# Ch10 Polynomial Interpolation.pdf - u2710 u2710 u2710...

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Chapter 10 Polynomial Interpolation Polynomial interpolants are rarely the end product of a numerical process. Their importance is more as building blocks for other, more complex algorithms in differentiation, integration, solution of dif- ferential equations, approximation theory at large, and other areas. Hence, polynomial interpolation arises frequently; indeed, it is one of the most ubiquitous tasks, both within the design of numerical algorithms and in their analysis. Its importance and centrality help explain the considerable length of the present chapter. Section 10.1 starts the chapter with a general description of approximation processes in one independent variable, arriving at polynomial interpolation as one such fundamental family of tech- niques. In Sections 10.2, 10.3, and 10.4 we shall see no less than three different forms (different bases) of interpolating polynomials. They are all of fundamental importance and are used exten- sively in the practical construction of numerical algorithms. Estimates and bounds for the error in polynomial interpolation are derived in Section 10.5. If the choice of locations for the interpolation data is up to the user, then a special set of abscissae (nodes) called Chebyshev points is an advantageous choice, and this is discussed in Section 10.6. Finally, Section 10.7 considers the case where not only function values but also derivative values are available for interpolation. 10.1 General approximation and interpolation Interpolation is a special case of approximation. In this section we consider different settings in which approximation problems arise, explain the need for fi nding approximating functions, describe a general form for interpolants and important special cases, and end up with polynomial interpola- tion. Discrete and continuous approximation in one dimension It is possible to distinguish between approximation techniques for two types of problems: 1. Data fitting (Discrete approximation problem): Given a set of data points { ( x i , y i ) } n i = 0 , fi nd a reasonable function v ( x ) that fi ts the data points. If the data are accurate it might make sense to require that v ( x ) interpolate 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. 295 Downloaded 10/19/18 to 132.174.255.3. Redistribution subject to SIAM license or copyright; see
296 Chapter 10. Polynomial Interpolation 0 1 2 3 4 5 6 7 0 0.5 1 1.5 2 2.5 x v (a) Reasonable. 0 1 2 3 4 5 6 7 0 0.5 1 1.5 2 2.5 x v (b) Unreasonable. Figure 10.1. Different interpolating curves through the same set of points. 2. Approximating functions : For a complicated function f ( x ) (which may be given explicitly, or only implicitly), fi nd a simpler function v ( x ) that approximates f ( x ). For instance, suppose we need to quickly fi nd an approximate value for sin(1.2) (that s 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, π / 6, π / 4, π / 3, and π / 2: how can we use these to estimate sin(1.2)?