Verifying DLMF with Maple and Mathematica

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This page presents the results of the publication: Comparative Verification of the Digital Library of Mathematical Functions and Computer Algebra Systems.

Bug Reports

You can find a PDF with commands that illustrate the encountered errors in Mathematica here: File:Mathematica Bugs Overview.pdf

We provide the same file in the Wolfram system notebook format (NB) here: File:Mathematica Bugs Notebook File.nb

DLMF Translations and Results

In the following, we present the translations of the DLMF equations to the CAS Maple and Mathematica.

DLMF Formula Translations
Maple		 Translations
Mathematica		 Symbolic Evaluation
Maple		 Symbolic Evaluation
Mathematica		 Numeric Evaluation
Maple		 Numeric Evaluation
Mathematica
DLMF 6,623 4,114 (62.1%) 4,713 (71.2%) 1,084 (26.3%) 1,235 (26.2%) 698 (26.7%) 784 (22.6%)

By clicking on a chapter of the DLMF below, you will see a large table that looks like this:

DLMF Formula Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
Link to DLMF Formula Translation to Maple Translation to Mathematica Symbolic Result in Maple Symbolic Result in Mathematica Numeric Result in Maple Numeric Result in Mathematica

The result tables do not contain every equation with a label in the DLMF since quite a few equations were skipped (see explanations in the paper).

Translations and Evaluations of the Digital Library of Mathematical Functions

  1. Algebraic and Analytic Methods
  2. Asymptotic Approximations
  3. Numerical Methods
  4. Elementary Functions
  5. Gamma Function
  6. Exponential, Logarithmic, Sine, and Cosine Integrals
  7. Error Functions, Dawson’s and Fresnel Integrals
  8. Incomplete Gamma and Related Functions
  9. Airy and Related Functions
  10. Bessel Functions I & Bessel Functions II
  11. Struve and Related Functions
  12. Parabolic Cylinder Functions
  13. Confluent Hypergeometric Functions
  14. Legendre and Related Functions
  15. Hypergeometric Function
  16. Generalized Hypergeometric Functions and Meijer G-Function
  17. q-Hypergeometric and Related Functions
  18. Orthogonal Polynomials
  19. Elliptic Integrals I & Elliptic Integrals II
  20. Theta Functions
  21. Multidimensional Theta Functions
  22. Jacobian Elliptic Functions
  23. Weierstrass Elliptic and Modular Functions
  24. Bernoulli and Euler Polynomials
  25. Zeta and Related Functions
  26. Combinatorial Analysis
  27. Functions of Number Theory
  28. Mathieu Functions and Hill’s Equation
  29. Lamé Functions
  30. Spheroidal Wave Functions
  31. Heun Functions
  32. Painlevé Transcendents
  33. Coulomb Functions
  34. 3j,6j,9j Symbols
  35. Functions of Matrix Argument
  36. Integrals with Coalescing Saddles
2C Translations Symbol Success Percent Symbolic Fails Numeric Success Percent Numeric Fail [Partially / Totally] Failed Abortion Error
1. AL 103 34 (33.0\%) 69 14 (20.3\%) 40 [ 10 / 30] 11 4
2. AS 65 6 ( 9.2\%) 59 4 ( 6.8\%) 38 [ 6 / 32] 7 9
3. NM 40 5 (12.5\%) 35 0 ( 0.0\%) 29 [ 8 / 21] 6 0
4. EF 564 304 (53.9\%) 260 110 (42.3\%) 146 [ 55 / 91] 2 0
5. GA 139 65 (46.8\%) 74 30 (40.5\%) 20 [ 9 / 11] 13 9
6. EX 77 18 (23.4\%) 59 23 (39.0\%) 32 [ 6 / 26] 4 0
7. ER 120 45 (37.5\%) 75 21 (28.0\%) 43 [ 13 / 30] 9 1
8. IG 163 65 (39.9\%) 98 22 (22.4\%) 44 [ 19 / 25] 16 15
9. AI 179 69 (38.5\%) 110 30 (27.3\%) 58 [ 38 / 20] 14 7
10. BS 486 115 (23.7\%) 371 90 (24.2\%) 151 [ 58 / 93] 92 18
11. ST 112 36 (32.1\%) 76 21 (27.6\%) 33 [ 8 / 25] 10 11
12. PC 90 18 (20.0\%) 72 13 (18.0\%) 43 [ 15 / 28] 12 3
13. CH 254 69 (27.2\%) 185 23 (12.4\%) 95 [ 59 / 36] 45 21
14. LE 229 30 (13.1\%) 199 59 (29.6\%) 92 [ 54 / 38] 41 5
15. HY 197 53 (26.9\%) 144 23 (16.0\%) 77 [ 52 / 25] 29 6
16. GH 25 2 ( 8.0\%) 23 1 ( 4.3\%) 10 [ 7 / 3] 9 2
17. QH 124 6 ( 4.8\%) 118 13 (11.0\%) 57 [ 52 / 5] 39 5
18. OP 288 101 (35.1\%) 185 47 (25.4\%) 66 [ 29 / 37] 52 12
19. EL 416 51 (12.2\%) 365 18 ( 4.9\%) 264 [ 49 / 215] 61 15
20. TH 98 1 ( 1.0\%) 97 33 (34.0\%) 40 [ 25 / 15] 24 0
21. MT 0 - - - - - - [ - / - ] - -
22. JA 238 30 (12.6\%) 206 22 (10.7\%) 131 [ 39 / 92] 51 0
23. WE 34 4 (11.8\%) 30 2 ( 6.7\%) 23 [ 9 / 14] 2 3
24. BP 148 23 (15.5\%) 125 78 (62.4\%) 33 [ 22 / 11] 14 0
25. ZE 120 48 (40.0\%) 72 22 (30.5\%) 22 [ 6 / 16] 22 3
26. CM 87 19 (21.8\%) 68 44 (64.7\%) 18 [ 10 / 8] 5 1
27. NT 15 6 (40.0\%) 9 3 (33.3\%) 6 [ 3 / 3] 0 0
28. MA 110 7 ( 6.4\%) 103 3 ( 2.9\%) 48 [ 13 / 35] 33 17
29. LA 22 0 ( 0.0\%) 22 0 ( 0.0\%) 21 [ 1 / 20] 0 1
30. SW 26 0 ( 0.0\%) 26 0 ( 0.0\%) 19 [ 2 / 17] 5 1
31. HE 15 2 (13.3\%) 13 0 ( 0.0\%) 8 [ 0 / 8] 5 0
32. PT 57 3 ( 5.3\%) 54 0 ( 0.0\%) 41 [ 2 / 39] 8 5
33. CW 11 0 ( 0.0\%) 11 0 ( 0.0\%) 11 [ 6 / 5] 0 0
34. TJ 37 0 ( 0.0\%) 37 14 (37.8\%) 10 [ 5 / 5] 13 0
35. FM 0 - - - - - - [ - / - ] - -
36. IC 24 0 ( 0.0\%) 24 3 (12.5\%) 13 [ 1 / 12] 1 6
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