The numerical inversion of functions from the plane to the plane

Abstract

Abstract. This paper contains a description of a program designed to find all the solutions of systems of two real equations in two real unknowns which uses detailed information about the critical set of the associated function from the plane to the plane. It turns out that the critical set and its image are highly structured, and this is employed in their numerical computation. The conceptual background and details of implementation are presented. The most important features of the program are the ability to provide global information about the function and the robustness derived from such topological information.

Prediction of azeotropic behaviour by the inversion of functions from the plane to the plane

“…Some Theoretical Ingredients of the Inversion of Functions from the Plane to the Plane Malta et al [26] and Saldanha and Tomei [27] analyzed and constructed algorithms to solve the 2 Â 2 nonlinear systems of equations F p ð Þ ¼ q, when F : R 2 ! R 2 is a nice function.…”

“…This has theoretical as well as numerical consequences and is one of the reasons that the methods of Malta et al [26] partially breaks down on the problem at hand and further ideas have to be implemented. It is stressed that F cannot be calculated for all u 2 R 2 and that not all u 2 D has physical meaning.…”

“…Malta et al [26] introduced a new and geometry-based numerical methodology to solve 2 Â 2 nonlinear algebraic problems: the numerical inversion of functions from the plane to the plane. Malta et al [26] introduced a new and geometry-based numerical methodology to solve 2 Â 2 nonlinear algebraic problems: the numerical inversion of functions from the plane to the plane.…”

Real perturbations of complex polynomials

“…In [DRRV] for example, it was proved that for generic real quadratic polynomials (each coordinates is a quadratic polynomial of x and y) of the plane the restriction of f to S f is injective. See also [MST1] and [MST2] where some examples are shown of maps f drawing S f homeomorphic to a circle but f S f not injective.…”

The geometry of the critical set of nonlinear periodic Sturm–Liouville operators

“…For instance, knowledge of C and F (C) yields substantial information about the number of solutions of the equation F (x) = b, b ∈ Y [12,14]. In this paper we consider a special but relevant example, the nonlinear periodic Sturm-Liouville operator and determine the topology of the pair (X, C) in the generic case.…”