Introduction to Scientific Computing Gauss-Seidel Method. Power ...

Institute of Scientific ComputingTechnische Universität BraunschweigProf. Hermann G. Matthies, PhDDr. Alexander LitvinenkoWinter Term 2009-2010January 14, 2010Introduction to Scientific ComputingGauss-Seidel Method. Power method.Assignment 11Exercise 1: Linear two-step methodConsider the linear two-step method(12 points)x n+2 + 4x n+1 − 5x n = h · (4f n+1 + 2f n ) (1)(a) Show that this method is consistent. Which order of consistency has the method ? (4 points)(b)Implement this method and solve the ODE˙u = uin the interval [0,1] with different step sizes h.u(0) = 1(2)(4 points)(c) Explain the numerical results (convergence). (4 points)Exercise 2: Gauss-Seidel MethodDepending on the real valued parameter α we have a matrix A defined as( )1 αA =−α 2(12 points)and b = (b 1 ,b 2 ) ⊤ an arbitrary (but given) vector. For solving the system of equations we use theGauss-Seidel method.(a) For which values of α does this method converge in general (using an arbitrary starting vector)and for which values does it diverge in general?(6 points)(b) Let b = (1,2) ⊤ . Solve the system of equations Ax = b numerically using the Gauss-Seidelmethod for α = − 3 2 + β100, β = 0,1,... ,300. Plot the numerical error and the number of iterationsvs. the different values of alpha.(6 points)Exercise 3: Power method (power iteration)(14 points)Sometimes it is very important to compute only the largest eigenvalue of a large matrix (withoutcomputing all other eigenvalues). The power method (or power iteration) performs this task.(a) Implement this method as a MATLAB procedure. (6 points)(b)Generate a 100*100 matrix in MATLAB and use your procedure to compute the largest eigenvalue.Use also MATLAB “eigs” procedure to check yourself. Pay attention that the power method

works not for all matrices.(6 points)(c) Compare computing times. (2 points)