1. Numerical Analysis
Lecture 24

2. Chapter 5 Interpolation

3. Finite Difference Operators Newton’s Forward Difference
Finite Difference Operators Newton’s Forward Difference Interpolation Formula Newton’s Backward Difference Interpolation Formula Lagrange’s Interpolation Formula Divided Differences Interpolation in Two Dimensions Cubic Spline Interpolation

4. Newton’s Forward Difference Interpolation Formula

5. The Newton’s forward difference formula for interpolation, which gives the value of f (x0 + ph) in terms of f (x0) and its leading differences.

6. An alternate expression is

7. NEWTON’S BACKWARD DIFFERENCE INTERPOLATION FORMULA

8. The formula is,

9. Alternatively, this formula can also be written as
Here

10. LAGRANGE’S INTERPOLATION FORMULA

11. Newton’s interpolation formulae can be used only when the values of the independent variable x are equally spaced. Also the differences of y must ultimately become small.

12. If the values of the independent variable are not given at equidistant intervals, then we have the Lagrange formula

13. Here the polynomial is of the form
or in the form

14. Here, the coefficients ak are so chosen as to satisfy this equation by the (n + 1) pairs (xi, yi). Thus we get
Therefore,

15. Similarly
and

16. Substituting the values of a0, a1, …, an we get
The Lagrange’s formula for interpolation

17. This formula can be used whether the values x0, x2, …, xn are equally spaced or not. Alternatively, this can also be written in compact form as

18. Thus introducing Kronecker delta notation
Where,
We can easily observe that,
and
Thus introducing Kronecker delta notation

19. Further, if we introduce the notation
That is is a product of (n + 1) factors. Clearly, its derivative contains a sum of (n + 1) terms in each of which one of the factors of will be absent.

20. We also define,
which is same as except that the factor (x–xk) is absent. Then
But, when x = xk, all terms in the above sum vanishes except Pk(xk)

21. Hence,

22. Finally, the Lagrange’s interpolation polynomial of degree n can be written as

23. DIVIDED DIFFERENCES

24. Let us assume that the function y = f (x) is known for several values of x, (xi, yi), for i=0,1,..n. The divided differences of orders 0, 1, 2, …, n are now defined recursively as:

25. is the zero-th order divided difference

26. The first order divided difference is defined as

27. Similarly, the higher order divided differences are defined in terms of lower order divided differences by the relations of the form

28. Generally

29. Standard format of the Divided Differences

30. We can easily verify that the divided difference is a symmetric function of its arguments. That is,

31. Now,

32. Therefore

33. This is symmetric form, hence suggests the general result as

34. NEWTON’S DIVIDED DIFFERENCE INTERPOLATION FORMULA

35. Let y = f (x) be a function which takes values y0, y1, …, yn corresponding to x = xi, i = 0, 1,…, n. We choose an interpolating polynomial, interpolating at x = xi, i = 0, 1, …, n in the following form

36. Here, the coefficients ak are so chosen as to satisfy above equation by the (n + 1) pairs (xi, yi). Thus, we have

37. The coefficients
a0, a1, …, an can be easily obtained from the above system of equations, as they form a lower triangular matrix.

38. The first equation gives
The second equation gives

39. Third equation yields
which can be rewritten as
that is

40. Thus, in terms of second order divided differences, we have
Similarly, we can show that

41. Newton’s divided difference interpolation formula can be written as

42. Newton’s divided differences can also be expressed in terms of forward, backward and central differences. They can be easily derived

43. Assuming equi-spaced values of abscissa, we have

44. By induction, we can in general arrive at the result

45. Similarly

46. In general, we have

47. Also, in terms of central differences, we have

48. In general, we have the following pattern

49. Numerical Analysis
Lecture 24