N12 - ME 510 NUMERICAL METHODS FOR ME II Norms Given A...

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ME – 510 NUMERICAL METHODS FOR ME II g51g85g82g73g17g3g39g85g17g3g41g68g85g88g78g3g36g85g213g81g111 Fall 2007 Norms Given : A function, f (x), or tabulated data points in a certain range, [a , b] Problem: A simple function, g(x), that represents or fits the given f (x) or the data points in [a , b] according to a certain "goodness" or "closeness" is required. How do we define "goodness" of fitting? Answer: There are basically three ways of defining "goodness" which are called "Norms" ME – 510 NUMERICAL METHODS FOR ME II g51g85g82g73g17g3g39g85g17g3g41g68g85g88g78g3g36g85g213g81g111 Fall 2007 Chebyshev, or L g146 Norm : Minimize maximum error, i.e., Max. g327 f (x) - g (x) g327 in [a , b] Least Deviation or L 1 Norm : Minimize average error, i.e., Least Squares or L 2 Norm : Minimize sum of the squares of the error, i.e. dx (x) g - (x) f b a g179 g62 g64 dx (x) g - (x) f b a 2 g179
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