A comparison of solution methods for Mandelbrot-like polynomials

We compare two different root-finding methods, eigenvalue methods and homotopy methods, using three test problems: Mandelbrot polynomials, Fibonacci-Mandelbrot polynomials, and Narayana-Mandelbrot polynomials. For the eigenvalue methods, using both MATLAB and Maple, we computed the eigenvalues of a specialized recursively-constructed, supersparse, upper Hessenberg matrix, inspired by Piers Lawrence\u27s original construction for the Mandelbrot polynomials, for all three families of polynomials. This led us to prove that this construction works in general. Therefore, this construction is genuinely a new kind of companion matrix. For the homotopy methods, we used a special-purpose homotopy, in which we used an equivalent differential equation to solve for the roots of all three families of polynomials. To solve these differential equations, we used our own ode solver, based on MATLAB\u27s ode45 routine, which has pole-vaulting capabilities. We had two versions of this ode solver: one in MATLAB, and the other in C++ that implements Bailey\u27s ARPREC package

Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices

Abstract The diagonals of regular n-gons for odd n are shown to form algebraic fields with the diagonals serving as the basis vectors. The diagonals are determined as the ratio of successive terms of generalized Fibonacci sequences. The sequences are determined from a family of triangular matrices with elements either 0 or 1. The eigenvalues of these matrices are ratios of the diagonals of the n-gons, and the matrices are part of a larger family of matrices that form periodic trajectories when operated on by the Mandelbrot operator. Generalized Mandelbrot operators are related to the Lucas polynomials have similar periodic properties

Golden ratios, Lucas Sequences and the Quadratic Family

It is conjectured that there is a converging sequence of some generalized Fibonacci ratios, given the difference between consecutive ratios, such as the Golden Ratio, $\varphi^1$, and the next golden ratio $\varphi^2$. Moreover, the graphic depiction of those ratios show some overlap with the quadratic family, and some numerical evidence suggest that everyone of those ratios in the finite set obtained, belong to at least one quadratic family.Comment: 8 pages, 4 figure

Upper Hessenberg and Toeplitz Bohemians

We also acknowledge the support of the Ontario Graduate Institution, The National Science & Engineering Research Council of Canada, the University of Alcala, the Rotman Institute of Philosophy, the Ontario Research Centre of Computer Algebra, and Western University. Part of this work was developed while R. M. Corless was visiting the University of Alcala, in the frame of the project Giner de los Rios. J.R. Sendra is member of the Research Group ASYNACS (Ref. CT-CE2019/683).A set of matrices with entries from a fixed finite population P is called “Bohemian”. The mnemonic comes from BOunded HEight Matrix of Integers, BOHEMI, and although the population P need not be solely made up of integers, it frequently is. In this paper we look at Bohemians, specifically those with population {−1,0,+1} and sometimes other populations, for instance {0,1,i,−1,−i}. More, we specialize the matrices to be upper Hessenberg Bohemian. We then study the characteristic polynomials of these matrices, and their height, that is the infinity norm of the vector of monomial basis coefficients. Focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give useful bounds on their height, and conjecture an accurate asymptotic formula. The lower bound for the maximal characteristic height is exponential in the order of the matrix; in contrast, the height of the matrices remains constant. We give theorems about the number of normal matrices and the number of stable matrices in these families.Agencia Estatal de Investigació

Golden ratios, Lucas Sequences and the Quadratic Family

8 pages, 4 figuresIt is conjectured that there is a converging sequence of some generalized Fibonacci ratios, given the difference between consecutive ratios, such as the Golden Ratio, $\varphi^1$, and the next golden ratio $\varphi^2$. Moreover, the graphic depiction of those ratios show some overlap with the quadratic family, and some numerical evidence suggest that everyone of those ratios in the finite set obtained, belong to at least one quadratic family