ZEAL: A mathematical software package for computing zeros of analytic functions

Kravanja, Peter; Barel, Marc Van; Ragos, Omiros; Vrahatis, M.N.; Zafiropoulos, F.A.

doi:10.1016/s0010-4655(99)00429-4

Cited by 32 publications

(26 citation statements)

References 3 publications

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“…Newton's identities are used to relate the moments (7) to (8), which can be solved by well known methods for polynomials. 1 Appendix A.1 describes this procedure in detail.…”

Section: Argument Principle Methodsmentioning

confidence: 99%

“…It is widely applicable, and requires only that the specified search domain be meromorphic, that is, free of branch cuts and essential singularities. Several freely available implementations of the argument principle method exist, 7,8 as well as related methods, 9 which are applicable to a wide class of transcendental equations. However, many variations of the argument principle method are possible, discussed in greater detail in Section 3, and specific choices must be made for each numerical implementation.…”

Section: Introductionmentioning

confidence: 99%

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Robust location of optical fiber modes via the argument principle method

Chen

Sivan

2017

Computer Physics Communications

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We implement a robust, globally convergent root search method for transcendental equations guaranteed to locate all complex roots within a specified search domain, based on Cauchy's residue theorem. Although several implementations of the argument principle already exist, ours has several advantages: it allows singularities within the search domain and branch points are not fatal to the method. Furthermore, our implementation is simple and is written in MATLAB, fulfilling the need for an easily integrated implementation which can be readily modified to accommodate the many variations of the argument principle method, each of which is suited to a different application. We apply the method to the step index fiber dispersion relation, which has become topical due to the recent proliferation of high index contrast fibers. We also find modes with permittivity as the eigenvalue, catering to recent numerical methods that expand the radiation of sources by eigenmodes.

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“…Newton's identities are used to relate the moments (7) to (8), which can be solved by well known methods for polynomials. 1 Appendix A.1 describes this procedure in detail.…”

Section: Argument Principle Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Robust location of optical fiber modes via the argument principle method

Chen

Sivan

2017

Computer Physics Communications

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“…Compared with the Mie series solution (3), the residue series solution (20) converges extremely fast at high frequencies. Hence, only the first several terms in (20) are required for numerically accurate solution.…”

Section: Watson Transformationmentioning

confidence: 99%

“…The pioneering work was done by Delves and Lyness [17]. Kravanja developed the ideas and used the formal orthogonal polynomials to approximate the zeros in the interior of a positively oriented Jordan curve [18][19][20]. Further, Protopopov proposed a method to locate the zeros of function with large values of derivatives [21].…”

Section: Computing Zeros Of Hankel Functionsmentioning

confidence: 99%

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High Frequency Scattering by an Impenetrable Sphere

Sha

Chew²

2009

PIER

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Abstract-The high frequency scattering of a scalar plane wave from an impenetrable sphere with a diameter of several thousand wavelengths is treated by the Sommerfeld-Watson transformation, the saddle-point technique (SPT), and the numerical steepest descent method (NSDM). Both the near and far fields for the sphere are computed within the observation angle range of 0 to 180 degree. First, with the aid of the Watson transformation, the fast-convergent residue series replacing the slow-convergent Mie series is derived. Second, a new algorithm for finding the zeros of the Hankel functions is developed. Third, a novel NSDM, which is adaptive to frequency and is hence frequency independent, is proposed to overcome the breakdown of the traditional SPT in the transition region. Numerical results show that when the observation angle is very small, the Mie series solution of the near-field will not be accurate due to error accumulation. Furthermore, using the proposed methods, the CPU times for both the near-field and far-field calculations are frequency independent with controllable error. This work can be used to benchmark future works for high-frequency scattering.

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Analysis of a discrete-time queue with general independent arrivals, general service demands and fixed service capacity

et al. 2015

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This paper analyzes a single-server discrete-time queueing model with general independent arrivals, where the service process of the server is characterized in two steps. Specifically, the model assumes that (1) each customer represents a random, arbitrarily distributed, amount of work for the server, the service demand, and (2) the server disposes of a fixed number of work units that can be executed per slot, the service capacity. For this non-classical queueing model, we obtain explicit closed-form results for the probability generating functions (pgf's) of the unfinished work in the system (expressed in work units) and the queueing delay of an arbitrary customer (expressed in time slots). Deriving the pgf of the number of customers in the system turns out to be hard, in general. Nevertheless, we derive this pgf explicitly in a number of special cases, i.e., either for geometrically distributed service demands, and/or for Bernoulli arrivals or geometric arrivals. The obtained results show that the tail distributions of the unfinished work, the customer delay and the system content all exhibit a geometric decay, with semi-analytic formulas for the decay rates available. Another interesting conclusion is that, for a given system load, the mean customer delay converges to constant limiting values when the service capacity per slot goes to infinity, and either the mean arrival rate or the mean service demand increases proportionally. Accurate approximative analytical expressions are available for these limiting values

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ZEAL: A mathematical software package for computing zeros of analytic functions

Cited by 32 publications

References 3 publications

Robust location of optical fiber modes via the argument principle method

Robust location of optical fiber modes via the argument principle method

High Frequency Scattering by an Impenetrable Sphere

Analysis of a discrete-time queue with general independent arrivals, general service demands and fixed service capacity

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