Problems and solutions by the application of Julia set theory to one-dot and multi-dots numerical methods

摘要

In 1977 Hubbard developed the ideas of Cayley (1879) and solvedin particular the Newton-Fourier imaginary problem. We solve theNewton-Fourier and the Chebyshev-Fourier imaginary problemscompletely. It is known that the application of Julia set theoryis possible to the one-dot numerical method like the Newton'smethod for computing solution of the nonlinear equations. Thesecants method is the two-dots numerical method and theapplication of Julia set theory to it is not demonstrated.Previously we have defined two one-dot combinations: theNewton's-secants and the Chebyshev's-secants methods and haveused the escape time algorithm to analyse the application ofJulia set theory to these two combinations in some special cases.We consider and solve the Newton's-secants andTchebicheff's-secants imaginary problems completely.