Summary
Applying Newton's method to a particular system of nonlinear equations we derive methods for the simultaneous computation of all zeros of generalized polynomials. These generalized polynomials are from a function space satisfying a condition similar to Haar's condition. By this approach we bring together recent methods for trigonometric and exponential polynomials and a well-known method for ordinary polynomials. The quadratic convergence of these methods is an immediate consequence of our approach and needs not to be proved explicitly. Moreover, our approach yields new interesting methods for ordinary, trigonometric and exponential polynomials and methods for other functions occuring in approximation theory.
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References
Alt, R.: Computing roots of polynomials on vector processing machines. Appl. Numer. Math.1, 299–308 (1985)
Angelova, E., Semerdzhiev, K.: Methods for the simultaneous approximate derivation of the roots of algebraic, trigonometric and exponential equations, USSR Comput. Maths. Math. Phys.22, 226–232 (1982)
Braess, D.: Nonlinear approximation theory. Berlin Heidelberg New York Tokyo: Springer (1986)
Dočev, K.: An alternative method of Newton for simultaneous calculation of all the roots of a given algebraic equation (Bulgarian). Phys. Math. J. Bulg. Acad. Sci.5, 136–139 (1962)
Durand, E.: Solutions numériques des équations algébriques, tome I. Paris: Masson (1960)
Kerner, I.: Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen. Numer. Math.8, 290–294 (1966)
Makrelov, I., Semerdzhiev, K.: On the convergence of two methods for the simultaneous finding of all roots of exponential polynomials. IMA J. Numer. Anal.5, 191–200 (1985)
Meinardus, G.: Approximation of functions: theory and numerical methods. Berlin Heidelberg New York: Springer (1967)
Murnaghan, F.D., Wrench, Jr., J.W.: The determination of the Chebyshev approximating polynomial for a differentiable function. Math. Tabl. Aids Comput.13, 185–193 (1959)
Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. New York: Academic Press (1970)
Weierstraß, K.: Neuer Beweis des Fundamentalsatzes der Algebra, Mathematische Werke Bd. III, Mayer u. Müller Berlin, pp. 251–269 (1903)
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Frommer, A. A unified approach to methods for the simultaneous computation of all zeros of generalized polynomials. Numer. Math. 54, 105–116 (1988). https://doi.org/10.1007/BF01403894
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DOI: https://doi.org/10.1007/BF01403894