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Unformatted text preview: Department of Mathematics University of Houston Numerical Analysis I Dr. Ronald H.W. Hoppe Numerical Analysis I (4th Homework Assignment) Exercise 13 ( Gradient method and semi-iterative Richardson iteration) ) Let A R n n be symmetric positive definite with the extreme eigenvalues := min ( A ) , := max ( A ) and let := / be the spectral condition of A . Further, let b R n and x R n . (i) Consider the semi-iterative Richardson iteration y m +1 = y m + m +1 ( b- Ay m ) , m . Specify the values of y R n and m +1 for which the sequence of iterates ( y m ) N corresponds to the sequence ( x m ) N obtained by the gradient method. (ii) Show that for any initial vector x R n the sequence ( x m ) N , obtained by the gradient method, converges to x * := A- 1 b . Moreover, verify the estimates F ( x m )- F ( x * ) ( - 1 + 1 ) 2 m [ F ( x )- F ( x * )] , k x m- x * k A ( - 1 + 1 ) m k x- x * k A , where F ( x ) := 1 2 < Ax,x >- < b,x > ....
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