A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain

Ying, Xingren; Katz, Irving

doi:10.1007/bf01395882

Cited by 47 publications

(21 citation statements)

References 5 publications

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“…The number of zeros is then the change of the argument divided by 2π. The segments in which the contour is divided have to be chosen small enough that the change of the argument over each segment is smaller than π. Adaptive procedures to achieve this have been proposed by, amongst others, Ying and Katz [31], but for simplicity we determined the partition of the contour by trial and error.…”

Section: Global Eigenvalue Search Methodsmentioning

confidence: 99%

Computing Stability of Multidimensional Traveling Waves

Ledoux

Malham²,

Niesen³

et al. 2009

SIAM J. Appl. Dyn. Syst.

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Abstract. We present a numerical method for computing the pure-point spectrum associated with the linear stability of multi-dimensional travelling fronts to parabolic nonlinear systems. Our method is based on the Evans function shooting approach. Transverse to the direction of propagation we project the spectral equations onto a finite Fourier basis. This generates a large, linear, one-dimensional system of equations for the longitudinal Fourier coefficients. We construct the stable and unstable solution subspaces associated with the longitudinal far-field zero boundary conditions, retaining only the information required for matching, by integrating the Riccati equations associated with the underlying Grassmannian manifolds. The Evans function is then the matching condition measuring the linear dependence of the stable and unstable subspaces and thus determines eigenvalues. As a model application, we study the stability of two-dimensional wrinkled front solutions to a cubic autocatalysis model system. We compare our shooting approach with the continuous orthogonalization method of Humpherys and Zumbrun. We then also compare these with standard projection methods that directly project the spectral problem onto a finite multi-dimensional basis satisfying the boundary conditions.

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Section: Global Eigenvalue Search Methodsmentioning

confidence: 99%

Computing Stability of Multidimensional Traveling Waves

Ledoux

Malham²,

Niesen³

et al. 2009

SIAM J. Appl. Dyn. Syst.

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“…19 To avoid any missing complex-valued roots, application of the Principle of the Argument is commonly favoured. [20][21][22] A numerical algorithm for obtaining the eigenvalues k p at each excitation frequency in the acousto-elastic layered system was written in the programming language Fortran (Intel Visual Fortran Compiler Release V11.0.062) by the authors. A detailed discussion of the algorithm falls outside the scope of the present paper.…”

Section: B the Complete Wave Spectrum Of An Acousto-elastic Waveguidementioning

confidence: 99%

The significance of the evanescent spectrum in structure-waveguide interaction problems

Tsouvalas

Dalen

Metrikine

2015

The Journal of the Acoustical Society of America

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Modal decomposition is often applied in elastodynamics and acoustics for the solution of problems related to propagation of mechanical disturbances in waveguides. One of the key elements of this method is the solution of an eigenvalue problem for obtaining the roots of the dispersion equation, which signify the wavenumbers of the waves that may exist in the system. For non-dissipative media, the wavenumber spectrum consists of a finite number of real roots supplemented by infinitely many imaginary and complex ones. The former refer to the propagating modes in the medium, whereas the latter constitute the so-called evanescent spectrum. This study investigates the significance of the evanescent spectrum in structure-waveguide interaction problems. Two cases are analysed, namely, a beam in contact with a fluid layer and a cylindrical shell interacting with an acousto-elastic waveguide. The first case allows the introduction of a modal decomposition method and the establishment of appropriate criteria for the truncation of the modal expansions in a simple mathematical framework. The second case describes structure-borne wave radiation in an offshore environment during the installation of a pile with an impact hammer-a problem that has raised serious concerns in recent years due to the associated underwater noise pollution.

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“…The localization of several roots closest to the origin using the argument principle followed by several Newton iterations can be faster than the localization of all the roots by some general-purpose method. Moreover for more general boundary conditions, e.g., involving multiplication by an analytic function of λ, the approximate characteristic function may not be a polynomial, and the argument principle becomes even more useful (see, e.g., [6,14,53]). The argument principle consists in the following, see, e.g., [10].…”

Section: 3mentioning

confidence: 99%

Spectral parameter power series for Sturm-Liouville equations with a potential polynomially dependent on the spectral parameter and Zakharov-Shabat systems

Kravchenko

Torba

Velasco-García

2015

Journal of Mathematical Physics

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Abstract. A spectral parameter power series (SPPS) representation for solutions of Sturm-Liouville equations of the formis obtained. It allows one to write a general solution of the equation as a power series in terms of the spectral parameter λ. The coefficients of the series are given in terms of recursive integrals involving a particular solution of the equation (pu 0 ) +qu0 = 0. The convenient form of the solution of ( * ) provides an efficient numerical method for solving corresponding initial value, boundary value and spectral problems. A special case of the Sturm-Liouville equation ( * ) arises in relation with the Zakharov-Shabat system. We derive an SPPS representation for its general solution and consider other applications as the one-dimensional Dirac system and the equation describing a damped string. Several numerical examples illustrate the efficiency and the accuracy of the numerical method based on the SPPS representations which besides its natural advantages like the simplicity in implementation and accuracy is applicable to the problems admitting complex coefficients, spectral parameter dependent boundary conditions and complex spectrum.

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A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain

Cited by 47 publications

References 5 publications

Computing Stability of Multidimensional Traveling Waves

Computing Stability of Multidimensional Traveling Waves

The significance of the evanescent spectrum in structure-waveguide interaction problems

Spectral parameter power series for Sturm-Liouville equations with a potential polynomially dependent on the spectral parameter and Zakharov-Shabat systems

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