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ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 22 CURVE FITTING Chapter 18 Function Interpolation and Approximation

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LAST TIME Curve Fitting Often we are faced with the problem… what value of y corresponds to x=0.935?

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LAST TIME Curve Fitting Question 1 : Is it possible to find a simple and convenient formula that reproduces the points exactly? e.g. Straight Line ? …or smooth line ? …or some other representation? Interpolation

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LAST TIME Curve Fitting Question 2 : Is it possible to find a simple and convenient formula that represents data approximately ? e.g. Best Fit ? Approximation

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LAST TIME Function Interpolation Linear Interpolation Simplest Form is to connect data points with a straight line (1 st Order Polynomial) x

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LAST TIME Linear Interpolation Slope of Line 1 st DIVIDED DIFFERENCE f [x i+1,x i ] First order interpolating polynomial

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LAST TIME Function Interpolation Higher Order Interpolation Newton Interpolating Polynomials x

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Last Time General Form of Newton’s Interpolating Polynomials

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Divided Differences Given a Set of Data xf(x) xoxo f(x o ) x1x1 f(x 1 ) x2x2 f(x 2 ) …… xnxn f(x n )

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Last Time Example

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Lagrange Interpolating Polynomials Reformulation of Newton’s Polynomials Avoid Calculation of Divided Differences xf(x) xoxo f(x o ) x1x1 f(x 1 ) x2x2 f(x 2 ) …… xnxn f(x n )

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Lagrange Interpolating Polynomial Cardinal Functions: Product of n-1 linear factors Skip x i Property:

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Example Write cardinal functions and give the Lagrange interpolating polynomial for

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Errors in Polynomial Interpolation It is expected that as number of nodes increases, error decreases, HOWEVER…. At all interpolation nodes x i Error=0 At all intermediate points Error: f(x)-f n-1 (x) f(x)

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Errors in Polynomial Interpolation Beware of Oscillations…. For Example: Consider f(x)=(1+x 2 ) -1 evaluated at 9 points in [-5,5] And corresponding p 8 (x) Lagrange Interpolating Polynomial P 8 (x) f(x)

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