Introducing the theory of residues to approximate complex roots of the polynomials of the arbitrary order

Authors: Abramov N.K.
Published in issue: #5(82)/2023
DOI: 10.18698/2541-8009-2023-5-891
Category: Mathematics | Chapter: Computational Mathematics
Keywords: polynomial, complex plane, contour integral, algorithm, root, Simpson’s rule, circle, programming, numerical methods
Published: 19.05.2023

The paper presents results of development and implementation of a method for finding all the roots of a polynomial with real coefficients based on the concept of a contour integral in the complex plane, deduction of the complex function and technique of the approximate Riemann integration by the Simpson’s rule. Mathematical substantiation of the proposed algorithm correctness is provided, software implementation in the C++ programming language is presented with certain assumptions, specifics of working with the resulting program are described, and examples of the algorithm are also presented. Separately, the key property of the obtained algorithm was considered and proved, i.e. the guaranteed convergence (excepting the case of all roots with the same radius, but this case could be very easily recognized analytically), even in the entire absence of information about relative position of the polynomial roots implemented by the method successive application with a reduced step in case of the incomplete roots finding.

References

[1] Shafarevich I.R. Osnovy algebraicheskoy geometrii [Fundamentals of Algebraic Geometry]. Moscow, MCCME Publ., 2007, 589 p. (In Russ.).

[2] Fine B., Rosenberger G. The Fundamental Theorem of Algebra. New York, Springer-Verlag, 1997.

[3] Aluffi P. Algebra: Chapter 0 (Graduate Studies in Mathematics) — American Mathematical Society, 2009. ISBN 0-8218-4781-3.

[4] Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P. Numerical Recipes: The Art of Scientific Computing. New York, Cambridge University Press, 2007, 1262 p.

[5] Golub G.H., Van Loan C.F. Matrix Calculations. JHU Press, 1996, 694 p. (Russ. ed.: Golub Dzh., Van Loun Ch. Matrichnye vychisleniya. Moscow, Mir Publ., 1999.).

[6] Aleksandrov P.S. Vvedenie v teoriyu mnozhestv i obshchuyu topologiyu [Introduction to Set Theory and General Topology]. Moscow, Lan Publ., 2010, 368 p. (In Russ.).

[7] Marsden J.E., Hoffman M.J. Basic Complex Analysis. W.H. Freeman, 516 p.

[8] Domrin A.V., Sergeev A.G. Lektsii po kompleksnomu analizu: V 2 ch. Ch. 1: Pervoe polugodie [Lectures on Complex Analysis: In 2 parts]. Moscow, MIAN Publ., 176 p. (In Russ.).

[9] Karmanov V.G. Matematicheskoe programmirovanie [Mathematical Programming]. Moscow, Publishing House of Phys.-Math. Literature, 2004, 264 p. (In Russ.).

[10] Stroustrup B. The C++ programming language. Special edition. AT&T Labs Murray Hill, New Jersey, 1997. (Russ. ed.: Straustrup B. Yazyk programmirovaniya C++. Spetsial’noe izdanie. Moscow, Binom-Press Publ., 2007, 1104 p.).