An Introduction of Multiple P-adic Data Type and Its Parallel Implementation
Our research group has been working on the P -adic theory and its implementation. Based on the Chinese Remainder theorem and the Hensel code a new data type, called Multiple P-adic Data Type, has been established to realize rational calculation. With this data type all rational number operations are...
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| Vydané v: | The International journal of networked and distributed computing (Online) Ročník 3; číslo 1; s. 51 - 59 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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Dordrecht
Springer Netherlands
01.01.2015
Springer |
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| ISSN: | 2211-7938, 2211-7946, 2211-7946 |
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| Abstract | Our research group has been working on the
P
-adic theory and its implementation. Based on the Chinese Remainder theorem and the Hensel code a new data type, called Multiple P-adic Data Type, has been established to realize rational calculation. With this data type all rational number operations are converted to integer calculations, and the fast integer multiplication of modern computer architectures can be fully taken advantage of. This data type can be significantly effective in the parallel and cloud computing environment due to its independent computation at each node during the calculation process. |
|---|---|
| AbstractList | Our research group has been working on the
P
-adic theory and its implementation. Based on the Chinese Remainder theorem and the Hensel code a new data type, called Multiple P-adic Data Type, has been established to realize rational calculation. With this data type all rational number operations are converted to integer calculations, and the fast integer multiplication of modern computer architectures can be fully taken advantage of. This data type can be significantly effective in the parallel and cloud computing environment due to its independent computation at each node during the calculation process. Our research group has been working on the P-adic theory and its implementation. Based on the Chinese Remainder theorem and the Hensel code a new data type, called Multiple P-adic Data Type, has been established to realize rational calculation. With this data type all rational number operations are converted to integer calculations, and the fast integer multiplication of modern computer architectures can be fully taken advantage of. This data type can be significantly effective in the parallel and cloud computing environment due to its independent computation at each node during the calculation process. |
| Author | Li, Xinkai Lu, Chao |
| Author_xml | – sequence: 1 givenname: Chao surname: Lu fullname: Lu, Chao email: clu@towson.edu organization: Department of Computer & Information Sciences, Towson University – sequence: 2 givenname: Xinkai surname: Li fullname: Li, Xinkai organization: Department of Computer & Information Sciences, Towson University |
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| Keywords | adic Parallel computing Computational efficiency Multiple modulus Chinese remainder theorem |
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| References | C. K. Koc, A Tutorial on P-adic Arithmetic, Oregon State University, Technical Report, April 2002. C. K. Koc, “Parallel P-adic Method for Solving Linear Systems of Equations”, Parallel Computing, Vol. 23(13), 1997. X. Li, M. Zhao, C. Lu and J. A. Sjogren, “Implementation of the Polynomial Method to Calculate eAt Using P-adic”, Proceedings of the 2012 ACM Research in Applied Computation Symposium, 2012. M. Newman, “Solving Equations Exactly”, Mathematics and Mathematical Physics, Vol. 71B, No. 4, Oct-Dec 1967. J. Morrison, “Parallel P-adic computation, Information Processing Letters”, Vol. 28, Issue 3, 1988. C. Lu, X. Li and L. Shan, “Periodicity of the P-adic Expansion after Arithmetic Operations in P-adic Field”, ACIS2012. P. Kornerup and D. W. Matula, Finite Precision Number Systems and Arithmetic, Cambridge University Press, 2010. KornerupPGregoryR T“Mapping Integers and Hensel Codes onto Farey Fractions”BIT198323920 M. Marques, G. Quintana-Orti and E. S. Quintana-Orti, Solving “Large” Dense Matrix Problems on Multi-Core Processors, In: 10th IEEE International Workshop on Parallel and Distributed Scientific and Engineering Computing-PDSEC 2009. E. V. Krishnamurthy, Matrix Processors Using P-adic Arithmetic for Exact Linear Computations, IEEE Transactions on computers, Vol. C-26, No, 7, July 1977. RaoT MSubramanianKKrishnamurthyE V“Residue Arithmetic Algorithm for Exact Computation of g-inverses of Matrices”SIAM J. NUMER. ANAL1976132155171 GregoryR T“The Use of Finite-segment P-adic Arithmetic for Exact Computation”BIT1978183282300 C. Limongelli and H. W. Loidl, “Rational Number Arithmetic by Parallel