Homotopy Map Research Articles

Algorithm 652

There are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map. HOMPACK provides three qualitatively different algorithms for tracking the homotopy zero curve: ordinary differential equation-based, normal flow, and augmented Jacobian matrix. Separate routines are also provided for dense and sparse Jacobian matrices. A high-level driver is included for the special case of polynomial systems.

A Differential Homotopy Continuation Method for Interlinked Solvent Extraction Cascades

Abstract A robust differential homotopy procedure is developed that is suitable for solving the bordered block tridiagonal systems of nonlinear equations that result from liquid-liquid extraction models. The procedure is capable of readily utilizing newton, fixed-point, and custom homotopy mapping. It is shown to have an efficiency far superior to Kubicek's method and comparable to that of Rheinboldt. Application of the method to liquid-liquid extraction cascade models revealed that the newton homotopy would converge to a solution as rapidly as a simultaneous correction procedure and that use of a custom homotopy resulted in faster convercrence.

Quantized oscillator chain and lattices

The problems of inhomogeneity, boundary conditions and tensor lattice activations are resolved using homotopy mapping.

Physical implications of multi-neighbor activations in the master equation

In this letter we show how multi-neighbor activations can arise from applications of the master equation to physical models, and how such randomized activations can be solved by homotopy mapping.

Homotopy equivalences and mapping torus projections

Homotopy equivalences and mapping torus projections