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CSC_349_HANDOUT#7

# CSC_349_HANDOUT#7 - COMPUTER SCIENCE 349A Handout Number 7...

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1 COMPUTER SCIENCE 349A Handout Number 7 STABILITY OF AN ALGORITHM Textbook (page 92 of the 5 th ed.; page 97 of the 6 th ): a computation is numerically unstable if the uncertainty of the input values is greatly magnified by the numerical method. The following is a more precise definition. Definition. An algorithm is said to be stable (for a class of problems) if it determines a computed solution (using floating-point arithmetic) that is close to the exact solution of some (small) perturbation of the given problem. given problem, specified by computed solution data { d i } floating-point { r i } computation perturbed problem, data exact solution { ˆ d i } = { d i + ε i } exact { ˆ r i } computation with i d i small If there exist data ˆ d i d i (small i for all i) such that ˆ r i r i (for all i ), then the algorithm is said to be stable. If there exists no set of data { ˆ d i } close to { d i } such that i i r r ˆ for all i , then the algorithm is said to be unstable . Meaning of numerical stability : the effect of uncertainty in the input data or of the floating-point arithmetic (the round-off error) is no worse than the effect of slightly perturbing the given problem, and solving the perturbed problem exactly. Example 1 Approximate x e when 5 . 5 = x using 5 , 10 = = k b rounding floating-point arithmetic and the Taylor polynomial approximation (expanded about 0 = a ) ! ! 4 ! 3 ! 2 1 4 3 2 n x x x x x e n x + + + + + + L .

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CSC_349_HANDOUT#7 - COMPUTER SCIENCE 349A Handout Number 7...

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