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Article title SOLUTION OF ALGEBRAIC EQUATIONS BY THE METHOD OF NIKIPORTS-RUTISHAUSER’S
Authors V.F. Guzik, G.A. Kirichenko, V.I. Shmoylov
Section SECTION II. MODELING OF COMPLEX SYSTEMS
Month, Year 06, 2015 @en
Index UDC 517.524
DOI
Abstract Presents analytical expressions representing all the roots of a random algebraic equation of the n-th degree through the ratio of the original equation. These formulas consist of two relations infinite Toeplitz determinants, diagonal elements of which are the coefficients of the algebraic equation. Such structures were called continuous fractions of Nikiports. For the efficient calculation of the values of continued fractions of Nikiports used recurrent algorithm of Rutishauser. When finding the roots of a polynomial algorithm for the summation of divergent continued fractions (r/φ-algorithm). Complex roots are determined from consideration of the values of a long series of suitable continuous fractions. The proposed algorithm for finding zeros of a polynomial has two features in comparison with the existing methods of solving algebraic equations. The first and, perhaps, a fundamentally important feature: a simple analytical method for recording all roots of an equation of n-th degree in the coefficients of the original equation. The second feature of the proposed algorithm for finding zeros of a polynomial of n-th degree, the simplicity and regularity of the information graph algorithm, which makes it attractive for hardware implementation in the final field of supercomputers with reconfigurable structure. As an example, consider the solution of the algebraic equation 39-th degree.

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Keywords Algebraic equations; infinite Toeplitz determinants; divergent continuous fraction; r/φ-algorithm.
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