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  • Numerical Analysis  (5)
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  • 1
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    The Quest for Pi (1996)
    In:  CASI
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    Publication Date: 2019-06-28
    Description: This article gives a brief history of the analysis and computation of the mathematical constant Pi=3.14159 ..., including a number of the formulas that have been used to compute Pi through the ages. Recent developments in this area are then discussed in some detail, including the recent computation of Pi to over six billion decimal digits using high-order convergent algorithms, and a newly discovered scheme that permits arbitrary individual hexadecimal digits of Pi to be computed.
    Keywords: Numerical Analysis
    Type: NASA-TM-112037 , NAS 1.15:112037 , NAS-96-015
    Format: application/pdf
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  • 2
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    On the Rapid Computation of Various Polylogarithmic Constants (1996)
    In:  CASI
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    Publication Date: 2019-06-28
    Description: We give algorithms for the computation of the d-th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log(2) or pi on a modest workstation in a few hours run time. We demonstrate this technique by computing the ten billionth hexadecimal digit of pi, the billionth hexadecimal digits of pi-squared, log(2) and log-squared(2), and the ten billionth decimal digit of log(9/10). These calculations rest on the observation that very special types of identities exist for certain numbers like pi, pi-squared, log(2) and log-squared(2). These are essentially polylogarithmic ladders in an integer base. A number of these identities that we derive in this work appear to be new, for example a critical identity for pi.
    Keywords: Numerical Analysis
    Type: NASA-TM-112039 , NAS 1.15:112039 , NAS-96-016
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  • 3
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    Algebraic Nonoverlapping Domain Decomposition Methods for Stabilized FEM and FV Discretizations (1998)
    In:  Other Sources
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    Publication Date: 2019-07-18
    Description: We consider preconditioning methods for convection dominated fluid flow problems based on a nonoverlapping Schur complement domain decomposition procedure for arbitrary triangulated domains. The triangulation is first partitioned into a number of subdomains and interfaces which induce a natural 2 x 2 partitioning of the p.d.e. discretization matrix. We view the Schur complement induced by this partitioning as an algebraically derived coarse space approximation. This avoids the known difficulties associated with the direct formation of an effective coarse discretization for advection dominated equations. By considering various approximations of the block factorization of the 2 x 2 system, we have developed a family of robust preconditioning techniques. A computer code based on these ideas has been developed and tested on the IBM SP2 using MPI message passing protocol. A number of 2-D CFD calculations will be presented for both scalar advection-diffusion equations and the Euler equations discretized using stabilized finite element and finite volume methods. These results show very good scalability of the preconditioner for various discretizations as the number of processors is increased while the number of degrees of freedom per processor is fixed.
    Keywords: Numerical Analysis
    Type: 10th Conference on Finite Element Methods in Fluids; Jan 08, 1998; Tucson, AZ; United States
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  • 4
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    Using Multithreading for the Automatic Load Balancing of 2D Adaptive Finite Element Meshes (1998)
    In:  Other Sources
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    Publication Date: 2019-07-18
    Description: In this paper, we present a multi-threaded approach for the automatic load balancing of adaptive finite element (FE) meshes. The platform of our choice is the EARTH multi-threaded system which offers sufficient capabilities to tackle this problem. We implement the question phase of FE applications on triangular meshes, and exploit the EARTH token mechanism to automatically balance the resulting irregular and highly nonuniform workload. We discuss the results of our experiments on EARTH-SP2, an implementation of EARTH on the IBM SP2, with different load balancing strategies that are built into the runtime system.
    Keywords: Numerical Analysis
    Type: 5th International Symposium on Solving Irregularly Structured Problems in Parallel; Aug 09, 1998 - Aug 11, 1998; Berkley, CA; United States
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  • 5
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    Finding New Math Identities by Computer (1996)
    In:  Other Sources
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    Publication Date: 2019-07-18
    Description: Recently a number of interesting new mathematical identities have been discovered by means of numerical searches on high performance computers, using some newly discovered algorithms. These include the following: pi = ((sup oo)(sub k=0))(Sigma) (1 / 16) (sup k) ((4 / 8k+1) - (2 / 8k+4) - (1 / 8k+5) - (1 / 8k+6)) and ((17 pi(exp 4)) / 360) = ((sup oo)(sub k=1))(Sigma) (1 + (1/2) + (1/3) + ... + (1/k))(exp 2) k(exp -2), zeta(3, 1, 3, 1, ..., 3, 1) = (2 pi(exp 4m) / (4m+2)! where m = number of (3,1) pairs. and where zeta(n1,n2,...,nr) = (sub k1 (is greater than) k2 (is greater than) ... (is greater than) kr)(Sigma) (1 / (k1 (sup n1) k2 (sup n2) ... kr (sup nr). The first identity is remarkable in that it permits one to compute the n-th binary or hexadecimal digit of pu directly, without computing any of the previous digits, and without using multiple precision arithmetic. Recently the ten billionth hexadecimal digit of pi was computed using this formula. The third identity has connections to quantum field theory. (The first and second of these been formally established; the third is affirmed by numerical evidence only.) The background and results of this work will be described, including an overview of the algorithms and computer techniques used in these studies.
    Keywords: Numerical Analysis
    Type: Jan 29, 1997; Portland, OR; United States
    Format: text
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