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July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 1 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 July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 2

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J. Res. NBS, 16(2),1973 July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 3

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NBS Internal Report, May 1978 July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 4

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July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 8 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 July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 9 1964 1965 2010

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July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 10 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/ July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 11

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July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 12 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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July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 13 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 July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 14

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First Capability Tables on Demand July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 15

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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. July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 18

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End Figures for sin(x) at x = 0.5, 1, 1.5 July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 19 40 th digit [71388 0817 98, 71388 0818 08] [98999 6225 54, 98999 6225 72] [87322 7066 41, 87322 7066 61] 40 th digit Interval interpretation:

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Raw Data for sin(x) Example July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 20

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Sin 1.5707 to 5,6,7,8 Digits July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 21 [.9999 98, 1.0000 02] [.99999 98, 1.00000 02] [.9999999 85, 1.000000 05] [.99999999 44,.99999999 64]

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Second Capability Comparison to Input Table of Function Values July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 22

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July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 24 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 July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 25 We can see the A&S 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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July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 26 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 figures underlined in lines 2 and 6) (with intentionally mistyped A&S values)

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Raw Data for Bessel Example July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 27

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sin 1.5707 Revisited: Comparison July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 28 5D 6D 7D Here, because sin x = x(1+O(x^2)), 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.

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Computational Limits July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 29 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 July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 30

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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 July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 31

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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 July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 32

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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 July 4-6, 2012Numerical Software Workshop: Design, Analysis and Verification 33

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