jacobi iteration in the form x ( k +1) = T x ( k ) + c .....
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 Fall '09
 Olver
 Matrices

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