P-adic Algorithms”, Springer Verlag, editor, Proc. Of Second International Conference of the Austrian Center for Parallel Computation (ACPC), Vol. 734 of LNCS, 1993. X. Li, C. Lu and J. A. Sjogren, “Parallel Implementation of Exact Matrix Computation Using Multiple P-adic Arithmetic”, International Journal of Networked and Distributed Computing (IJNDC) - Atlantis Press, 1(3), August, 2013. LiXLuCSjogrenJ A“A Method for Hensel Code Overflow Detection”ACM SIGAPP Applied Computing Review2012121611 HenselK“Theorie der Algebraischen Zahlen”1908TeubnerLeipzig-Stuttgart M. Miola, “The conversion of Hensel Codes to Their Rational Equivalents: or How to Solve the Gregory’s Open Problem”, ACM Sigsam bulletin, vol. 16, Issue 4, November 1982. BachmanGIntroduction to P-adic Numbers and Valuation Theory1964New YorkAcdemic Press YoungD MGregoryR T“A Survey of Numberical Mathematics”1973MassAddison Wesley, Reading J. D. Dixon, “Exact Solution of Linear Equations Using P-adic Expansions”, Number. Math. 40, 137–141(1982), Springer- Verlag. X. Li, M, Zhao and C. Lu, “Efficient Algorithms and Implementation for Error-free Computation Using P-adic”, CSNI2011. |
| References_xml | – reference: J. D. Dixon, “Exact Solution of Linear Equations Using P-adic Expansions”, Number. Math. 40, 137–141(1982), Springer- Verlag. – reference: KornerupPGregoryR T“Mapping Integers and Hensel Codes onto Farey Fractions”BIT198323920 – reference: X. Li, M. Zhao, C. Lu and J. A. Sjogren, “Implementation of the Polynomial Method to Calculate eAt Using P-adic”, Proceedings of the 2012 ACM Research in Applied Computation Symposium, 2012. – reference: RaoT MSubramanianKKrishnamurthyE V“Residue Arithmetic Algorithm for Exact Computation of g-inverses of Matrices”SIAM J. NUMER. ANAL1976132155171 – reference: C. Limongelli and H. W. Loidl, “Rational Number Arithmetic by Parallel P-adic Algorithms”, Springer Verlag, editor, Proc. Of Second International Conference of the Austrian Center for Parallel Computation (ACPC), Vol. 734 of LNCS, 1993. – reference: LiXLuCSjogrenJ A“A Method for Hensel Code Overflow Detection”ACM SIGAPP Applied Computing Review2012121611 – reference: C. K. Koc, “Parallel P-adic Method for Solving Linear Systems of Equations”, Parallel Computing, Vol. 23(13), 1997. – reference: M. Newman, “Solving Equations Exactly”, Mathematics and Mathematical Physics, Vol. 71B, No. 4, Oct-Dec 1967. – reference: C. Lu, X. Li and L. Shan, “Periodicity of the P-adic Expansion after Arithmetic Operations in P-adic Field”, ACIS2012. – reference: C. K. Koc, A Tutorial on P-adic Arithmetic, Oregon State University, Technical Report, April 2002. – reference: P. Kornerup and D. W. Matula, Finite Precision Number Systems and Arithmetic, Cambridge University Press, 2010. – reference: GregoryR T“The Use of Finite-segment P-adic Arithmetic for Exact Computation”BIT1978183282300 – reference: E. V. Krishnamurthy, Matrix Processors Using P-adic Arithmetic for Exact Linear Computations, IEEE Transactions on computers, Vol. C-26, No, 7, July 1977. – reference: M. Miola, “The conversion of Hensel Codes to Their Rational Equivalents: or How to Solve the Gregory’s Open Problem”, ACM Sigsam bulletin, vol. 16, Issue 4, November 1982. – reference: M. Marques, G. Quintana-Orti and E. S. Quintana-Orti, Solving “Large” Dense Matrix Problems on Multi-Core Processors, In: 10th IEEE International Workshop on Parallel and Distributed Scientific and Engineering Computing-PDSEC 2009. – reference: HenselK“Theorie der Algebraischen Zahlen”1908TeubnerLeipzig-Stuttgart – reference: BachmanGIntroduction to P-adic Numbers and Valuation Theory1964New YorkAcdemic Press – reference: YoungD MGregoryR T“A Survey of Numberical Mathematics”1973MassAddison Wesley, Reading – reference: X. Li, M, Zhao and C. Lu, “Efficient Algorithms and Implementation for Error-free Computation Using P-adic”, CSNI2011. – reference: X. Li, C. Lu and J. A. Sjogren, “Parallel Implementation of Exact Matrix Computation Using Multiple P-adic Arithmetic”, International Journal of Networked and Distributed Computing (IJNDC) - Atlantis Press, 1(3), August, 2013. – reference: J. Morrison, “Parallel P-adic computation, Information Processing Letters”, Vol. 28, Issue 3, 1988. |
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P
-adic theory and its implementation. Based on the Chinese Remainder theorem and the Hensel code a new data type,... Our research group has been working on the P-adic theory and its implementation. Based on the Chinese Remainder theorem and the Hensel code a new data type,... |
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| SubjectTerms | Chinese remainder theorem Computational efficiency Multiple modulus P-adic Parallel computing Research Article |
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| Title | An Introduction of Multiple P-adic Data Type and Its Parallel Implementation |
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