Academic Journal
Algorithm 1006: Fast and accurate evaluation of a generalized incomplete gamma function
| العنوان: | Algorithm 1006: Fast and accurate evaluation of a generalized incomplete gamma function |
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| المؤلفون: | Abergel, Rémy, Moisan, Lionel |
| المساهمون: | Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques (INSMI-CNRS)-Centre National de la Recherche Scientifique (CNRS) |
| المصدر: | ISSN: 0098-3500 ; ACM Transactions on Mathematical Software ; https://hal.science/hal-01329669 ; ACM Transactions on Mathematical Software, 2020, 46 (1), pp.1 -- 24. ⟨10.1145/3365983⟩. |
| بيانات النشر: | HAL CCSD Association for Computing Machinery |
| سنة النشر: | 2020 |
| مصطلحات موضوعية: | incomplete gamma function, continued fraction, floating-point arithmetic, Romberg's method, numerical cancellation, incomplete gamma integral, ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.2: Approximation/G.1.2.9: Special function approximations, [INFO.INFO-MS]Computer Science [cs]/Mathematical Software [cs.MS], [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] |
| الوصف: | International audience ; We present a computational procedure to evaluate the integral ∫ x y s p-1 e -μs ds, for 0 ≤ x < y ≤+∞, μ = ±1, p > 0, which generalizes the lower (x=0) and upper (y=+∞) incomplete gamma functions. To allow for large values of x, y, and p while avoiding under/overflow issues in the standard double precision floating point arithmetic, we use an explicit normalization that is much more efficient than the classical ratio with the complete gamma function. The generalized incomplete gamma function is estimated with continued fractions, integrations by parts, or, when x ≈ y, with the Romberg numerical integration algorithm. We show that the accuracy reached by our algorithm improves a recent state-of-the-art method by two orders of magnitude, and is essentially optimal considering the limitations imposed by the floating point arithmetic. Moreover, the admissible parameter range of our algorithm (0 ≤ p,x,y ≤ 10 15 ) is much larger than competing algorithms and its robustness is assessed through massive usage in an image processing application. |
| نوع الوثيقة: | article in journal/newspaper |
| اللغة: | English |
| Relation: | hal-01329669; https://hal.science/hal-01329669; https://hal.science/hal-01329669v2/document; https://hal.science/hal-01329669v2/file/gammainc_revised_preprint.pdf |
| DOI: | 10.1145/3365983 |
| الاتاحة: | https://hal.science/hal-01329669 https://hal.science/hal-01329669v2/document https://hal.science/hal-01329669v2/file/gammainc_revised_preprint.pdf https://doi.org/10.1145/3365983 |
| Rights: | info:eu-repo/semantics/OpenAccess |
| رقم الانضمام: | edsbas.E9C9379B |
| قاعدة البيانات: | BASE |
| DOI: | 10.1145/3365983 |
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