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aimz5 - AIMS Exercise Set 5 Peter J Olver 1 Use the power...

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AIMS Exercise Set # 5 Peter J. Olver 1. Use the power method to find the dominant eigenvalue and associated eigenvector of the following matrices: ( a ) - 2 0 1 - 3 - 2 0 - 2 5 4 , ( b ) 4 1 0 1 1 4 1 0 0 1 4 1 1 0 1 4 . 2. Use Newton’s Method to find all points of intersection of the following pairs of plane curves: x 3 + y 3 = 3 , x 2 - y 2 = 2 . 3. The system x 2 + xz = 2 , xy - z 2 = - 1 , y 2 + z 2 = 1 , has a solution x ? = 1 , y ? = 0 , z ? = 1. Consider a fixed point iteration scheme with g ( x, y, z ) = ( x + α ( x 2 + xz - 2) , y + α ( xy - z 2 + 1) , z + α ( y 2 + z 2 - 1) ) T , where α is a constant. ( a ) For which values of α does the iterative scheme converge to the solution when the initial guess is nearby? (
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