Computer algebra systems, mathematical representation, and the DLMF

1. Computer algebra systems, mathematical representation, and the DLMF Richard Fateman, Bruce Char, Jeremy Johnson University of California, Berkeley Drexel University, Philadelphia National Institute of Standards and Technology DLMF Seminar Series, November 6, 2000

2. Desiderata for the Digital Library of Mathematical Functions • Traditional usage • New modes of interaction • Examples • New ambitions Computer Algebra and DLMF

3. Non-digital tradition: Finding Out Stuff • Individually owned reference works • Access to libraries’ references works • Access to colleagues by letter, phone, email • Paper and pencil exploration • Numerical experimentation Computer Algebra and DLMF

4. Wolfram Research’s Special Functions site: 3 versions • Huge posters • Interactive web site/ Mathematica notebooks • Printed form (or the equivalent PDF) Computer Algebra and DLMF

5. The posters Computer Algebra and DLMF

6. The web site (here, the Arcsin page) Computer Algebra and DLMF

7. continued fractions generating functions group representations differential equations difference equations transformations addition formulas etc operations integral transforms identities representations through more general functions relations with other functions zeros inequalities theorems other information history and applications references WRI’s Categories/ Some Subcategories primary definition specific values general characteristics series representations generalized power series at various points q-series exponential fourier series dirichlet series asymptotic series other series integral reprsentations on the real axis contour integrals multiple integral representation analytic continuations product representations limit representations Computer Algebra and DLMF

8. Click on “Series Representations”… Computer Algebra and DLMF

9. This is not very useful • These are blurry pictures of math formulas. • The most plausible next step seems to be to copy them down on paper and check by hand. • There is a possibility of making typos or fresh algebra mistakes. • The notation might be different from what you are using. • Sparse (or no) info on singularities, regions of validity. • To run some numbers through, you need to write a computer program (Fortran? Matlab? C++?) Computer Algebra and DLMF

10. Notebook form (I) Input form ArcSin[z] == z^3/6 + z + (3*z^5)/40 + \[Ellipsis] == Sum[(Pochhammer[1/2, k]*z^(2*k + 1))/((2*k + 1)*k!), {k, 0, Infinity}] == z*Hypergeometric2F1[1/2, 1/2, 3/2, z^2] /; Abs[z] < 1 One could imagine that a “system independent” language such as proposed by the OpenMath consortium would replace this language. Note however that agreement on the semantics of \[Ellipsis] would be difficult. Computer Algebra and DLMF

11. Notebook form (II) Displayed form (one version) In reality, Mathematica does not look quite as good as this in the interactive mode. Computer Algebra and DLMF

12. Notebook form (III) TeX form {Condition}(\arcsin (z) = {\frac{{{\Mfunction{z}}^3}}{6}} + z + {\frac{3\,{z^5}}{40}} + \ldots = \Mfunction{\sum}_{k = 0}^{\infty } {\frac{\Mfunction{Pochhammer}({\frac{1}{2}},k)\, {{\Mfunction{z}}^{2\,k + 1}}}{\left( 2\,k + 1 \right) \,k!}} = \Mfunction{z}\,\Mfunction{Hypergeometric2F1}( {\frac{1}{2}},{\frac{1}{2}},{\frac{3}{2}},{z^2}), \Mfunction{Abs}(z) < 1)) Useful in case you wanted to paste/edit this into another paper, using Mathematica TeX macros. Computer Algebra and DLMF

13. Computing Inside the Notebook How good is the 3-term approximation at z= ½ ? ArcSin[z] == z + z^3/6 + (3*z^5)/40 + ... /. z -> 1/2  Pi/6 == 2009/3840 + ... Surprised? N[Pi/6 == 2009/3840 + ...]  0.523599 == 0.523177 + ... N[Pi/6 == 2009/3840 + ..., 30]  0.523599 == 0.523177 + ... N[Pi/6 == 2009/3840 + ..., 30]  0.52359877559829887307710723055 == 0.52317708333333333333333333333 + ... Computer Algebra and DLMF

14. Simplification Inside the Notebook In[30] := z* Hypergeometric2F1[1/2, 1/2, 3/2, z^2] Note: this is how Mathematica interactive output looks. This should be the same as ArcSin[z] for |z|<1. And yes, z/Sqrt[z^2] is not the same as 1. Computer Algebra and DLMF

15. All commercial computer algebra systems (CAS) have essentially the same notebook paradigm • Macsyma • Maple • Mathematica • Axiom • MuPad • Scientific Word / Maple • Derive Computer Algebra and DLMF

16. Advice on coding a reference chapter Computer Algebra and DLMF

17. What about legacy “knowledge”? Can we convert from scanned text? Example from integral table • In practice, we can do some parsing using OCR if we know about the domains. • But in general, we cannot read “with understanding”. Computer Algebra and DLMF

