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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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- Fall '10