Highly Parallel, High-Precision Numerical Integration

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This paper describes a scheme for rapidly computing numerical values of definite integrals to very high accuracy, ranging from ordinary machine precision to hundreds or thousands of digits, even for functions with singularities or infinite derivatives at endpoints. Such a scheme is of interest not only in computational physics and computational chemistry, but also in experimental mathematics, where high-precision numerical values of definite integrals can be used to numerically discover new identities. This paper discusses techniques for a parallel implementation of this scheme, then presents performance results for 1-D and 2-D test suites. Results are also given for a certain ... continued below

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Bailey, David H. & Borwein, Jonathan M. April 22, 2005.

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Description

This paper describes a scheme for rapidly computing numerical values of definite integrals to very high accuracy, ranging from ordinary machine precision to hundreds or thousands of digits, even for functions with singularities or infinite derivatives at endpoints. Such a scheme is of interest not only in computational physics and computational chemistry, but also in experimental mathematics, where high-precision numerical values of definite integrals can be used to numerically discover new identities. This paper discusses techniques for a parallel implementation of this scheme, then presents performance results for 1-D and 2-D test suites. Results are also given for a certain problem from mathematical physics, which features a difficult singularity, confirming a conjecture to 20,000 digit accuracy. The performance rate for this latter calculation on 1024 CPUs is 690 Gflop/s. We believe that this and one other 20,000-digit integral evaluation that we report are the highest-precision non-trivial numerical integrations performed to date.

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  • SC2005, Seattle, WA, Nov 12-18,2005

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  • Report No.: LBNL--57491
  • Grant Number: DE-AC02-05CH11231
  • Office of Scientific & Technical Information Report Number: 860346
  • Archival Resource Key: ark:/67531/metadc781446

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  • April 22, 2005

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  • Dec. 3, 2015, 9:30 a.m.

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  • April 1, 2016, 8:04 p.m.

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Bailey, David H. & Borwein, Jonathan M. Highly Parallel, High-Precision Numerical Integration, article, April 22, 2005; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc781446/: accessed August 23, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.