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Unformatted text preview: Advanced Mathematical Programming IE417 Lecture 15 Dr. Ted Ralphs IE417 Lecture 15 1 Reading for This Lecture Sections 8.18.5 IE417 Lecture 15 2 Numerical Analysis IE417 Lecture 15 3 Numerical Analysis Numerical analysis is the study of algorithms for problems from continuous mathematics. A problem is a map from f : X Y , where X and Y are normal vector spaces. A numerical algorithm is a procedure which calculates F ( x ) Y , an approximation of f ( x ) . As we have already discussed, we can define these algorithms in terms of an algorithmic map. Because we have to use floating point arithmetic and other approximations, our answers will not be exact. IE417 Lecture 15 4 Conditioning A problem is wellconditioned if x x f ( x ) f ( x ) . Otherwise, it is illconditioned . Notice that wellconditioned requires all small perturbations to have a small effect. Illconditioned only requires some small perturbations to have a large effect. Condition number of a problem Absolute Relative IE417 Lecture 15 5 Stability An algorithm is stable if F ( x ) f ( x ) for some x x . This says that a stable algorithm computes nearly the right answer to nearly the right question . Notice the contrast between conditioning and stability: Conditioning applies to problems. Stability applies to algorithms. IE417 Lecture 15 6 Accuracy Stability plus good conditioning implies accuracy ....
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 Fall '08
 Linderoth

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