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lecture24 - CMPSC/MATH 451 Numerical Computations Lecture...

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CMPSC/MATH 451 Numerical Computations Lecture 24 October 17, 2011 Prof. Kamesh Madduri
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Class Overview: Interpolation Motivation Issues in Interpolation Polynomial Interpolation Basis functions Monomial basis Scaled monomial basis Horner’s nested evaluation 2
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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
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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
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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
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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?
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