july 4-6, 2012numerical software workshop: design, analysis and verification 1 dlmf tables: a new...

July 4-6, 2012 Numerical Software Workshop: Design, Analysis and Verification

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DLMF Tables: A New Source of Data for Mathematical Software

Developers

Daniel LozierMathematical Software Group

Applied & Computational Math Division

Information Technology Laboratory

National Instititute of Standards and Technology

Gaithersburg, MD, 20899-8910 USA

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 Servicefor Special Functions, in Quality of NumericalSoftware: Assessment and Enhancement,R.F. Boisvert, ed., Chapman and Hall, London,1997, pp.

The DLMF Project


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1964 1965

20100

500

1000

1500

2000

2500

1971 1977 1983 1989 1995 2001 2007


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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,…

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: d^n/?^n)– Links

• Internal to symbol definitions, bib items, help…• External to online articles, reviews, software…

– Cut & paste tex, png, MathML– Sample applications

Kelvin’s shipwave

Hankel function


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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

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 Capability

Tables 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 n+2 digits are displayed. No rounding is done.

• The line-by-line absolute errors are indicated by two digits following the sign after digit n+2.


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110 nu

u

End Figures for sin(x) at x = 0.5, 1, 1.5


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40th digit

[71388 0817 98, 71388 0818 08][98999 6225 54, 98999 6225 72][87322 7066 41, 87322 7066 61]

40th digit

Interval interpretation:

Raw Data for sin(x) Example


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Sin 1.5707 to 5,6,7,8 Digits


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[.9999 98, 1.0000 02]

[.99999 98, 1.00000 02]

[.9999999 85, 1.000000 05]

[.99999999 44, .99999999 64]

Second Capability

Comparison to Input Tableof Function Values


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Values on lines 2,4,6,8,10,12are taken from Abramowitzand Stegun, Table 9.4, page 407

Previous Slide Zoomed


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We can see the A&Svalues are correctlyrounded, except wecannot see this for .)100(100J

Cannot tell if is correctly roundedbecause of ambiguityin 12D interval value.In fact, it is.

)100(100J


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End figures from A&S, 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 figuresunderlined in lines 2 and 6)

(with intentionallymistyped A&S values)

Raw Data for Bessel Example


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sin 1.5707 Revisited: Comparison


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5D

6D

7D

Here, becausesin x = x(1+O(x^2)),the comparison valuemust be given to atleast 7D to get thecorrect leading digits inthe interval computation.This leads to the redfigures for 5D and 6D.

Computational Limits


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Per-request execution time limit exists.Excessive repeated requests from same IP are denied.Other limits may become necessary.

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 n=500– Table values displayed to n+2 digits– Error bounded by 1 unit in digit n+1– 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 A&S)– 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 n+2 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


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july 4-6, 2012numerical software workshop: design, analysis and verification 1 dlmf tables: a new...

Documents

numerical software workshop

quality of numerical

usa slide

overall design antwerp

previous slide zoomed

cambridge edition slide

short nonnumerical tables

proposed software test