Chapter 8 Interpolation (1)

1. Chapter 8Interpolation (1) Copyleft  2005 by MediaLab

2. Table of Contents • 8.1 Polynomial Interpolation • 8.1.1 Lagrange Interpolation • 8.1.2 Newton Interpolation • 8.1.3 Difficulties with Polynomial Interpolation • 8.2 Hermite Interpolation • 8.3 Rational-Function Interpolation Copyleft  2005 by MediaLab

3. Lagrange Interpolation Polynomials • Basic concept • The Lagrange interpolating polynomial is the polynomial of degree n-1 that passes through the n points. • Using given several point, we can find Lagrange interpolation polynomial. Copyleft  2005 by MediaLab

4. General Form of Lagrange • The general form of the polynomial is p(x) = L1y1 + L2y2 + … + Lnyn where the given points are (x1,y1), ….. , (xn,yn). • The equation of the line passing through two points (x1,y1) and (x2,y2) is • The equation of the parabola passing through three points (x1,y1), (x2,y2), and (x3,y3) is • http://math.fullerton.edu/mathews/n2003/lagrangepoly/LagrangePolyProof.pdf Copyleft  2005 by MediaLab

5. Example of Lagrange Interpolation • Example • Given points (x1,y1)=(-2,4), (x2,y2)=(0,2), (x3,y3)=(2,8) Copyleft  2005 by MediaLab

6. MATLAB Function for Lagrange Interpolation • We can represent the Lagrange polynomial with coefficient ck. • p(x)=c1N1+c2N2+ … +cnNn Copyleft  2005 by MediaLab

7. Higher Order Interpolation Polynomials • Higher order interpolation polynomials x = [ -2 -1 0 1 2 3 4], y = [ -15 0 3 0 -3 0 15] Copyleft  2005 by MediaLab

8. Review and Discussion • In Lagrange interpolation polynomial, it always go through given points. Think with equation below • The Lagrange form of polynomial is convenient when the same abscissas may occur in different applications. • It is less convenient than the Newton form when additional data points may be added to the problem. Copyleft  2005 by MediaLab

9. Newton Interpolation Polynomials • Newton form of the equation of a straight line passing through two points (x1, y1) and (x2, y2) is • Newton form of the equation of a parabola passing through three points (x1, y1), (x2, y2), and (x3, y3) is • the general form of the polynomial passing through n points (x1, y1), …,(xn, yn) is Copyleft  2005 by MediaLab

10. Newton Interpolation Polynomials (cont’d) • Substituting (x1, y1) into • Substituting (x2, y2) into • Substituting (x3, y3) into Copyleft  2005 by MediaLab

11. Newton Interpolation Parabola • Passing through the points (x1, y1)=(-2, 4), (x2, y2)=(0, 2), and (x3, y3)=(2, 8). • The equations is • Where the coefficients are • thus Copyleft  2005 by MediaLab

12. Newton Interpolation Parabola (cont’d) • Passing through the points (x1, y1)=(-2, 4), (x2, y2)=(0, 2), and (x3, y3)=(2, 8). Copyleft  2005 by MediaLab

13. Additional Data Points • We extend the previous example, adding the points (x4, y4) = (-1, -1) and (x5, y5) = (1, 1) • Divided-difference table becomes (with new entries shown in bold) • Newton interpolation polynomial is Copyleft  2005 by MediaLab

14. Higher Order Interpolation Polynomials • Consider again the data from Example 8.4 with Lagrange form. x = [ -2 -1 0 1 2 3 4 ], y = [ -15 0 3 0 -3 0 15] • Do it again with Newton form. That the polynomial is cubic is clear. Copyleft  2005 by MediaLab

15. Higher Order Interpolation Polynomials (cont’d) • If the y values are modified slightly, the divided-difference table shows the small contribution from the higher degree terms: Copyleft  2005 by MediaLab

16. MATLAB functions – Finding the coefficients >> x = [-2 -1 0 1 2 3 4]; >> y = [-15 0 3 0 -3 0 15]; >> Newton_Coef(x,y); d = 15 -6 1 0 0 0 3 -3 1 0 0 0 -3 0 1 0 0 0 -3 3 1 0 0 0 3 6 0 0 0 0 15 0 0 0 0 0 >> function a = Newton_Coef(x, y) n = length(x); %Calculate coeffiecients of Newton interpolating polynomial a(1) = y(1); for k=1 : n-1 d(k,1) = (y(k+1) - y(k))/(x(k+1) - x(k)); %1st divided diff end for j=2 : n-1 for k=1 : n-j d(k,j) = (d(k+1,j-1) - d(k,j-1))/(x(k+j) - x(k)); %jth divided diff end end d for j=2:n a(j) = d(1, j-1); end Copyleft  2005 by MediaLab

