Slides_2010_03_05

# Slides_2010_03_05 - Applied linear algebra and numerical...

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Unformatted text preview: Applied linear algebra and numerical analysis Session 25 Prof. Ulrich Hetmaniuk Department of Applied Mathematics March 4, 2010 Approximation of functions • Given some function f ( t ) , we want to find a function g ( t ) from some finite dimensional function space that approximates f ( t ) over some interval [ a , b ] . • A function space of dimension n can be described by n basis functions φ 1 ( t ) ,..., φ n ( t ) . The problem is to find the coefficients c 1 ,..., c n in g ( t ) = c 1 φ 1 ( t )+ ... + c n φ n ( t ) . • The best choice of “basis functions” φ j ( t ) depends on where the function f or the data comes from. Approximation of functions We will study three kinds of approximation. 1 Interpolation: Given a set of n data points ( t i , y i ) , the interpolation function will agree exactly at those data points. 2 Data fitting: Given a set of n data points ( t i , y i ) , the interpolation function will approximate those data points. 3 Global approximation: Instead of fitting m data points, we can try to find a function g ( t ) that is as close as possible to f ( t ) at all points in [ a , b ] . Interpolation • Given a set of n data points ( t i , y i ) , we look for a function g ( t ) that is a linear combination of n given functions, φ 1 ( t ) , ..., φ n ( t ) : g ( t ) = c 1 φ 1 ( t )+ c 2 φ 2 ( t )+ ··· + c n φ n ( t ) . • We can solve the least-squares problem n ∑ i = 1 ( y i- c 1 φ 1 ( t i )-···- c n φ n ( t i )) 2 = k y- Ac k 2 2 where A = φ 1 ( t 1 ) ··· φ n ( t 1 ) . . . . . . φ 1 ( t n ) ··· φ n ( t n ) c = c 1 . . . c n y = y 1 ....
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Slides_2010_03_05 - Applied linear algebra and numerical...

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