1.5-Approximating Functions of a Single Variable

1.5-Approximating Functions of a Single Variable -...

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Approximating Functions of a Single Variable A continuous function f x () can be approximated by a polynomial to any accuracy desired (Weierstrass Approximation Theorem) uniformly over any interval ab , . (For multiple-dimensional continuous functions the Stone-Weierstrass theorem guarantees this using coordinate variables over a closed and bounded set in E n . (Karl Weierstrass (1815- 1897) a German mathematician)) Just how do we go about doing this approximation? There are several approaches depending on the type of approximation desired. Let’s work with a quadratic approximation of the form . If we want the approximation to give good values for the function over a range of x values, then we need to provide several data points such as fx a b x c x d x () ≈+ + + 23 r xfx ii i n ,() bg m = 1 to use to fit the function parameters: (, . The best least-squares fit to this data is an optimization problem ,) abc min ( ) ,,, abcd ii i i i n x d x af di ++ + − = 1 2 . Rather than get ahead of ourselves, that is, this material is being developed so that we can discuss optimization problems, we will restrict ourselves to exact fits. So with four parameters to estimate , there are multiple ways that we can develop the approximation. ,, ) ( i ) If we want the function values to be exact at four points spread out along the x-axis, then, we need four-data points that we require the approximation function to yield exact results for these points: .
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This note was uploaded on 10/26/2011 for the course ISEN 620 taught by Professor Curry during the Fall '10 term at Texas A&M.

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1.5-Approximating Functions of a Single Variable -...

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