Modified bounded homotopies to enable a narrow bounding zone

The aim of this paper is to introduce modifications that enhance the usability of the bounded homotopy methods proposed by Paloschi [1995. Bounded homotopies to solve systems of algebraic nonlinear equations. Computers and Chemical Engineering 19, 1243–1254; 1997. Bounded homotopies to solve systems of sparse algebraic nonlinear equations. Computers and Chemical Engineering 21, 531–541], especially in the area of chemical engineering. In modified bounded homotopies, the homotopy path is tracked by exploiting mapped variables instead of unmapped ones. Path tracking based on mapped variables makes it significantly easier to track the bounded homotopy path even though the bounding zone has to be narrow. Mapping also improves the bounding effect of bounded homotopies and makes it possible to avoid unreasonable variable values in homotopy path tracking. The performance of the modifications is illustrated with test cases. These examples clearly show that the modifications enlarge the capability and accuracy of bounded homotopies when solving both small- and large-scale sets of nonlinear equations describing chemical engineering problems.