DLMF Tables: A New Source of Data for Mathematical Software Developers

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DLMF Tables A New Source of Data for
Mathematical Software Developers

• Daniel Lozier
• Mathematical Software Group
• Applied Computational Math Division
• Information Technology Laboratory
• National Instititute of Standards and Technology
• Gaithersburg, MD, 20899-8910 USA

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Outline

• A Short History of Function Testing at NBS/NIST,
  1971-1997
• The DLMF Project, 1997-2010
• The DLMF Tables Project, 2010-present
• Closing Remarks

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J. Res. NBS, 16(2),1973
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NBS Internal Report, May 1978
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D.W. Lozier, A Proposed Software Test Service for
Special Functions, in Quality of
Numerical Software Assessment and
Enhancement, R.F. Boisvert, ed., Chapman and
Hall, London, 1997, pp.
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The DLMF Project
1965
1964
2010
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Scope of Coverage

• Methods (3 chapters)
• Algebraic and analytical methods
• Asymptotic approximations
• Numerical methods
• Mathematical Functions (33 chapters)
• Elementary
• Airy, Bessel, Legendre,
• Orthogonal polynomials
• Elliptic integrals and functions
• Combinatorics, number theory
• Mathieu, Lamé, Heun, Painlevé, Coulomb,

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http//dlmf.nist.gov/
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Website vs. Book

• enhanced superset of the book with cutting-edge
  IT capabilities
• Color visualizations
• Equation search (example dn/?n)
• Links
• Internal to symbol definitions, bib items, help
• External to online articles, reviews, software
• Cut paste tex, png, MathML
• Sample applications

Hankel function
Kelvins shipwave
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Some Numbers

• 36 chapters
• 876 sections
• 1996 subsections
• 2302 bibliographic items
• 9896 numbered equations
• 608 graphs
• 119 short non-numerical tables
• 976 pages in Cambridge edition

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DLMF Tables Project

• NIST collaboration with U. Antwerp
• 3 workshops at NIST
• 2009, 2010, 2011
• Telephone conference calls
• Task leaders
• NIST and Antwerp overall design
• Antwerp compute engine
• NIST web interface

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First CapabilityTables on Demand
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Error Bounds

• Let denote relative error and
  .
• With n digits requested, values are computed with
  uniformly bounded relative error ,
  and n2 digits are displayed. No rounding is
  done.
• The line-by-line absolute errors are indicated by
  two digits following the sign after digit n2.

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End Figures for sin(x) at x 0.5, 1, 1.5
40th digit
Interval interpretation
71388 0817 98, 71388 0818 08 98999 6225 54,
98999 6225 72 87322 7066 41, 87322 7066 61
40th digit
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Raw Data for sin(x) Example
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Sin 1.5707 to 5,6,7,8 Digits
.9999 98, 1.0000 02
.99999 98, 1.00000 02
.9999999 85, 1.000000 05
.99999999 44, .99999999 64
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Second CapabilityComparison to Input Tableof
Function Values
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Values on lines 2,4,6,8,10,12 are taken from
Abramowitz and Stegun, Table 9.4, page 407
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Previous Slide Zoomed
We can see the AS values are correctly rounded,
except we cannot see this for .
Cannot tell if is correctly
rounded because of ambiguity in 12D interval
value. In fact, it is.
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(with intentionally mistyped AS values)
End figures from AS, Table 9.4 (see previous
slide) 6866 (line 2) 9756 (line 4) 2647 (line
6) 3144 (line 8) 3528 (line 10) 7330 (line
12) (intentionally mistyped figures underlined in
lines 2 and 6)
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Raw Data for Bessel Example
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sin 1.5707 Revisited Comparison
5D
Here, because sin x x(1O(x2)), the comparison
value must be given to at least 7D to get
the correct leading digits in the interval
computation. This leads to the red figures for 5D
and 6D.
6D
7D
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Computational Limits
Per-request execution time limit
exists. Excessive repeated requests from same IP
are denied. Other limits may become necessary.
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Recap 1

• NIST has long history
• with special functions
• with software testing, especially of functions
• NIST released the DLMF in May 2010
• DLMF Tables not yet accessible to public
• Public release will be fully supported by NIST
• Users will have two new capabilities

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Recap 2 Tables on Demand

• Specifiable precision n digits, up to n500
• Table values displayed to n2 digits
• Error bounded by 1 unit in digit n1
• Bounds certified accurate by NIST
• Table values not rounded to n digits but
• Rounding to n digits often discernible
• Leading n digits usually correct
• Big improvement over static tables (as in AS)
• Visual display for people
• Raw output for computers

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Recap 3 Software Verification

• Verify by comparison to standard values
• Tables on Demand generates standard values
• User inputs test values manually or from file
• System generates standard values to n digits,
  displays n2 digits, where n is determined by
  input set of test values
• Test digits that differ from standard digits are
  colored red in visual display
• Raw output is available for computers

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Acknowledgments

• Antwerp group for the back end
• Annie Cuyt, Franky Backeljauw
• Stefan Becuwe, Joris Van Deun
• NIST group for front end
• Bonita Saunders, Bruce Miller,
• Marjorie McClain, Daniel Lozier
• Other colleagues, too many to name, who have
  provided valuable advice and constructive
  criticism