Lecture15

# Lecture15 - Advanced Mathematical Programming IE417 Lecture...

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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 8.1-8.5 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 well-conditioned if x ≈ x ⇒ f ( x ) ≈ f ( x ) . • Otherwise, it is ill-conditioned . • Notice that well-conditioned requires all small perturbations to have a small effect. • Ill-conditioned 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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Lecture15 - Advanced Mathematical Programming IE417 Lecture...

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