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chap07_8up

# chap07_8up - Interpolation Polynomial Interpolation...

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Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Scientific Computing: An Introductory Survey Chapter 7 – Interpolation Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientific Computing 1 / 56 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Outline 1 Interpolation 2 Polynomial Interpolation 3 Piecewise Polynomial Interpolation Michael T. Heath Scientific Computing 2 / 56 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Motivation Choosing Interpolant Existence and Uniqueness Interpolation Basic interpolation problem: for given data ( t 1 , y 1 ) , ( t 2 , y 2 ) , . . . ( t m , y m ) with t 1 < t 2 < · · · < t m determine function f : R R such that f ( t i ) = y i , i = 1 , . . . , m f is interpolating function , or interpolant , for given data Additional data might be prescribed, such as slope of interpolant at given points Additional constraints might be imposed, such as smoothness, monotonicity, or convexity of interpolant f could be function of more than one variable, but we will consider only one-dimensional case Michael T. Heath Scientific Computing 3 / 56 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Motivation Choosing Interpolant Existence and Uniqueness Purposes for Interpolation Plotting smooth curve through discrete data points Reading between lines of table Differentiating or integrating tabular data Quick and easy evaluation of mathematical function Replacing complicated function by simple one Michael T. Heath Scientific Computing 4 / 56 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Motivation Choosing Interpolant Existence and Uniqueness Interpolation vs Approximation By definition, interpolating function fits given data points exactly Interpolation is inappropriate if data points subject to significant errors It is usually preferable to smooth noisy data, for example by least squares approximation Approximation is also more appropriate for special function libraries Michael T. Heath Scientific Computing 5 / 56 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Motivation Choosing Interpolant Existence and Uniqueness Issues in Interpolation Arbitrarily many functions interpolate given set of data points What form should interpolating function have? How should interpolant behave between data points? Should interpolant inherit properties of data, such as monotonicity, convexity, or periodicity? Are parameters that define interpolating function meaningful? If function and data are plotted, should results be visually pleasing? Michael T. Heath Scientific Computing 6 / 56 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Motivation Choosing Interpolant Existence and Uniqueness Choosing Interpolant Choice of function for interpolation based on How easy interpolating function is to work with

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