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Advanced Numerical Analysis: Interpolation and Extrapolation Techniques at Hanyang University

This comprehensive course covers Exact Fit, Interpolation, Extrapolation, Approximation, Weierstrass Approximation Theorem, Error Analysis, Lagrange Interpolating Polynomial, Divided Differences, Newton’s Interpolating Polynomials, Forward Difference Interpolating Polynomials, Error Analysis, Multivariate Function Interpolation, Inverse Interpolation, Spline Interpolation, and Cubic Spline Interpolation. Dive into the world of numerical analysis and sharpen your skills in fitting functions accurately.

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Advanced Numerical Analysis: Interpolation and Extrapolation Techniques at Hanyang University

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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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