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Locating and computing in parallel all the simple roots of special functions using PVM
, 2001
"... An algorithm is proposed for locating and computing in parallel and with certainty all the simple roots of any twice continuously differentiable function in any specific interval. To compute with certainty all the roots, the proposed method is heavily based on the knowledge of the total number of ro ..."
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Cited by 8 (7 self)
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An algorithm is proposed for locating and computing in parallel and with certainty all the simple roots of any twice continuously differentiable function in any specific interval. To compute with certainty all the roots, the proposed method is heavily based on the knowledge of the total number of roots within the given interval. To obtain this information we use results from topological degree theory and, in particular, the KroneckerPicard approach. This theory gives a formula for the computation the total number of roots of a system of equations within a given region, which can be computed in parallel. With this tool in hand, we construct a parallel procedure for the localization and isolation of all the roots by dividing the given region successively and applying the above formula to these subregions until the final domains contain at the most one root. The subregions with no roots are discarded, while for the rest am#zxxv;Tqz of the wellknown bisectionmsect isem#### for thecomxzWv;T of the contained root. The new aspect of the present contribution is that the comqTqv;T# of the total numlv of zeros using the KroneckerPicard integral as well as the localization and comWIx#xv; of all the roots isperformv in parallel using the parallel virtual mrtual (PVM). PVM is an integrated set of software tools and libraries that emtv### a generalpurpose, flexible, heterogeneous concurrent comcurre framurre on interconnected comrconn of varied architectures. The proposed algorithm has large granularity and low synchronization, and is robust. It has been imv#Wq#v; and tested and our experience is that it canmnvq## comq# with certainty all the roots in a certain interval. Performance information from momrm comrmv##W related to a recently proposed conjecture due to Elbert (this issue, J.ComI#q Appl. Math. 1...) is reported.
A Proposed Software Test Service for Special Functions
 Quality of Numerical Software: Assessment and Enhancement
, 1997
"... This is a proposal to develop a software test service at the National Institute of Standards and Technology for use in testing the accuracy, or numerical precision, of mathematical software for special functions. The service would use the World Wide Web to receive test requests and return test resul ..."
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Cited by 2 (1 self)
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This is a proposal to develop a software test service at the National Institute of Standards and Technology for use in testing the accuracy, or numerical precision, of mathematical software for special functions. The service would use the World Wide Web to receive test requests and return test results. The tests would be run on a network of workstations at the Institute. It is hoped that such a service will be of practical utility to anyone who uses special functions in physics or other applications, and that it will stimulate the interest of applied mathematicians who are interested in the computation of special functions as well as computer scientists who are interested in innovative uses of the Internet. The author solicits comments on any aspect of the proposed service. 1 Introduction Mathematical software is deeply embedded in the computing environment. Since this environment is evolving rapidly, its impact on mathematical software needs to be revisited regularly. Progress in p...
Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528,
, 2006
"... The zzeros of the modified Bessel function of the third kind Kν(z), also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order ν. Approximate expressions for the zeros, applicable in the cases of very small or very large ν, are given. Th ..."
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The zzeros of the modified Bessel function of the third kind Kν(z), also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order ν. Approximate expressions for the zeros, applicable in the cases of very small or very large ν, are given. The behaviour of the zeros for varying ν  or arg ν, obtained numerically, is illustrated by means of some graphics. Key words: Macdonald function, modified Bessel function of the third kind, Hankel function, zeros
Zeros of the Macdonald function of complex order
, 2006
"... The zzeros of the modified Bessel function of the third kind Kν(z), also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order ν. Approximate expressions for the zeros, applicable in the cases of very small or very large ν, are given. Th ..."
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The zzeros of the modified Bessel function of the third kind Kν(z), also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order ν. Approximate expressions for the zeros, applicable in the cases of very small or very large ν, are given. The behaviour of the zeros for varying ν  or arg ν, obtained numerically, is illustrated by means of some graphics. Key words: Macdonald function, modified Bessel function of the third kind, Hankel function, zeros
c ○ 2005 Society for Industrial and Applied Mathematics EFFICIENTLY COMPUTING MANY ROOTS OF A FUNCTION ∗
"... Abstract. We present a new bisection based method for counting and computing roots of a function in a given interval. Our method is focused on very large problems, i.e., instances with the number of roots of the order of hundreds. The method draws its power from the fact that the roots are expected ..."
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Abstract. We present a new bisection based method for counting and computing roots of a function in a given interval. Our method is focused on very large problems, i.e., instances with the number of roots of the order of hundreds. The method draws its power from the fact that the roots are expected to be many, and is able to discover a large percentage of them very efficiently. Its main advantage, apart from its efficiency, is the fact that it requires only the sign of the function at a certain point and not its actual value. Also, its simplicity makes it a suitable preprocessing step for reducing the size of the problem, prior to more robust but also more demanding methods. The algorithm is accompanied by a probabilistic analysis of its behavior, which shows that a simple existence criterion like Bolzano’s rule can be a powerful tool in the zerofinding process.
unknown title
, 1999
"... www.elsevier.com/locate/cam Locating and computing in parallel all the simple roots of special functions using PVM ..."
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www.elsevier.com/locate/cam Locating and computing in parallel all the simple roots of special functions using PVM