Chemical Engineering Hand Written_Notes_Part_91

# Chemical - 184 5 OPTIMIZATION AND RELATED NUMERICAL SCHEMES Figure 5 The equation can be rearranged as(5.9 p= a aT y aT a = 1 aT a aaT y = Pr.y T 1

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5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES Figure 5 The equation can be rearranged as (5.9) p = a μ a T y a T a = 1 a T a ¸ £ aa T ¤ y = P r . y where P r = 1 a T a aa T is a n × n matrix is called as projection matrix, which projects vector y into its column space. 5.2. Distance of a point from Subspace. The situation is exactly same when we are given a point y R 3 and plane S in R 3 passing through origin , we want to f nd distance of y from S, i.e. a point p S such that k p y k 2 is minimum (see Figure 5). Again, from school geometry, we know that such point can be obtained by drawing a perpendicular from y to S ; p is the point where this perpendicular meets S (see Figure 5). We would like to formally derive this result using optimization. More generally, given a point y R m and subspace S of R m ,thep rob lem is to f nd a point p in subspace S such that it is closest to vector y .L e t S = span © a (1) , a (2) , .... , a ( m ) ª and as

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## This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

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Chemical - 184 5 OPTIMIZATION AND RELATED NUMERICAL SCHEMES Figure 5 The equation can be rearranged as(5.9 p= a aT y aT a = 1 aT a aaT y = Pr.y T 1

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