numerical analysis in the digital library of mathematical functions

ICNAAM, September 2008, Kos 1

Numerical Analysisin the Digital Library of Mathematical Functions

Dan LozierMath and Computational Sciences Division

National Institute of Standards and TechnologyGaithersburg, MD 20899-8910 USA

http://dlmf.nist.gov

[email protected]


Outline

• Introduction: The DLMF Project

• Part I: The Hardcopy DLMF

• Part II: The Online DLMF

• Part III: The Chapter on Numerical Methods

• Closing Remarks


Introduction

The DLMF Project


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 …


Frank OlverEditor-in-ChiefMathematics Editor


Frank OlverEditor-in-ChiefMathematics Editor

Nico TemmeAuthor, Num. Meth.Assoc. Editor


Part I

The Hardcopy DLMF


Annual Citations 1974-1995

Blue: 1964 NBS HandbookRed: Total SCI (normalized)


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


Part II

The Online DLMF


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)


Math Search


Interactive Graphics


Links from the Software Section of the Chapter on Airy Functions


Part III

Numerical Methods


Quadrature


Here, complex gauss quadrature is useless when λ is large(due to massive cancellation). Another approach is to deformthe path into a steepest descent path passing through a saddlepoint, and then to choose an effective quadrature rule.

This approach has been used successfully to compute specialfunctions, for example Scorer (or inhomogenous Airy)functions.


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.)

0w


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 backwardrecurrence are both very unstable for its computation.


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.


The Boundary Value Problem


Olver’s Algorithm

Formulation: Given 0w calculate

Advantage: Combines the tridiagonal solutionwith the optimal determination of N.


3.6.9


(Continued)

The results of this computation are shown on the next slide.


hgf and , , 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. Wepartition the path by successive points and use Taylor series to advance from point to point.

sss hgf and , ,

bzzza P ,..., , 10


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.


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.


Closing Remarks


• 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.


Thanks to National Science Foundation NIST Manufacturing Engineering Laboratory NIST Physics Laboratory NIST Standard Reference Data Program NIST Information Technology Laboratory

for financial support.


http://dlmf.nist.gov

Chapters on Asymptotic ApproximationsGamma and Related Functions Airy FunctionsFunctions of Number Theory3j, 6j, 9j Symbols

exist now for public review.Please try it and send us your comments!

numerical analysis in the digital library of mathematical functions

Documents

dlmf projectpart

math reviews

joint math meetings

hardcopy dlmfpart

chiefmathematics editoricnaam

kosfrank olvereditor

online dlmfpart

backward recurrence