18. What about using LaTeX as source and then converting to OpenMath/ CAS? Generally speaking: not automatically TeX does not distinguish semantically between 1*2*3 and 123. Or between x cos x and xfoox. It has no notion of precedence of operators Gradshteyn and Rhyzik, Table of Integrals and Series (Academic Press) was re-typeset completely in TeX TWICE, because the first version did not reflect semantics. MathML, XML, and OpenMath are inadequate. Computer Algebra and DLMF

19. Using OpenMath as original source is pretty much out of the question. • Intent is to code: • x cos x • <OMOBJ> • <OMA> • <OMS cd = "arith1" name="times"/> • <OMV name="x"/> • <OMA> • <OMS cd="transc1" name="cos"/> • <OMV name="x"/> • </OMA> • </OMA> • </OMOBJ> Computer Algebra and DLMF

20. Using MathML as original source is pretty much out of the question, too. • <math> • <msqrt> • <mfrac> • <mrow><mn>2</mn><mi>&pi;</mi></mrow> • <mrow><mi>&kappa;</mi></mrow> • </mfrac> • <mfenced open="(" close=")"> • <mn>1</mn> • <mi>&minus;</mi> • <mi>&beta;</mi> • <msup> • <mrow><mn>2</mn></mrow> • </msup> <mi>/</mi><mn>2</mn></mfenced></msqrt></math> Computer Algebra and DLMF

21. What can a CAS do better? • Semantics for what makes sense to the CAS is immediate. • Presentation for what doesn’t make sense to the CAS • Advantage: There is an immediate computational ontology • Immediate syntactic disambiguation • Easy translation into MathML for display • Easy translation into OpenMath, if anyone else cares. Computer Algebra and DLMF

22. What about using Java Applets? • Pro: • an applet provides more intimate interaction with a user. • Examples from Math Forum … • Con: • Only Java-enabled clients can use such applets. • Java standardization is problematical • High quality numerical software in Java? • Symbolic computation in Java? • Poor access to underlying computer. Computer Algebra and DLMF

23. What about Server Side software? • Pro: • arbitrarily powerful – could be huge database and super-fast computer • always up-to-date • controlled by validation team? • Can collect / re-distribute new data Computer Algebra and DLMF

24. What about Server Side software? • Con: • Risk/cost of computation at server • Communication requirement • Cost • Connection reliability Computer Algebra and DLMF

25. What about no software? • Pro: • You can “run” DLMF without a computer • You can work on a desert island • Con: • Anyone with a computer or electricity will be disappointed Computer Algebra and DLMF

26. What about only browser software? • Pro: • You can “run” DLMF on an appliance ($300) • Con: • Loss in the marketplace of ideas • In some respects it will suffer from comparison with software, some of which is free today • Students who are used to (say) Mathematica will use other resources, even if less authoritative Computer Algebra and DLMF

27. What about numeric-only software? • Pro: • You print numeric tables as needed • Con: • Symbolic data is endlessly tabulated instead. etc Computer Algebra and DLMF

28. What about symbolic software • Now we can consider including algorithms for • trig(n/m ) • Indefinite integrals (replacing 10-20,000) • Summation, Limits • Definite integrals (a challenge still) • Implicit application of identities, reduction of argument or order by recursion, etc. • Generation of any number of terms in series • Expansion in Chebyshev or other polynomials • Exact arithmetic or “bigfloat” arithmetic Computer Algebra and DLMF

29. A challenge: Include a CAS in DLMF • Free Macsyma (c. 1982) • (buy) Commercial system • Macsyma, Maple, Mathematica, Axiom … alternatively • Require the user to have a CAS separately (like requiring a Fortran compiler to use GAMS) Computer Algebra and DLMF

30. What do we want? What can we attempt? • Exhaustive hierarchical hyperlinks to everything known • Human readable form • Computer usable form + ALGORITHMS • Searchable form /Unique identifiers for formulas • Provenance of information • Annotations from other users • Corrections current or past • Cross-reference hyperlinks to all of mathematics • Applications • A bridge across paper/pencil computer gap • A tireless, accurate and efficient robot to help us Computer Algebra and DLMF

31. Computers do more than arithmetic Many persons who are not conversant with mathematical studies imagine that because the business of [Babbage's Analytical Engine] is to give its results in numerical notation, the nature of its processes must consequently be arithmetical and numerical, rather than algebraical and analytical. This is an error. The engine can arrange and combine its numerical quantities exactly as if they were letters or any other general symbols; and in fact it might bring out its results in algebraic notation, were provisions made accordingly. -- Ada Augusta, Countess of Lovelace, (1844) Computer Algebra and DLMF