Nešetřil and Ossona de Mendez introduced the notion of first order convergence, which unifies the notions of convergence for sparse and dense graphs. They asked whether if (Gi)i∈N is a sequence of graphs with M being their first order limit and v is a vertex of M , then there exists a sequence (vi)i∈N of vertices such that the graphs Gi rooted at vi converge to M rooted at v. We show that this ...

The model theory based notion of the first order convergence unifies the notions of the left-convergence for dense structures and the BenjaminiSchramm convergence for sparse structures. It is known that every first order convergent sequence of graphs with bounded tree-depth can be represented by an analytic limit object called a limit modeling. We establish the matroid counterpart of this resul...

One-parameter families of iterative methods for the simultaneous determination of multiple complex zeros of a polynomial are considered. Acceleration of convergence is performed by using Newton’s and Halley’s corrections for a multiple zero. It is shown that the convergence order of the constructed total-step methods is five and six, respectively. By applying the Gauss-Seidel approach, further ...

Starting from a suitable fixed point relation, we derive higher order iterative methods for the simultaneous inclusion of polynomial multiple zeros in circular complex interval arithmetic. Each of the resulting disks contain one and only one zero in every iteration. This convenient inclusion property, together with very fast convergence, ranks these methods among the most powerful iterative met...

Iterative methods of Laguerre’s type for the simultaneous inclusion of all zeros of a polynomial are proposed. Using Newton’s and Halley’s corrections, the order of convergence of the basic method is increased from 4 to 5 and 6, respectively. Further improvements are achieved by the Gauss-Seidel approach. Using the concept of the R-order of convergence of mutually dependent sequences, we presen...

When nonlinear equation solvers are applied to parameter-dependent problems, their iterates can be interpreted as functions of these variable parameters. The derivatives (if they exist) of these iterated functions can be recursively evaluated by the forward mode of automatic diierentiation. Then one may ask whether and how fast these derivative values converge to the derivative of the implicit ...

Kung-Traub’s conjecture states that an optimal iterative method based on d function evaluations for finding a simple zero of a nonlinear function could achieve a maximum convergence order of 2d−1. During the last years, many attempts have been made to prove this conjecture or develop optimal methods which satisfy the conjecture. We understand from the conjecture that the maximum order reached b...