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  • NUMERICAL ANALYSIS  (4)
  • MATHEMATICS AND INFORMATION SCIENCES  (2)
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  • 1
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    Numerical results on the transcendence of constants involving pi, e, and Euler's constant (1988)
    In:  Other Sources
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    Publication Date: 2011-08-19
    Description: The existence of simple polynomial equations (integer relations) for the constants e/pi, e + pi, log pi, gamma (Euler's constant), e exp gamma, gamma/e, gamma/pi, and log gamma is investigated by means of numerical computations. The recursive form of the Ferguson-Fourcade algorithm (Ferguson and Fourcade, 1979; Ferguson, 1986 and 1987) is implemented on the Cray-2 supercomputer at NASA Ames, applying multiprecision techniques similar to those described by Bailey (1988) except that FFTs are used instead of dual-prime-modulus transforms for multiplication. It is shown that none of the constants has an integer relation of degree eight or less with coefficients of Euclidean norm 10 to the 9th or less.
    Keywords: NUMERICAL ANALYSIS
    Type: Mathematics of Computation (ISSN 0025-5718); 50; 275-281
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  • 2
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    Numerical results on relations between fundamental constants using a new algorithm (1989)
    In:  Other Sources
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    Publication Date: 2011-08-19
    Description: An efficient algorithm is described for finding whether or not certain fundamental mathematical constants satisfy simple algebraic polynomials. The algorithm, which finds whether an integer relation exists for a vector of real numbers, or else establishes bounds within which no relation can exist. The algorithm is implemented on high-speed computers, using multiprecision arithmetic. Numerical results are summarized, and other possible applications for the algorithm are discussed.
    Keywords: NUMERICAL ANALYSIS
    Type: Mathematics of Computation (ISSN 0025-5718); 53; 649-656
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  • 3
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    The computation of pi to 29,360,000 decimal digits using Borweins' quartically convergent algorithm (1988)
    In:  Other Sources
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    Publication Date: 2011-08-19
    Description: The quartically convergent numerical algorithm developed by Borwein and Borwein (1987) for 1/pi is implemented via a prime-modulus-transform multiprecision technique on the NASA Ames Cray-2 supercomputer to compute the first 2.936 x 10 to the 7th digits of the decimal expansion of pi. The history of pi computations is briefly recalled; the most recent algorithms are characterized; the implementation procedures are described; and samples of the output listing are presented. Statistical analyses show that the present decimal expansion is completely random, with only acceptable numbers of long repeating strings and single-digit runs.
    Keywords: NUMERICAL ANALYSIS
    Type: Mathematics of Computation (ISSN 0025-5718); 50; 283-296
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  • 4
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    The fractional Fourier transform and applications (1991)
    In:  Other Sources
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    Publication Date: 2019-07-12
    Description: This paper describes the 'fractional Fourier transform', which admits computation by an algorithm that has complexity proportional to the fast Fourier transform algorithm. Whereas the discrete Fourier transform (DFT) is based on integral roots of unity e exp -2(pi)i/n, the fractional Fourier transform is based on fractional roots of unity e exp -2(pi)i(alpha), where alpha is arbitrary. The fractional Fourier transform and the corresponding fast algorithm are useful for such applications as computing DFTs of sequences with prime lengths, computing DFTs of sparse sequences, analyzing sequences with noninteger periodicities, performing high-resolution trigonometric interpolation, detecting lines in noisy images, and detecting signals with linearly drifting frequencies. In many cases, the resulting algorithms are faster by arbitrarily large factors than conventional techniques.
    Keywords: NUMERICAL ANALYSIS
    Type: SIAM Review (ISSN 0036-1445); 33; 389-404
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  • 5
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    NASKERN Program Tests Computers (1988)
    In:  Other Sources
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    Publication Date: 2019-07-12
    Description: Computer program, NASKERN developed to aid in evaluation of performances of supercomputers. Consists of seven test-kernel programs performing computations typical of supercomputer calculations. Written in FORTRAN 77.
    Keywords: MATHEMATICS AND INFORMATION SCIENCES
    Type: ARC-11726 , NASA Tech Briefs (ISSN 0145-319X); 12; 5; P. 60
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  • 6
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    Calculating Numbers To Arbitrarily High Precision (1988)
    In:  Other Sources
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    Publication Date: 2019-07-12
    Description: PITEST program is package of fast routines for performing multiprecision arithmetic. Written in FORTRAN 77.
    Keywords: MATHEMATICS AND INFORMATION SCIENCES
    Type: ARC-11725 , NASA Tech Briefs (ISSN 0145-319X); 12; 5; P. 61
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