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Unformatted text preview: BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Interpolation BME 5020 Computer and Mathematical Application in Bioengineering © 2006 Jingwen Hu October 10, 2006 BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Interpolation and Extrapolation • Often scientific experimentation or numerical computation results in values for a function, f(x), only at discrete points • These values of f(x) may be spaced either evenly or unevenly along x • Interpolation is to find the value of f(x) between the known points • Extrapolation is to find find the value of f(x) outside the known points BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Interpolation • Estimation of intermediate values between precise data points. The most common method is: • Although there is one and only one n thorder polynomial that fits n+1 points, there are a variety of mathematical formats in which this polynomial can be expressed: • The Newton polynomial • The Lagrange polynomial n n x a x a x a a x f + + + + = L 2 2 1 ) ( BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Interpolation BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Newton’s DividedDifference Interpolating Polynomials Linear Interpolation/ • Is the simplest form of interpolation, connecting two data points with a straight line. • f 1 (x) designates that this is a firstorder interpolating polynomial. ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 1 1 x x x x x f x f x f x f x x x f x f x x x f x f − − − + = − − = − − Linearinterpolation formula Slope and a finite divided difference approximation to 1 st derivative BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Newton’s DividedDifference Interpolating Polynomials BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Quadratic Interpolation/ • If three data points are available, the estimate is improved by introducing some curvature into the line connecting the points. • A simple procedure can be used to determine the values of the coefficients. Newton’s DividedDifference Interpolating Polynomials ) )( ( ) ( ) ( 1 2 1 2 x x x x b x x b b x f − − + − + = 2 1 1 1 2 1 2 2 2 1 1 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( x x x x x f x f x x x f x f b x x x x x f x f b x x x f b x x − − − − − − = = − − = = = = BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Newton’s DividedDifference Interpolating Polynomials • General Form of Newton’s Interpolating Polynomials/ 2 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 2 1 1 ] , , , [ ] , , , [ ] , , , , [ ] , [ ] , [ ] , , [ ) ( ) ( ] , [ ] , , , , [ ] , , [ ] , [ ) ( ] , , , [ ) ( ) )( ( ] , , [ ) )( ( ] , [ ) ( ) ( ) ( x x x x x f x x x f x x x x f x x x x f x x f x x x f x x x f x f x x f x x x x f b x x x...
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 Fall '06
 King
 Biomedical Engineering

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