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Unformatted text preview: Chapter 10 Polynomial Interpolation Polynomial interpolation is rarely the end product of a numerical process. But it arises frequently, indeed it is one of the most ubiquitous tasks, both within the design of numerical algorithms and in their analysis. Its im portance and centrality help explain the considerable length of the present chapter. In the next four chapters plus Chapter 6 we develop approximation tech niques for two types of problems: 1. Data fitting (Discrete approximation problem): Given a set of data points { ( x i , y i ) } n i =0 , find a reasonable function v ( x ) that fits the data points. If the data are accurate it might make sense to require that v ( x ) in terpolate the data, i.e. that the curve pass through the data exactly, satisfying v ( x i ) = y i , i = 0 , 1 , . . . , n. See Figure 10.1. 2. Approximating known functions : For a complicated function f ( x ) (which may be given explicitly, or only implicitly), find a simpler function v ( x ) that approximates f ( x ). For instance, suppose we need to quickly find an approximate value for sin(1 . 2) (thats 1 . 2 in radians, not degrees) with only a primitive calculator at hand. From basic trigonometry we know the values of sin( x ) for x = 0 , / 6 , / 4 , / 3 and / 2: How can we use these to estimate sin(1 . 2)? For another instance, suppose we have a complex, expensive program that calculates the final point (say, the landing location) of the trajec tory of a space shuttle for each given value of a certain control param eter. We perform this calculation, i.e., invoke the program, for several 323 324 Chapter 10: Polynomial Interpolation 1 2 3 4 5 6 7 0.5 1 1.5 2 2.5 x v (a) reasonable 1 2 3 4 5 6 7 0.5 1 1.5 2 2.5 x v (b) unreasonable Figure 10.1: Different interpolating curves through the same set of points. parameter values. But then, we may want to use this computed infor mation to have an idea of the resulting landing location for other values of the control parameter without resorting to the full calculation for each parameter value. The good news is that interpolation techniques for such function ap proximation are identical to those of data fitting once we specify the data points { ( x i , y i = f ( x i )) } n i =0 . The difference between fuction inter polation and data fitting interpolation is that in the former we have some freedom to choose x i cleverly, and we can consider the global interpolation error. Why do we want to find an approximating function v ( x ) in general? For prediction : we can use v ( x ) to find approximate values of the un derlying function at values x other than the data abscissae x , . . . , x n . If x is inside the smallest interval containing all the data abscissae then this is called interpolation ; if x is outside that interval then we have extrapolation . For instance, we may have data regarding the perfor mance of some stock at each trading weeks end during the past year....
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This note was uploaded on 01/27/2010 for the course CS 303 taught by Professor Greif during the Winter '10 term at University of Bristol.
 Winter '10
 Greif
 Algorithms, The Land

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