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Numerical Analysis – Interpolation

Numerical Analysis – Interpolation. Hanyang University Jong-Il Park. Fitting. Exact fit Interpolation Extrapolation Approximate fit. Extrapolation. x. Interpolation. x. x. x. x. Weierstrass Approximation Theorem. Approximation error. Better approximation.

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Numerical Analysis – Interpolation

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  1. Numerical Analysis –Interpolation Hanyang University Jong-Il Park

  2. Fitting • Exact fit • Interpolation • Extrapolation • Approximate fit Extrapolation x Interpolation x x x x

  3. Weierstrass Approximation Theorem

  4. Approximation error Better approximation

  5. Lagrange Interpolating Polynomial

  6. Illustration of Lagrange polynomial • Unique • Too much complex

  7. Error analysis for intpl. polynml(I)

  8. Error analysis for intpl. polynml(II)

  9. Differences • Difference • Forward difference : • Backward difference : • Central difference : f

  10. Divided Differences ; 1st order divided difference ; 2nd order divided difference

  11. N-th divided difference

  12. Newton’s Intpl. Polynomials(I)

  13. Newton’s Intpl. Polynomials(II)

  14. Newton’s Forward Difference Interpolating Polynomials(I) • Equal Interval h • Derivation n=1 n=2

  15. Newton’s Forward Difference Interpolating Polynomials(II) Generalization • Error Analysis Binomial coef.

  16. 1 1 Intpl. of Multivariate Function • Successive univariate polynomial • Direct mutivariate polynomial 2 direct multivariate Successive univariate

  17. Inverse Interpolation = finding x(f) • Utilization of Newton’s polynomial Solve for x 1st approximation 2nd approximation Repeat until a convergence

  18. spline polynomial Spline Interpolation • Why spline? Linear spline Quadratic spline Cubic spline Continuity • Good approximation !! • Moderate complexity !!

  19. 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

  20. Cubic spline interpolation(II) • Determining boundary condition Method 1 : Method 2 : Method 3 :

  21. Eg. CG modeling Non-Uniform Rational B-Spline

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