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A History of Numerical Analysis Ideas

A History of Numerical Analysis Ideas. Prepared for CS 378 History of Computing October 14, 2003. Alan Kaylor Cline Department of Computer Sciences The University of Texas at Austin. What is Different in Numerical Computing?. What is Different in Numerical Computing?. Well, it’s numbers.

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A History of Numerical Analysis Ideas

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

  2. What is Different in Numerical Computing?

  3. What is Different in Numerical Computing? 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. Where’s the problem? 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 3.136548490545940 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 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 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. Not so boring if the result of this computation affects • 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. So what are the common problems of numerical analysis? 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. Didn’t we study that stuff in math classes? 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. Why were computers used primarily for numerical problems initially? • 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. What were the major computing ideas that arose in numerical analysis? • 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 true operation input output

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

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

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