1. ICNAAM, September 2008, Kos1 Numerical Analysis in the Digital Library of Mathematical Functions Dan Lozier Math and Computational Sciences Division National Institute of Standards and Technology Gaithersburg, MD 20899-8910 USA http://dlmf.nist.gov daniel.lozier@nist.gov

2. ICNAAM, September 2008, Kos2 Outline Introduction: The DLMF Project Part I: The Hardcopy DLMF Part II: The Online DLMF Part III: The Chapter on Numerical Methods Closing Remarks

3. ICNAAM, September 2008, Kos3 Introduction The DLMF Project

4. ICNAAM, September 2008, Kos4 Digital Library of Math Functions A new reference work: –For scientists and mathematicians –Notation, definitions, graphics, special values –Differential equations, integrals, series, sums –Recurrence relations, generating functions –Continued fractions, integral representations –Analytic continuation, zeros, limiting forms –Approximations, methods of computation –References to proofs or proof hints –And more …

5. ICNAAM, September 2008, Kos5 Frank Olver Editor-in-Chief Mathematics Editor

6. ICNAAM, September 2008, Kos6 Frank Olver Editor-in-Chief Mathematics Editor Nico Temme Author, Num. Meth. Assoc. Editor

7. ICNAAM, September 2008, Kos7 Part I The Hardcopy DLMF

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10. 10 Annual Citations 1974-1995 Blue: 1964 NBS Handbook Red: Total SCI (normalized)

11. ICNAAM, September 2008, Kos11 Impending Publication Complete replacement for the old NBS Handbook Twice as many formulas Almost no tables 1000 pages Publisher to be announced at Joint Math Meetings, Washington, DC, January 2009

12. ICNAAM, September 2008, Kos12 Part II The Online DLMF

13. ICNAAM, September 2008, Kos13 The DLMF Web Site Math search Interactive graphics Links –Within DLMF (cross-references) –External (articles, math reviews, software) Basis for further math-on-the-web work –Extend search & graphics beyond DLMF –Support “interoperability” (communication among computer algebra systems, for example)

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15. ICNAAM, September 2008, Kos15 Math Search

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19. ICNAAM, September 2008, Kos19 Interactive Graphics

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22. ICNAAM, September 2008, Kos22 Links from the Software Section of the Chapter on Airy Functions

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25. ICNAAM, September 2008, Kos25 Part III Numerical Methods

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27. ICNAAM, September 2008, Kos27 Quadrature

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32. ICNAAM, September 2008, Kos32 Here, complex gauss quadrature is useless when λ is large (due to massive cancellation). Another approach is to deform the path into a steepest descent path passing through a saddle point, and then to choose an effective quadrature rule. This approach has been used successfully to compute special functions, for example Scorer (or inhomogenous Airy) functions.

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35. ICNAAM, September 2008, Kos35 Stable Computation of Solutions Forward recurrence from two consecutive starting values is stable if w grows as rapidly as any other solution. Backward recurrence is stable if w grows as rapidly as any other solution in the backward direction. Here the starting values might come from an asymptotic approximation. Instead of backward recurrence, Miller’s algorithm is often used. Here purely arbitrary starting values are taken. Using them, a trial solution is computed by backward recurrence. This is normalized using the value of computed, for example, by the Maclaurin series. (Other normalizations, e.g. summation formulas, are also possible and often superior.)

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37. ICNAAM, September 2008, Kos37 Complementary solutions are the Bessel functions The Weber function grows much more slowly than one of these Bessel functions, and much faster than the other, so forward and backward recurrence are both very unstable for its computation.

38. ICNAAM, September 2008, Kos38 Stable Computation of an Intermediate Solution Assume there exist complementary solutions such that so neither forward nor backward recurrence is a stable for. But a stable computation is possible with a boundary-value method. It leads to a sequence of tridiagonal linear systems.

39. ICNAAM, September 2008, Kos39 The Boundary Value Problem

40. ICNAAM, September 2008, Kos40 Olver’s Algorithm Formulation: Given calculate Advantage: Combines the tridiagonal solution with the optimal determination of N.

41. ICNAAM, September 2008, Kos41 3.6.9

42. ICNAAM, September 2008, Kos42 (Continued) The results of this computation are shown on the next slide.

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45. ICNAAM, September 2008, Kos45 are analytic functions, and for applications to special functions they are often simple rational functions. By repeated differentiation, all derivatives are expressed as where are generated by simple recurrences. We wish to integrate along a finite path from a to b. We partition the path by successive points and use Taylor series to advance from point to point.

46. ICNAAM, September 2008, Kos46 Taylor series formulation: where A is a 2 by 2 matrix and b is a 2-vector. Stability: It is stable if w grows in magnitude at least as fast as all other solutions of the differential equation. Parallelizability: The A’s and b’s can be computed in parallel, and the product of matrices by cascading.

47. ICNAAM, September 2008, Kos47 The previous method solves an initial-value problem, and it is stable for recessive or dominant solutions. It can be reformulated as a boundary-value problem in terms of a linear system of order 2P+2. The matrix is a block band matrix with 2 by 2 blocks (which are A matrices as before) on the diagonal and 2 by 2 identity matrices on the superdiagonal. The system is solved by transforming into tridiagonal form and applying Olver’s algorithm. This process is stable for intermediate solutions.

48. ICNAAM, September 2008, Kos48 Closing Remarks

49. ICNAAM, September 2008, Kos49 The DLMF is a resource for applications of math in science and engineering. It is also an experiment in support of math on the Web. The Numerical Methods chapter is oriented toward computation of special functions. All methods discussed have been applied successfully to computation of complex-valued special functions. Other methods, e.g. using continued fractions, have also been applied successfully.

50. ICNAAM, September 2008, Kos50 Thanks to National Science Foundation NIST Manufacturing Engineering Laboratory NIST Physics Laboratory NIST Standard Reference Data Program NIST Information Technology Laboratory for financial support.

51. ICNAAM, September 2008, Kos51 http://dlmf.nist.gov Chapters on Asymptotic Approximations Gamma and Related Functions Airy Functions Functions of Number Theory 3j, 6j, 9j Symbols exist now for public review. Please try it and send us your comments!