(315h) A Novel Homotopy Continuation Technique to Locate All Real Solutions of a Nonlinear System of Algebraic Equations

Saeed Khaleghi Rahimian^a,*, J.D. Seader^b

^a San Jose, CA 95129, USA

^b Department of Chemical Engineering, University of Utah, Salt Lake City, UT 84112, USA

* Email: s.khaleghi.r@gmail.com (AIChE Member)

One of the most critical limitations of all homotopy continuation methods, in dealing with solving a system of nonlinear equations, is to track all the real solutions including those located on isolated branches called “Isolas” [1]. To tackle this difficulty, a new system of equations is developed where all real roots are also bifurcation points, through which new stemming branches connect all the solutions. The homotopy function is defined as: H(x,t) = 1+g[f(x)]-t, where f(x) and t are the system of equations and homotopy parameter, respectively. Function g[f(x)] requires two properties. First, it must only vanish where f(x)=0 (i.e. g[f(x)]=0, iff f(x)=0), thus g[f(x)]=0 has the same roots as f(x)=0. The derivative of g(x) at roots also needs to be zero (i.e. g'(0)=0). The latter property makes the Jacobian matrix H_x singular at all real roots with dim N{H_x(0)}=n, where n is number of equations. Thus, we would have 2^n-1 maximum unique bifurcated branches at all roots [2]. The initial point on the homotopy paths is found readily by solving a nonlinear system of equations with the degree of freedom equal to one. To show the robustness of the new algorithm, two smooth (i.e. twice differentiable) functions are used for g(f) including square functions (i.e. g(f)=f²) [3] and a combination of cosine and linear functions. The new approach with both g functions was able to find successfully all the reported roots for more than 20 test problems, which included a variety of algebraic and transcendental terms.

[1] Choi, S. H.; Book, N. L. Unreachable roots for global homotopy continuation methods. AIChE J. 1991, 37, 1093–109.

[2] Decker, D.W.; Keller, H.B. Multiple limit point bifurcation, J. Math. Anal. Appl., 1980, 75, 417–430.

[3] Khaleghi Rahimian, S.; Jalali, F.; Seader, J. D.; White, R. E. A robust homotopy continuation method for seeking all real roots of unconstrained systems of nonlinear algebraic and transcendental equations. Ind. Eng. Chem. Res. 2011, 50, 8892−8900.