finalf09 - Math 570A Midterm Fall 2009 Name Show your work...

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Math 570A Midterm Fall 2009 Name Show your work and write neatly Only write on one side of the paper Number your pages and put your name on each page Avoid writing too close to the corners of the page 1) Let A = 111 110 101 Answer yes or no to the following. A is (a) diagonally dominant (b) diagonalizable (c) positive deFnite (d) orthogonal Next Fnd the Jacobi iteration matrix for A and determine whether Jacobi iteration con- verges using it. 2) (a) Let A = 010 Compute the Gauss-Seidel iteration matrix and use it to determine whether Gauss-Seidel iteration converges for A . (b) Show that Gauss-Seidel iteration converges for every nonsingular upper triangular matrix (Hint: the eigenvalues of an upper triangular matrix are on the diagonal). 3) Let A be an n × n matrix and A = M - N be a splitting for A . Let x be the solution to Ax = b . Consider the iterative scheme Mx ( k +1) = Nx ( k ) + b . Let the error be given by e ( k ) = x - x ( k ) . Give a careful proof the if ρ ( G ) < 1, where G = M - 1 N and ρ ( G ) is the spectral radius, then the iterative scheme converges (i.e.
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finalf09 - Math 570A Midterm Fall 2009 Name Show your work...

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