BackErrGE - = max ij | u ij | max ij | a ij | . The...

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Errors in Gaussian Elimination Many would mark the birth of modern numerical analysis as a branch of mathematics with the 1947 paper of von Neumann and Goldstine: “Numerical Inverting of Matrices of High Order”. The perspective, hinted at, if not explicitly stated there, of viewing the computed solution as the exact solution to another problem, is called backward error analysis . If we can guarantee that our computed solution is always the exact solution to a nearby problem, then we call the method backward stable. Without pivoting, G.E. is not stable. Here is a backward error result that applies when no zero pivots are encountered. If ˜ L and ˜ U are the computed versions of L and U , respectfully, then there exists an δA R n × n for which ˜ L ˜ U = A + δA, where k δA k k A k = k L kk U k k A k O( μ ) . The reason this result does not imply backward stability is that k L k or k U k (and often both) can be very large. But with partial pivoting k L k = O(1) and the only concern is with k U k . Turning to U we define the growth factor for G.E. as
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Unformatted text preview: = max ij | u ij | max ij | a ij | . The backward error result for G.E.P.P. is L U = P ( A + A ) , where k A k k A k = O( ) . This implies G.E.P.P. is backward stable if = O(1). We have a beautiful delimma here that makes further discussion dicult, so we will ever so simply say that G.E.P.P. is backward stable, but not really, . ..but eectively is. Backward and forward substitution, on the other hand, are clearly backward stable. The result for back substitution is that the computed x satises ( R + R ) x = b, where k R k k R k = O( ) . Combining the results above, we can say the that the computed solution, x to Ax = b , using G.E.P.P with forward and backward substitution, satises ( A + A ) x = b, where k A k k A k = n 3 O( ) . The n 3 term above appears quite pessimistic, for in practice we see k A k k A k = n O( ) ....
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