A multiple-precision division algorithm

Smith, David M.

doi:10.1090/s0025-5718-96-00688-6

Cited by 15 publications

(6 citation statements)

References 7 publications

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“…Instead, the normalization operation may be performed only occasionally. See [8] for details. On an IBM RS6000/590 workstation, this new division routine is as much as four times faster than the previous routine.…”

Section: The Fortran-90 Mpfun Packagementioning

confidence: 99%

A Fortran 90-based multiprecision system

Bailey

1995

ACM Trans. Math. Softw.

190

199

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The author has developed a new version of his Fortran multiprecision computation system that is based on the Fortran-90 language. With this new approach, a translator program is not required -translation of Fortran code for multiprecision is accomplished by merely utilizing advanced features of Fortran-90, such as derived data types and operator extensions. This approach results in more reliable translation and also permits programmers of multiprecision applications to utilize the full power of the Fortran-90 language.Three multiprecision datatypes are supported in this system: multiprecision integer, real and complex. All the usual Fortran conventions for mixed mode operations are supported, and many of the Fortran intrinsics, such as SIN, EXP and MOD, are supported with multiprecision arguments. This paper also briefly describes an interesting application of this software, wherein new number-theoretic identities have been discovered by means of multiprecision computations. IntroductionReaders may be familiar with the author's previous multiprecision system [3], which consists of the TRANSMP translator program and the MPFUN package of multiprecision (MP) computation routines. Together they permit one to write straightforward Fortran-77 code that can be executed using an arbitrarily high level of numeric precision.From its inception, the TRANSMP program was intended only as an interim tool until Fortran-90 was available. This is because advanced Fortran-90 features such as derived data types and operator extensions permit one to implement multiprecision translation in a much more natural way. Now that day has arrived -Fortran-90 is currently available on several computer systems, and it soon will be available from all major vendors of scientific computers. Accordingly, the author has written a set of Fortran-90 modules that permit the user to handle MP data like any other Fortran data type.With the new Fortran-90 based system, one declares variables to be of type MP integer, MP real or MP complex using Fortran-90 type statements. With a few exceptions, one can then write ordinary Fortran-90 code involving these variables. In particular, arithmetic operations involving these variables are performed with a numeric precision level that can be set to an arbitrarily high level. Also, most of the Fortran intrinsic functions, such as SIN, EXP and MOD, are defined with MP arguments.In comparison to the TRANSMP approach, there are a few disappointments. To begin with, one has to give up the ability to run MP source code, without change, as a standard single precision or double precision program. Also, features such as read/write statements are not as elegant in the new system -subroutines must now be called for formatted MP read and write.On the other hand, features such as generic functions work much better in the Fortran-90 version. Also, the coverage of Fortran features is more complete with the Fortran-90 version than with TRANSMP -programmers can now utilize the full power of the Fortran-90 language in a M...

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Section: The Fortran-90 Mpfun Packagementioning

confidence: 99%

A Fortran 90-based multiprecision system

Bailey

1995

ACM Trans. Math. Softw.

190

199

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“…A computer program utilizing multiple-precision arithmetic based on methods described by Smith (1996Smith ( , 2003 was written to evaluate the exact formula, Equation (2), employing the Wronskian in forward recursion, which provides completely stable computations. This way, function values of γ (β) were obtained up to an argument of β = 20,000.…”

Section: Coefficientsmentioning

confidence: 99%

Evaluation of the Exact Coagulation Kernel under Simultaneous Brownian Motion and Gravitational Settling

Sajo

2008

Aerosol Science and Technology

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The analytic representation of the exact coagulation kernel for combined Brownian motion and gravitational settling entails an infinite sum of ratios of modified Bessel functions. Heretofore, the evaluation of this sum has been limited to relatively small arguments due to computational difficulties, hindering the numerical solution of the general dynamic equation in computational aerosol transport simulations. An approximation and an asymptotic extension have been available, but their accuracy for large arguments has not been established, and they have been shown to lead to numerical diffusion and oscillation. Using multiple-precision arithmetic, exact values for the kernel are presented in this paper for large arguments, and it is shown that both the asymptotic and approximating formulae provide a poor fit in the range of practical interest. Based on the results, the asymptotic formula is rectified and it is shown that the approximation is no longer needed. Practical formulae are given to facilitate rapid numerical evaluation of the exact kernel for all arguments.

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“…Due to the higher dimensionality of the problem it is not possible to obtain simple analytical expressions for the series related to the observables ŴX (q; t) and S (X) P (q; t), requiring a numerical approach. The numerical evaluation of series is a nontrivial problem and in the application of the next section we used the Smith [37] routines package of multiple precise computation.…”

Section: S (X)mentioning

confidence: 99%