1. A History of Numerical Analysis Ideas Alan Kaylor Cline Department of Computer Sciences The University of Texas at Austin Prepared for CS 378 History of Computing October 14, 2003

2. What is Different in Numerical Computing?

3. Well, it’s numbers

4. Scientific Computing vs. Numerical Analysis

5. Your Original CS Department 1966

6. Your Original CS Department 1966 10 faculty with 2 numerical analysts

7. Is there anything special with numerical computing?

8. A Small Example A computation of π

9. Simple iteration:

10. 2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545941 6 3.140331156954739 7 3.141277250932757 8 3.141513801144145 9 3.141572940367883 10 3.141587725279961 11 3.141591421504635 12 3.141592345611077 13 3.141592576545004 14 3.141592633463248 15 3.141592654807589 16 3.141592645321215 17 3.141592607375720 18 3.141592910939673 19 3.141594125195191 20 3.141596553704820 21 3.141596553704820 22 3.141674265021758 23 3.141829681889202 24 3.142451272494134 25 3.142451272494134 26 3.162277660168380 27 3.162277660168380 28 3.464101615137754 29 4.000000000000000 30 0.000000000000000 31 0.000000000000000 Result of 15 digit computation Red digits are correct White and green digits are incorrect

11. 2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545941 6 3.140331156954739 7 3.141277250932757 8 3.141513801144145 9 3.141572940367883 10 3.141587725279961 11 3.141591421504635 12 3.141592345611077 13 3.141592576545004 14 3.141592633463248 15 3.141592654807589 16 3.141592645321215 17 3.141592607375720 18 3.141592910939673 19 3.141594125195191 20 3.141596553704820 21 3.141596553704820 22 3.141674265021758 23 3.141829681889202 24 3.142451272494134 25 3.142451272494134 26 3.162277660168380 27 3.162277660168380 28 3.464101615137754 29 4.000000000000000 30 0.000000000000000 31 0.000000000000000... Result of 15 digit computation Red digits are correct White and green digits are incorrect π = 0 ?

12. Where’s the problem?

13. is calculated as zero

14. Let’s replace with the algebraically identical expression

15. New iteration: results in …

16. 2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545941 6 3.140331156954739 7 3.141277250932757 8 3.141513801144145 9 3.141572940367883 10 3.141587725279961 11 3.141591421504635 12 3.141592345611077 13 3.141592576545004 14 3.141592633463248 15 3.141592654807589 16 3.141592645321215 17 3.141592607375720 18 3.141592910939673 19 3.141594125195191 20 3.141596553704820 21 3.141596553704820 22 3.141674265021758 23 3.141829681889202 24 3.142451272494134 25 3.142451272494134 26 3.162277660168380 27 3.162277660168380 28 3.464101615137754 29 4.000000000000000 30 0.000000000000000 31 0.000000000000000 2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545940 6 3.140331156954753 7 3.141277250932773 8 3.141513801144301 9 3.141572940367091 10 3.141587725277160 11 3.141591421511200 12 3.141592345570118 13 3.141592576584872 14 3.141592634338563 15 3.141592648776985 16 3.141592652386591 17 3.141592653288992 18 3.141592653514593 19 3.141592653570993 20 3.141592653585093 21 3.141592653588618 22 3.141592653589499 23 3.141592653589719 24 3.141592653589774 25 3.141592653589788 26 3.141592653589792 27 3.141592653589793 28 3.141592653589793 29 3.141592653589793 30 3.141592653589793 31 3.141592653589793 π correct to all digits

17. Boring… Is that all there is to numerical analysis?

18. Not so boring if the result of this computation affects

19. The ability of the next plane you fly to stay in the air

20. Not so boring if the result of this computation affects The ability of the next plane you fly to stay in the air The integrity of the next bridge you cross

21. Not so boring if the result of this computation affects The ability of the next plane you fly to stay in the air The integrity of the next bridge you cross The state of the economy on which you live

22. Not so boring if the result of this computation affects The ability of the next plane you fly to stay in the air The integrity of the next bridge you cross The state of the economy on which you live The path of a missile that isn’t intended to strike you

