 Numerical Analysis –Interpolation

Presentation on theme: "Numerical Analysis –Interpolation"— Presentation transcript:

Numerical Analysis –Interpolation
Hanyang University Jong-Il Park

Fitting Exact fit Approximate fit Interpolation Extrapolation

Weierstrass Approximation Theorem

Approximation error Better approximation

Lagrange Interpolating Polynomial

Illustration of Lagrange polynomial
Unique 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 Function
Successive 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