1. Scientific Computing: Does Anyone Care? Alan Kaylor Cline Department of Computer Sciences The University of Texas at Austin October 30, 2008 ACM 101 Lecture Series

2. Maximum volume sphere to fit between the wall, floor, and cylinder

3. Maximum volume sphere to fit between the wall, floor, and cylinder

4. Remember the old algebraic trick where expressions of the form can be multiplied by to rationalize the denominator and get

5. … but also expressions of the form can be multiplied by to rationalize the NUMERATOR and get

6. So can be written in many ways: 1. 2. 3. 4. 5.

7. Errors associated with replacing with 1.41 (2.98% error) =1.4121356237…

8. Errors associated with replacing with 1.41 (2.98% error) Correct value is 5.05063388… x10 -3

9. Errors associated with replacing with 1.41 (2.98% error) Correct value is 5.05063388… x10 -3

10. Errors associated with replacing with 1.41 (2.98% error) Correct value is 5.05063388… x10 -3

11. Boring… Is that all there is to scientific computing?

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

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

14. 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

15. 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 in which you live

16. 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 in which you live The path of a missile that isn’t intended to strike you

17. But this shouldn’t be boring:

18. The $25,000,000,000 Eigenvector

19. The Page Rank Problem 1.Label the sites on the World Wide Web 1,2,…,n. 2.Build an n x n matrix M so that M i,j is 1 if web site j links to web site i (or if web site j links to no web site) and is zero otherwise. 3.Let n x n matrix A be determined from M by dividing each column by the number of ones in the column. 4.Let E be an n x n matrix of all ones. 5.For some value of p, set G = p M + (1-p)/n E Solve G x = x Assign web site i page rank k if x i is the kth largest element of x

20. The steps of scientific computation

21. 1.Observation of nature

22. The steps of scientific computation 1.Observation of nature 2.Construction of a mathematical model

23. The steps of scientific computation 1.Observation of nature 2.Construction of a mathematical model 3.Selection of a computational method to solve the mathematical formulation

24. The steps of scientific computation 1.Observation of nature 2.Construction of a mathematical model 3.Selection of a computational method to solve the mathematical formulation 4.Program the method

25. The steps of scientific computation 1.Observation of nature 2.Construction of a mathematical model 3.Selection of a computational method to solve the mathematical formulation 4.Program the method 5.Execute the program and display the results

26. The steps of scientific computation 1.Observation of nature 2.Construction of a mathematical model 3.Selection of a computational method to solve the mathematical formulation 4.Program the method 5.Execute the program and display the results 6.Interpret the results and compare to observations

27. The steps of scientific computation 1.Observation of nature 2.Construction of a mathematical model 3.Selection of a computational method to solve the mathematical formulation 4.Program the method 5.Execute the program and display the results 6.Interpret the results and compare to observations 7.(Possibly) Refine

28. So what is scientific computing? The study of computational methods for solving common scientific and engineering problems that are both accurate and efficient.

29. So what are the common problems of these “common problem” areas?

30. Linear Equations

31. So what are the common problems of these “common problem” areas? Linear Equations Nonlinear equations - single and systems

32. So what are the common problems of these “common problem” areas? Linear Equations Nonlinear equations - single and systems Optimization

33. So what are the common problems of these “common problem” areas? Linear Equations Nonlinear equations - single and systems Optimization Data Fitting - interpolation and approximation

34. So what are the common problems of these “common problem” areas? Linear Equations Nonlinear equations - single and systems Optimization Data Fitting - interpolation and approximation Integration

35. So what are the common problems of these “common problem” 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 Sphere Example shows, math classes are just the beginning

38. What’s a great computing idea that arose in scientific computing?

39. Let’s look at integration (we certainly spent a lot of time on that in calculus classes)

40. Let’s look at integration (we certainly spent a lot of time on that in calculus classes) 1. Almost all problems do not have antiderivatives expressible as “simple” functions. For example,

41. Let’s look at integration (we certainly spent a lot of time on that in calculus classes) 1. Almost all problems do not have antiderivatives expressible as “simple” functions. For example, 2.Many functions are not known as algebraic expressions but as blackboxes f xf(x)

42. Let’s look at integration (we certainly spent a lot of time on that in calculus classes) 1. Almost all problems do not have antiderivatives expressible as “simple” functions. For example, 2.Many functions are not known as algebraic expressions but as blackboxes 3.Even when there is an simple antiderivative known, using it may incur large error. f xf(x)

43. An Example:

44. for p = -1.00001 with six decimal digit arithmetic

45. An Example: for p = -1.00001 with six decimal digit arithmetic

46. An Example: for p = -1.00001 with six decimal digit arithmetic Correct answer is.693144778… This is.989% error.

47. OK, SmartyPants can you do better?

48. Backward error analysis What’s a great computing idea that arose in scientific computing?

49. Backward error analysis What’s a great computing idea that arose in scientific computing? It allows us to analyze errors without worrying about the loss of associativity, commativity, and distributivity.

50. What is backward error analysis ? input output

51. input output true operation What is backward error analysis ?

52. input output true operation approximate operation What is backward error analysis ?

53. input output true operation approximate operation error What is backward error analysis ?

54. input output true operation approximate operation What is backward error analysis ?

55. input output true operation approximate operation backward error What is backward error analysis ?