An Introduction to the Quality of Computed Solutions

1. An Introduction to the Quality of Computed Solutions Sven Hammarling NAG Ltd, Oxford sven@nag.co.uk

2. Plan of Talk • Introduction • Floating point numbers and IEEE arithmetic • Condition, stability and error analysis with examples • Implications for software. • LAPACK and NAG • Other approaches • Summary

3. Introduction to NAG

4. NAG History • 1970 - Nottingham Algorithms Group • 1971 - Mark 1 NAG Library • 1973 - NAG moved to Oxford, renamed Numerical Algorithms Group • 1976 - NAG Ltd, a non-profit company • 1978 - NAG Inc established in USA • 1980 - NAG Ltd financially self-sufficient • 1990 - NAG GmbH established • 1998 - Nihon NAG KK established

5. What does non-profit mean? No shareholders, no owner surplus is re-invested in the Company

6. How many people work at NAG? Distributors worldwide

7. NAG Products and Services • Product Lines • Numerical Libraries • Statistical Systems • NAGWare: Compiler and tools • Visualization and Graphics • PDE Solutions • Consultancy • Customer Support www.nag.co.uk or www.nag.com

8. NAG Numerical Libraries F77 FL90Plus F90 C Parallel SMP MPI

9. Quality of Computed Solutions

10. Quality of Computed Solutions The quality of computed solutions is concerned with assessing how good a computed solution is in some appropriate measure

11. Software Quality Quality software should implement reliable algorithms and should provide measures of solution quality

12. Floating Point Numbers Floating point numbers are a subset of the real numbers that can be conveniently represented in the finite word length of a computer, without unduly restricting the range of numbers represented For example, the IEEE standard uses 64 bits to represent double precision numbers in the approximate range

13. Floating Point Numbers - Representation

14. Floating Point Numbers - Example

15. Floating Point Numbers - Example (cont’d) 0 1 2 3

16. IEEE Arithmetic Standard ANSI/IEEE Standard 754-1985 is a standard for binary arithmetic (b = 2). The standard specifies: • Floating point number formats • Results of the basic floating point operations • Rounding modes • Signed zero, infinity ( ) and not-a-number (NaN) • Floating point exceptions and their handling • Conversion between formats Most modern machines use IEEE arithmetic.

17. IEEE Arithmetic Formats

18. Why Worry about Computed Solutions? • Tacoma bridge collapse • North Sea oil rig collapse • Vancouver stock exchange index, Jan 1982 to Nov 1983 • Ariane 5 rocket, flight 501, failure, 4 June 1996 • Patriot missile, 25 Feb 1991 • Auckland Bridge • London millennium bridge

19. Tacoma Bridge

20. Auckland Bridge, 1975

21. Millennium Bridge, 2000

22. Web Sites • Disasters attributable to bad numerical computing:http://www.math.psu.edu/dna/disasters/ • Numerical problems: RISKS-LIST: http://catless.ncl.ac.uk/Risks/ • London millennium bridge: http://www.arup.com/MillenniumBridge/

23. Example - Means

24. Example - Sample Variance

25. Excel Example:Standard Deviation

26. Overflow/Underflow Example: Modulus of a Complex Number

27. Excel Example:Standard Deviation - Overflow

28. Stability The stability of a method for solving a problem is concerned with the sensitivity of the method to (rounding) errors in the solution process A method that guarantees as accurate a solution as the data warrants is said to be stable, otherwise the method is unstable

29. Condition The condition of a problem is concerned with the sensitivity of the problem to perturbations in the data A problem is ill-conditioned if small changes in the data cause relatively large changes in the solution. Otherwise a problem is well-conditioned

30. Condition Examples

33. Condition Examples (cont’d)

35. Condition Examples (cont’d)

37. Condition Number for Linear Equations

38. Condition Number Example

39. Stability Examples

40. Stability Examples (Contd) recursive.m

41. Stability Examples (Contd)

42. Stability Examples (cont’d) brecursive.m

43. Error Analysis Error analysis is concerned with establishing whether or not an algorithm is stable for the problem in hand A forward error analysis is concerned with how close the computed solution is to the exact solution A backward error analysis is concerned with how well the computed solution satisfies the problem to be solved

44. Example

45. The Purpose of Error Analysis “The clear identification of the factors determining the stability of an algorithm soon led to the development of better algorithms. The proper understanding of inverse iteration for eigenvectors and the development of the QR algorithm by Francis are the crowning achievements of this line of research. “For me, then, the primary purpose of the rounding error analysis was insight.” Wilkinson, 1986 Bulletin of the IMA, Vol 22, p197

47. Backward Error and Perturbation Analysis

48. Condition and Error Analysis

49. LAPACK Linear Algebra PACKage for high-performance computers • Systems of linear equations • Linear least squares problems • Eigenvalue and singular value problems, including generalized problems • Matrix factorizations • Condition and error estimates • The BLAS as a portability layer Dense and banded linear algebra for Shared Memory