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

2. Fitting Exact fit Approximate fit Interpolation Extrapolation

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

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

18. Spline Interpolation Why spline? Good approximation !!
Linear spline
Quadratic spline
Cubic spline
spline
polynomial
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