A unified approach to methods for the simultaneous computation of all zeros of generalized polynomials

Frommer, Andreas

doi:10.1007/bf01403894

Cited by 16 publications

(10 citation statements)

References 7 publications

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“…Variants and generalizations of this method have been discussed, e.g., by Samelson (1958), Kerner (1966), Schro der (1969), Grau (1971), Aberth (1973), Green et al (1976), Pasquini and Trigiante (1985), Frommer (1988), andCarstensen (1992). These publications analyze the Newton algorithm from the usual point of view of numerical analysis, where floating point operations are counted.…”

Section: Related Research and Open Problemsmentioning

confidence: 97%

Partial Fraction Decomposition in C(z) and Simultaneous Newton Iteration for Factorization in C[z]

Kirrinnis

1998

Journal of Complexity

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The subject of this paper is fast numerical algorithms for factoring univariate polynomials with complex coefficients and for computing partial fraction decompositions (PFDs) of rational functions in C(z). Numerically stable and computationally feasible versions of PFD are specified first for the special case of rational functions with all singularities in the unit disk (the``bounded case'') and then for rational functions with arbitrarily distributed singularities. Two major algorithms for computing PFDs are presented: The first one is an extension of the``splitting circle method'' by A. Scho nhage (``The Fundamental Theorem of Algebra in Terms of Computational Complexity,'' Technical Report, Univ. Tu bingen, 1982) for factoring polynomials in C[z] to an algorithm for PFD. The second algorithm is a Newton iteration for simultaneously improving the accuracy of all factors in an approximate factorization of a polynomial resp. all partial fractions of an approximate PFD of a rational function. Algorithmically useful starting value conditions for the Newton algorithm are provided. Three subalgorithms are of independent interest. They compute the product of a sequence of polynomials, the sum of a sequence of rational functions, and the modular representation of a polynomial. All algorithms are described in great detail, and numerous technical auxiliaries are provided which are also useful for the design and analysis of other algorithms in computational complex analysis. Combining the splitting circle method with simultaneous Newton iteration yields favourable time bounds (measured in bit operations) for PFD, polynomial factoring, and root calculation. In particular, the time bounds for computing high accuracy PFDs, high accuracy factorizations, and high accuracy approximations for zeros of squarefree polynomials are linear in the output size (and hence optimal) up to logarithmic factors.

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Section: Related Research and Open Problemsmentioning

confidence: 97%

Partial Fraction Decomposition in C(z) and Simultaneous Newton Iteration for Factorization in C[z]

Kirrinnis

1998

Journal of Complexity

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“…Firstly, the continuation method of [3] is sketched which gives method (MN) using the Taylor method of N-th order for solving a certain initial value problem. Secondly, it is proved that (M2) is method (M) in the unified approach of [7] which is Newton-Raphson's method for a certain system of nonlinear equations.…”

Section: Continuation Process Unified Method Euler Methodsmentioning

confidence: 99%

“…(1)). [] In [3], [9] and [7] only (M2) is considered and convergence is proved only for particular cases. We stress that Theorem 2 gives convergence for all methods simultaneously and also for all methods of higher order N.…”

Section: Z~v) (T) 2~i 0v~ (1 -T)q(z) + Tf(z)/mentioning

confidence: 99%

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On a class of higher order methods for simultaneous rootfinding of generalized polynomials

Carstensen

Reinders

1993

Numer. Math.

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Summary. Using the argument principle higher order methods for simultaneous computation of all zeros of generalized polynomials (like algebraic, trigonometric and exponential polynomials or exponential sums) are derived. The methods can also be derived following the continuation principle from [3]. Thereby, the unified approach of I-7] is enlarged to arbitrary order N. The local convergence as well as a-priori and a-posteriori error estimates for these methods are treated on a general level. Numerical examples are included Subject Classification (1991): 65H05, 65H10 Mathematics

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“…Other reformulations of this method are also known as Durand-Kerner or Dochev's method. Nowadays, there are several methods that generalize this approach in the univariate case (see Frommer, 1988;Bellido, 1992;Sendov et al, 1994, Bini, 1996, Pan, 1997or Fortune, 2001 for a good survey). There are several ways to introduce this method.…”

Section: Univariate Casementioning

confidence: 98%

Relations Between Roots and Coefficients, Interpolation and Application to System Solving

Mourrain

Ruatta

2002

Journal of Symbolic Computation

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We propose an algebraic framework to represent zero-dimensional algebraic systems. In this framework, we give new interpolation formulae. We use this good representation of the algebraic systems to develop a generalization of Weierstrass's method to the multivariate systems. This method allows us to approximate simultaneously all the roots of an algebraic system. We obtain an effective iteration function with a quadratic convergence in a neighbourhood of the solutions. We use this Weierstrass iteration function in a continuation method to obtain a global method. Experiments are exposed to underline the efficiency of the approach.

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A unified approach to methods for the simultaneous computation of all zeros of generalized polynomials

Cited by 16 publications

References 7 publications

Partial Fraction Decomposition in C(z) and Simultaneous Newton Iteration for Factorization in C[z]

Partial Fraction Decomposition in C(z) and Simultaneous Newton Iteration for Factorization in C[z]

On a class of higher order methods for simultaneous rootfinding of generalized polynomials

Relations Between Roots and Coefficients, Interpolation and Application to System Solving

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