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  • COMPUTER PROGRAMMING AND SOFTWARE  (5)
  • NUMERICAL ANALYSIS  (4)
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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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    NASA TECHNICAL REPORTS
  • 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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    NASA TECHNICAL REPORTS
  • 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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    A high-performance FFT algorithm for vector supercomputers (1988)
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    Publication Date: 2011-08-19
    Description: Many traditional algorithms for computing the fast Fourier transform (FFT) on conventional computers are unacceptable for advanced vector and parallel computers because they involve nonunit, power-of-two memory strides. A practical technique for computing the FFT that avoids all such strides and appears to be near-optimal for a variety of current vector and parallel computers is presented. Performance results of a program based on this technique are given. Notable among these results is that a FORTRAN implementation of this algorithm on the CRAY-2 runs up to 77-percent faster than Cray's assembly-coded library routine.
    Keywords: COMPUTER PROGRAMMING AND SOFTWARE
    Type: International Journal of Supercomputer Applications (ISSN 0890-2720); 2; 82-87
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  • 5
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    Floating point arithmetic in future supercomputers (1989)
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    Publication Date: 2011-08-19
    Description: Considerations in the floating-point design of a supercomputer are discussed. Particular attention is given to word size, hardware support for extended precision, format, and accuracy characteristics. These issues are discussed from the perspective of the Numerical Aerodynamic Simulation Systems Division at NASA Ames. The features believed to be most important for a future supercomputer floating-point design include: (1) a 64-bit IEEE floating-point format with 11 exponent bits, 52 mantissa bits, and one sign bit and (2) hardware support for reasonably fast double-precision arithmetic.
    Keywords: COMPUTER PROGRAMMING AND SOFTWARE
    Type: International Journal of Supercomputer Applications (ISSN 0890-2720); 3; 86-90
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  • 6
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    Using Strassen's algorithm to accelerate the solution of linear systems (1990)
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    Publication Date: 2019-07-13
    Description: Strassen's algorithm for fast matrix-matrix multiplication has been implemented for matrices of arbitrary shapes on the CRAY-2 and CRAY Y-MP supercomputers. Several techniques have been used to reduce the scratch space requirement for this algorithm while simultaneously preserving a high level of performance. When the resulting Strassen-based matrix multiply routine is combined with some routines from the new LAPACK library, LU decomposition can be performed with rates significantly higher than those achieved by conventional means. We succeeded in factoring a 2048 x 2048 matrix on the CRAY Y-MP at a rate equivalent to 325 MFLOPS.
    Keywords: COMPUTER PROGRAMMING AND SOFTWARE
    Type: NAS 2-12961 , The Journal of Supercomputing (ISSN 0920-8542); p. 357-371
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  • 7
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    FFTs in external or hierarchical memory (1989)
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    Publication Date: 2019-07-13
    Description: A description is given of advanced techniques for computing an ordered FFT on a computer with external or hierarchical memory. These algorithms (1) require as few as two passes through the external data set, (2) use strictly unit stride, long vector transfers between main memory and external storage, (3) require only a modest amount of scratch space in main memory, and (4) are well suited for vector and parallel computation. Performance figures are included for implementations of some of these algorithms on Cray supercomputers. Of interest is the fact that a main memory version outperforms the current Cray library FFT routines on the Cray-2, the Cray X-MP, and the Cray Y-MP systems. Using all eight processors on the Cray Y-MP, this main memory routine runs at nearly 2 Gflops.
    Keywords: COMPUTER PROGRAMMING AND SOFTWARE
    Type: Supercomputing ''89; Nov 13, 1989 - Nov 17, 1989; Reno, NV; United States
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    NASA TECHNICAL REPORTS
  • 8
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    The fractional Fourier transform and applications (1991)
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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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  • 9
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    A Strassen-Newton algorithm for high-speed parallelizable matrix inversion (1988)
    In:  Other Sources
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    Publication Date: 2019-07-13
    Description: Techniques are described for computing matrix inverses by algorithms that are highly suited to massively parallel computation. The techniques are based on an algorithm suggested by Strassen (1969). Variations of this scheme use matrix Newton iterations and other methods to improve the numerical stability while at the same time preserving a very high level of parallelism. One-processor Cray-2 implementations of these schemes range from one that is up to 55 percent faster than a conventional library routine to one that is slower than a library routine but achieves excellent numerical stability. The problem of computing the solution to a single set of linear equations is discussed, and it is shown that this problem can also be solved efficiently using these techniques.
    Keywords: COMPUTER PROGRAMMING AND SOFTWARE
    Type: Supercomputing ''88; Nov 14, 1988 - Nov 18, 1988; Orlando, FL; United States
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