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tut3 - 57‘ 2008/09 MT3802 Numerical Analysis Tutorial...

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Unformatted text preview: 57‘ 2008/09 MT3802 - Numerical Analysis Tutorial Sheet 3 1. The recurrence relation WU“) : Wm (21 7 AWW) where A and WU‘) are N X N matrices and I is the unit matrix of order N , generates a sequence of matrices from some initial Wm) . Show that 2 I 1 AW<T+1> : (I 1 AW<">) and deduce that when HI 7 AWm) H < 1 lim WW : A’l THOO The above is an iterative method for finding the inverse of a matrix and requires no more than ma— trix multiplication and some suitable starting matrix. If the matrix A does not possess an inverse, det(A) : 0, the sequence cannot converge and HI 7 AWm) H 2 1 for all matrices Wm). 2. Suppose the matrix W is an approximation to A71. A simple way of assessing the accuracy of W is to compute R : I — AW (the better the approximation, the closer to the zero matrix R becomes). Show that HRH W 3 HA’1 *WH S “W“ H(I , R)’1|| HR||~ Note that when HRH < 1, as in Question 1, the right hand side can be simplified further by observing that H(1 , Rflll S (1 , HRH)? 3. If a real N X N matrix B is diagonisable, show that B” —> 0 as n —> 00 if and only if p(B) < 1 . Hint: Remember, if BNxN is diagonisable, then P’lBP = A {07‘ B = PAPA), where A is a diagonal matrix with the eigenvalues A7: of B on the diagonal: A1 0 ...
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