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

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Desiderata for the Digital Library of
Mathematical Functions

• Traditional usage
• New modes of interaction
• Examples
• New ambitions

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

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Wolfram Researchs Special Functions site 3
versions

• Huge posters
• Interactive web site/ Mathematica notebooks
• Printed form (or the equivalent PDF)

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The posters
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The web site (here, the Arcsin page)
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WRIs Categories/ Some Subcategories

• 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

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
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Click on Series Representations
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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?)

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Notebook form (I)
Input form
ArcSinz z3/6 z (3z5)/40 \Ellipsis
Sum(Pochhammer1/2, kz(2k 1))/((2k
1)k!), k, 0, Infinity
zHypergeometric2F11/2, 1/2, 3/2, z2 / Absz
lt 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.
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Notebook form (II)
Displayed form (one version)
In reality, Mathematica does not look quite as
good as this in the interactive mode.
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Notebook form (III)
TeX form
Condition(\arcsin (z) \frac\Mfunction
z36 z \frac3\,z540
\ldots \Mfunction\sum_k 0\infty
\frac\MfunctionPochhammer(\frac12
,k)\, \Mfunctionz2\,k
1\left( 2\,k 1 \right) \,k!
\Mfunctionz\,\MfunctionHypergeometric2F1
( \frac12,\frac12,\frac32
,z2), \MfunctionAbs(z) lt 1))
Useful in case you wanted to paste/edit this into
another paper, using Mathematica TeX macros.
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Computing Inside the Notebook
How good is the 3-term approximation at z ½ ?
ArcSinz z z3/6 (3z5)/40 ... /. z
-gt 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 ...
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Simplification Inside the Notebook
In30 z Hypergeometric2F11/2, 1/2, 3/2, z2
Note this is how Mathematica interactive output
looks. This should be the same as ArcSinz for
zlt1. And yes, z/Sqrtz2 is not the same as 1.
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All commercial computer algebra systems (CAS)
have essentially the same notebook paradigm

• Macsyma
• Maple
• Mathematica
• Axiom
• MuPad
• Scientific Word / Maple
• Derive

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Advice on coding a reference chapter
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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.

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What about using LaTeX as source and then
converting to OpenMath/ CAS?

• Generally speaking not automatically

• TeX does not distinguish semantically between
  123 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.

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Using OpenMath as original source is pretty much
out of the question.
Intent is to code x cos x ltOMOBJgt ltOMAgt
ltOMS cd "arith1" name"times"/gt ltOMV
name"x"/gt ltOMAgt ltOMS cd"transc1"
name"cos"/gt ltOMV name"x"/gt lt/OMAgt
lt/OMAgt lt/OMOBJgt
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Using MathML as original source is pretty much
out of the question, too.
ltmathgt ltmsqrtgt ltmfracgt ltmrowgtltmngt2lt/mngtltmi
gtpilt/migtlt/mrowgt ltmrowgtltmigtkappalt/migtlt/mrow
gt lt/mfracgt ltmfenced open"(" close")"gt
ltmngt1lt/mngt ltmigtminuslt/migt
ltmigtbetalt/migt ltmsupgt ltmrowgtltmngt2lt/mngtlt/m
rowgt lt/msupgt ltmigt/lt/migtltmngt2lt/mngtlt/mfencedgtlt/ms
qrtgtlt/mathgt
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What can a CAS do better?

• Semantics for what makes sense to the CAS is
  immediate.
• Presentation for what doesnt 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.

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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.

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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

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What about Server Side software?

• Con
• Risk/cost of computation at server
• Communication requirement
• Cost
• Connection reliability

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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

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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

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What about numeric-only software?

• Pro
• You print numeric tables as needed
• Con
• Symbolic data is endlessly tabulated instead.

etc
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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

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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)

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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

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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)