23. So what are the common problems of numerical analysis?

24. Application areas:

25. So what are the common problems of numerical analysis? Application areas: Petroleum modeling

26. So what are the common problems of numerical analysis? Application areas: Petroleum modeling Atomic energy – including weapons

27. So what are the common problems of numerical analysis? Application areas: Petroleum modeling Atomic energy – including weapons Weather modeling

28. So what are the common problems of numerical analysis? Application areas: Petroleum modeling Atomic energy – including weapons Weather modeling Other modeling such as aircraft and automobile

29. So what are the common problems of numerical analysis? Algorithm areas:

30. So what are the common problems of numerical analysis? Algorithm areas: Linear Equations

31. So what are the common problems of numerical analysis? Algorithm areas: Linear Equations Nonlinear equations - single and systems

32. So what are the common problems of numerical analysis? Algorithm areas: Linear Equations Nonlinear equations - single and systems Optimization

33. So what are the common problems of numerical analysis? Algorithm areas: Linear Equations Nonlinear equations - single and systems Optimization Data Fitting - interpolation and approximation

34. So what are the common problems of numerical analysis? Algorithm areas: Linear Equations Nonlinear equations - single and systems Optimization Data Fitting - interpolation and approximation Integration

35. So what are the common problems of numerical analysis? Algorithm areas: Linear Equations Nonlinear equations - single and systems Optimization Data Fitting - interpolation and approximation Integration Differential Equations - ordinary and partial

36. Didn’t we study that stuff in math classes?

37. Yes, but as the Pi Example shows, math classes are just the beginning

38. Let’s get back to history

39. Why were computers used primarily for numerical problems initially?

40. Mathematicians and engineers designed them

41. Why were computers used primarily for numerical problems initially? Mathematicians and engineers designed them A history of algorithms in that area

42. Why were computers used primarily for numerical problems initially? Mathematicians and engineers designed them A history of algorithms in that area Immediate war-time and post-war-time applications

43. Why were computers used primarily for numerical problems initially? Mathematicians and engineers designed them A history of algorithms in that area Immediate war-time and post-war-time applications Applications did not depend upon having a large number of computers

44. Why were computers used primarily for numerical problems initially? Mathematicians and engineers designed them A history of algorithms in that area Immediate war-time and post-war-time applications Applications did not depend upon having a large number of computers However, there were non-numerical examples ENIGMA

45. What were the major computing ideas that arose in numerical analysis?

46. Backward error analysis

47. What were the major computing ideas that arose in numerical analysis? Backward error analysis input output

48. What were the major computing ideas that arose in numerical analysis? Backward error analysis input output true operation

49. What were the major computing ideas that arose in numerical analysis? Backward error analysis input output true operation approximate operation

50. What were the major computing ideas that arose in numerical analysis? Backward error analysis input output true operation approximate operation error

51. What were the major computing ideas that arose in numerical analysis? Backward error analysis input output true operation approximate operation

52. What were the major computing ideas that arose in numerical analysis? Backward error analysis input output true operation approximate operation backward error

53. What were the major computing ideas that arose in numerical analysis? Backward error analysis FORTRAN

54. What were the major computing ideas that arose in numerical analysis? Backward error analysis FORTRAN Mathematical based Computationally Efficient Portable Standardized – 3 times

55. What were the major computing ideas that arose in numerical analysis? Backward error analysis FORTRAN Mathematical software packages

56. What were the major computing ideas that arose in numerical analysis? Backward error analysis FORTRAN Mathematical software packages IMSL Eispack Linpack

57. What were the major computing ideas that arose in numerical analysis? Backward error analysis FORTRAN Mathematical software packages NANET

58. What were the major computing ideas that arose in numerical analysis? Backward error analysis FORTRAN Mathematical software packages NANET Weekly information about people, problems, and papers Software repository

59. What were the major computing ideas that arose in numerical analysis? Backward error analysis FORTRAN Mathematical software packages NANET Matlab, Mathematica

60. What were the major computing ideas that arose in numerical analysis? Backward error analysis FORTRAN Mathematical software packages NANET Matlab, Mathematica Scientific computing environments

61. What were the major computing ideas that arose in numerical analysis? Backward error analysis FORTRAN Mathematical software packages NANET Matlab, Mathematica Super computers - Parallelism