17. MATLAB functions – Evaluate the polynomials >> a = [-15 15 -6 1 0 0 0]; >> x = [-2 -1 0 1 2 3 4]; >> t = [0 1 2]; >> Newton_Eval(t, x, a); t = 0 1 2 p = 3 0 -3 function p = Newton_Eval(t,x,a) % t : input value of the polynomial (x) % x : x values of interpolating points % a : answer of previous MATLAB function, i.e, the coefficients of Newton polynomial. n = length(x); hold on; for i =1 : length(t) ddd(1) = 1; %Compute first term c(1) = a(1); for j=2 : n ddd(j) = (t(i) - x(j-1)).*ddd(j-1); % Compute jth term c(j) = a(j).*ddd(j); end; p(i) = sum(c); %plot(t(i),p(i)); grid on; end t p Copyleft  2005 by MediaLab

18. Humped and Flat Data • The data • x = [ -2 -1.5 -1 -0.5 0 – 0.5 1 1.5 2] • y = [ 0 0 0 0.87 1 0.87 0 0 0] illustrate the difficulty with using higher order polynomials to interpolate a moderately large number of points. Copyleft  2005 by MediaLab

19. Noisy Straight Line • The data • x = [ 0.00 0.20 0.80 1.00 1.20 1.90 2.00 2.10 2.95 3.00] • y = [ 0.01 0.22 0.76 1.03 1.18 1.94 2.01 2.08 2.90 2.95] • Not well suited with noisy straight line. Copyleft  2005 by MediaLab

20. Runge Function • The function is a famous example of the fact that polynomial interpolation does not produce a good approximation for some functions and that using more function values (at evenly spaced x values) does not necessarily improve the situation. • Example 1. • x = [ -1 -0.5 0.0 0.5 1.0 ] • y = [0.0385 0.1379 1.0000 0.1379 0.0385] Copyleft  2005 by MediaLab

21. Runge Function (cont’d) • Example 2. • x = [-1.000 -0.750 -0.500 -0.250 0.000 0.250 0.500 0.750 1.000 ] • y = [0.0385 0.0664 0.138 0.3902 1.000 0.3902 0.138 0.0664 0.0385] • The interpolation polynomial overshoots the true polynomial muchmore severely than the polynomial formed by using only five points. Copyleft  2005 by MediaLab

22. 8.2Hermite Interpolation Copyleft  2005 by MediaLab

23. Hermite Interpolation • Hermite interpolation allows us to find a ploynomial that matched both function value and some of the derivative values Copyleft  2005 by MediaLab

24. More data for Product concentration Copyleft  2005 by MediaLab

25. MATLAB Code for Hermite interpolation Copyleft  2005 by MediaLab

26. Difficult Data • As with lower order polynomial interpolation, trying to interpolate in humped and flat regions cause overshoots. Copyleft  2005 by MediaLab

27. 8.3Rational-Function Interpolation Copyleft  2005 by MediaLab

28. Rational-Function Interpolation • Why use rational-function interpolation? • Some functions are not well approximated by polynomials.(runge-function) • but are well approximated by rational functions, that is quotients of polynomials. Copyleft  2005 by MediaLab

29. Bulirsch-Stoer algorithm • Bulirsch-Stoer algorithm • The approach is recursive, based on tabulated data(in a manner similar to that for the Newton form of polynomial interpolation). • Given a set of m+1 data points (x1,y1), … , (xm+1, ym+1), we seek an interpolation function of the form <Bulirsch-Stoer algorithm general pattern> The proof is in J.Stoer and R.Bulirsch, 'Introduction to Numerical Analysis' Copyleft  2005 by MediaLab

30. Bulirsch-Stoer algorithm(cont’d) • Bulirsch-Stoer method for three data points Copyleft  2005 by MediaLab

31. Third stage Second stage data First stage x1 y1 R1= y1 x2 y2 R2= y2 x3 y3 R3=y3 x4 y4 R4=y4 x5 y5 R5=y5 Bulirsch-Stoer algorithm(cont’d) • Bulirsch-Stoer method for five data points Copyleft  2005 by MediaLab

32. Fifth stage Forth stage Bulirsch-Stoer algorithm(cont’d) Copyleft  2005 by MediaLab

33. Bulirsch-Stoer Rational-Function(cont’d) Copyleft  2005 by MediaLab

34. Example 8.13 rational-function interpolation data points: x = [-1 -0.5 0.0 0.5 1.0] y = [0.0385 0.1379 1.0000 0.1379 0.0385] Copyleft  2005 by MediaLab

35. Example 8.13 rational-function interpolation(cont’d) Copyleft  2005 by MediaLab