Newton's Backward Interpolation Formula with Example

Newton's Backward Interpolation explained with example. History of interpolation along with it's advantages and disadvantages. Applications of interpolation in computer sciences.

1. Numerical Analysis Newton’s Backward Interpolation Formula Presented By: Muhammad Usman Ikram (F2018266065 )

2. What is Interpolation ? 2

3. Interpolation is a type of estimation, a method of constructing new data points within the range of a discrete set of known data points.

4. Example Year (x) 1990 1995 2000 2005 2010 Sales (y) (in millions) 57 63 64 68 70 4

5. History 5

6. History 300 BC Babylonian astronomers used linear and higher-order interpolation to fill gaps in ephemerides of the sun, moon, and the then-known planets. 6

7. History 1000 A.D The Arabian scientist Al- Biruni writes his major work Al- Qanun'l - Mas'udi , in which he describes a method for second-order interpolation. 7

8. Types of Interpolation (For equally-spaced d ata) Newton Forward Interpolation Newton Backward Interpolation Stirling’s Interpolation Gauss’s Forward Interpolation Formula Gauss’s Backward Interpolation Formula 8

9. Newton’s Backward Interpolation 9

10. The Backward Difference Table 10 x y y x y

11. The Backward Difference Table 11 x y y x y

12. The Backward Difference Table 12 x y y x y

13. The Backward Difference Table 13 x y y x y =  

14. The Backward Difference Table 14 x y y x y =   =  

15. The Backward Difference Table 15 x y y x y =   =   =  

16. The Backward Difference Table 16 x y y x y =   =   =   =  

17. The Backward Difference Table 17 x y y x y =   =   =   =  

18. The Backward Difference Table 18 x y y x y =   =   =   =  

19. The Backward Difference Table 19 x y y x y =   =   =   =  

20. The Backward Difference Table 20 x y y x y =   =   =   =    

21. The Backward Difference Table 21 x y y x y =   =   =   =      

22. The Backward Difference Table 22 x y y x y =   =   =   =      

23. The Backward Difference Table 23 x y y x y =   =   =   =      

24. Newton Backward Interpolation Formula 24 f(x) = + P + + + + + _ _ _ _ _ _ _ _ _ _ _ Where   = last value in column x = last value in column y   = difference b/w values of x =  

25. Example Question 25

26. Question x 20 25 30 35 40 45 F(x) 354 332 291 260 231 204 26 Estimate f(42) for the following data

27. Step 1 27 x y y 20 25 30 35 40 45 x y 20 25 30 35 40 45

28. Step 1 28 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204

29. Step 1 29 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204  

30. Step 1 30 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204    

31. Step 1 31 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204      

32. Step 1 32 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204        

33. Step 1 33 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204          

34. Step 1 34 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204          

35. Step 1 35 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204          

36. Step 1 36 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204          

37. Step 1 37 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204          

38. Step 1 38 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204            

39. Step 1 39 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204              

40. Step 1 40 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204                

41. Step 1 41 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204                

42. Step 1 42 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204                

43. Step 1 43 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204                  

44. Step 1 44 x y y 20 354 25 332 30 291 35 260 40 231 45 204 x y 20 354 25 332 30 291 35 260 40 231 45 204                  

45. Step 2 45 x= 42 h= 5 = 45 = 204   = = = = - 0.6  

46. Formula 46 f(x) = + P + + + +  

47. Putting Values in Formula 47 f(42) = + (-0.6)(-27) + (2)+ (0) + (8) + (45) = f(42 ) = 216.68  

48. Advantages and Disadvantages 48

49. Advantages Helpful in estimation between given set of data. Simple and intuitive. Quick and easy. Helpful in images enhancing (image resizing) Helpful in Digital Signal Processing. 49

50. Disadvantages 50 Not always precise. Sometimes due to the fault in program used, image after resizing are blurry.

51. Applications in Computer Sciences 51

52. Applications in Computer Sciences Digital Image Processing Image interpolation works in two directions, and tries to achieve a best approximation of a pixel's intensity based on the values at surrounding pixels. 52 Original Image Enlarging Image to 183 % With Interpolation Without Interpolation

53. Applications in Computer Sciences Game Development and Graphics Linear interpolation (commonly known as 'lerp') is a really handy function for creative coding, game development and generative art . It ensures the smooth movement of objects in games. 53

54. Sources Cited Interpolation - https://en.wikipedia.org/wiki/Interpolation A Chronology of Interpolation: From Ancient Astronomy to Modern Signal and Image Processing - http:// bigwww.epfl.ch/publications/meijering0201.pdf Resizing Images - https://sisu.ut.ee/imageprocessing/book/3 Digital Image Interpolation - https :// www.cambridgeincolour.com/tutorials/image-interpolation.htm A Brief Introduction to Lerp - https:// www.gamedev.net/tutorials/programming/general-and-gameplay-programming/a-brief-introduction-to-lerp-r4954 Linear interpolation - https:// en.wikipedia.org/wiki/Linear_interpolation 54

55. 55 JazakAllah