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Numerical Analysis –InterpolationHanyang University Jong-Il Park
Fitting Exact fit Approximate fit Interpolation Extrapolation
Weierstrass Approximation Theorem
Approximation error Better approximation
Lagrange Interpolating Polynomial
Illustration of Lagrange polynomialUnique Too much complex
Error analysis for intpl. polynml(I)
Error analysis for intpl. polynml(II)
Differences f Difference Forward difference : Backward difference :Central difference : f
Divided Differences ; 1st order divided difference; 2nd order divided difference
N-th divided difference
Newton’s Intpl. Polynomials(I)
Newton’s Intpl. Polynomials(II)
Newton’s Forward Difference Interpolating Polynomials(I)Equal Interval h Derivation n=1 n=2
Newton’s Forward Difference Interpolating Polynomials(II)Generalization Error Analysis Binomial coef.
Intpl. of Multivariate FunctionSuccessive univariate polynomial Direct mutivariate polynomial 2 1 1 Successive univariate direct multivariate
Inverse Interpolation= finding x(f) Utilization of Newton’s polynomial Solve for x 1st approximation 2nd approximation Repeat until a convergence
Spline Interpolation Why spline? Good approximation !!Linear spline Quadratic spline Cubic spline spline polynomial Continuity Good approximation !! Moderate complexity !!
Cubic spline interpolation(I)Cubic Spline Interpolation at an interval 4 unknowns for each interval 4n unknowns for n intervals Conditions 1) 2) 3) continuity of f’ 4) continuity of f’’ n n n-1 n-1
Cubic spline interpolation(II)Determining boundary condition Method 1 : Method 2 : Method 3 :
Eg. CG modeling Non-Uniform Rational B-Spline
Interpolation A standard idea in interpolation now is to find a polynomial pn(x) of degree n (or less) that assumes the given values; thus (1) We call.
CSE 330: Numerical Methods
Numerical Methods. Polynomial interpolation involves finding a polynomial of order n that passes through the n+1 points. Several methods to obtain.
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1 Chapter 4 Interpolation and Approximation Lagrange Interpolation The basic interpolation problem can be posed in one of two ways: The basic interpolation.
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Curve-Fitting Polynomial Interpolation
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ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 22 CURVE FITTING Chapter 18 Function Interpolation and Approximation.
Bezier and Spline Curves and Surfaces Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico.
Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Chapter 181 Interpolation Chapter 18 Estimation of intermediate.
ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 23 CURVE FITTING Chapter 18 Function Interpolation and Approximation.
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