Cyclic and Non-Cyclic Iterations of Linear-Fractional Functions
https://doi.org/10.56304/S2304487X20020029
Abstract
Functional iterative processes fk(x) = f(fk-1(x)), k = 2, 3, where the initial function belongs to the class of non-degenerate linear fractional functions are considered. The aim of this work is to study all types of iterative processes that arise when the parameters are varied. To solve the appearing recurrence relations, matrix methods and complex numbers are used. Formulas for the coefficients of the kth iteration for any k depending on the coefficients of the initial function are obtained in the general form. Two invariants of iterative processes are defined. It is shown that cycles of the length n > 2 can exist only for complex conjugate eigenvalues of the coefficient matrix of a linear-fractional function. All initial functions that generate cycles of an arbitrary given length are found and explicit expressions are obtained for the coefficients of any element of the cycle in terms of the coefficients of the initial function. An example of a cycle of the maximum length n = 6, where all the coefficients of each iteration are integers, is given. For non-cyclic processes, the behavior of the kth iteration is studied for k → and limit functions are determined in the cases of convergence. Non-cyclic iterative processes are divided into converging (real eigenvalues) and diverging (complex conjugate values that do not satisfy cyclic conditions). Converging iterations have a constant function as their limit function.
and limit functions are determined in the cases of convergence. Non-cyclic iterative processes are divided into converging (real eigenvalues) and diverging (complex conjugate values that do not satisfy cyclic conditions). Converging iterations have a constant function as their limit function.About the Author
V. P. Cherniavsky
Sarov Physical Technical Institute, National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
607190
Sarov
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Review
For citations:
Cherniavsky V.P. Cyclic and Non-Cyclic Iterations of Linear-Fractional Functions. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2020;9(2):139-146. (In Russ.) https://doi.org/10.56304/S2304487X20020029
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