For solving Ax = b, consider the stationary Richardson iteration Xk+1=Xk+a (b Ax). For this...

 

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For solving Ax = b, consider the stationary Richardson iteration Xk+1=Xk+a (b − Ax). For this question, we assume that all eigenvalues of A are real. (a) Find the iteration matrix. (b) Show that if Amin < 0 < Amax, the method will always be divergent for some initial iterate. (c) So assuming that all eigenvalues are positive, find the spectral radius of the iteration matrix in terms of a, Amin and Xmax? (d) Find amin < amax such that the method converges for any amin < a < amax. In other words, find the largest range for a that ensures the method always stays convergent. (e) What is the optimal value for a? In other words find aopt that minimizes the spectral radius of the iteration matrix. (f) Suppose A > 0. Using this optimal step-size, obtain the optimal convergence rate in terms of the matrix condition number. For solving Ax = b, consider the stationary Richardson iteration Xk+1=Xk+a (b − Ax). For this question, we assume that all eigenvalues of A are real. (a) Find the iteration matrix. (b) Show that if Amin < 0 < Amax, the method will always be divergent for some initial iterate. (c) So assuming that all eigenvalues are positive, find the spectral radius of the iteration matrix in terms of a, Amin and Xmax? (d) Find amin < amax such that the method converges for any amin < a < amax. In other words, find the largest range for a that ensures the method always stays convergent. (e) What is the optimal value for a? In other words find aopt that minimizes the spectral radius of the iteration matrix. (f) Suppose A > 0. Using this optimal step-size, obtain the optimal convergence rate in terms of the matrix condition number.

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The iteration matrix is I aA The iteration matrix M for the stationary Richardson iteration is given ... View the full answer