(142g) Reaching Isolas Using Homotopy to Find All Real Roots of System of Nonlinear Equations

Abstract A new method is presented to find all real solutions of a system of nonlinear equations by reaching all isolas of the homotopy path. The original system of equations is first changed to the new system of equations by multiplying the (x-x0) term. It is mathematically proven that when x0 is selected as the starting point and a FPN (Fixed-Point Newton) homotopy is applied, bifurcation points are generated. In addition, we have proven that if each equation of the original system is squared, bifurcation points at all roots besides the above bifurcation points are located. Therefore, all isolas that contain the roots of the system of equations are reachable. It is necessary to switch to other branches, which are bifurcated from all bifurcation points that include the roots. We first find the bifurcation points where one of its unknown variables equals to its unknown variable at the initial point. These points are obtained by solving the (n-1) nonlinear equations, and the bifurcation points of the system of (n-1) equations are obtained by solving the (n-2) nonlinear equations. After repeating this procedure, we obtain one equation and one unknown variable. Solving this equation, the coordinates of the bifurcation points of a system of two nonlinear equations are found. Finally we can obtain all bifurcation points of n nonlinear equations where one of its unknown variables is equal to its unknown variable of the initial point. The other bifurcation points for the other unknown variables are equal to the other unknown variables at the initial point and are found in the same way as mentioned above. After finding all bifurcation points, it is necessary to switch to other branches, which are bifurcated from these points, to seek all solutions to the new system of equations. Now we have to switch to other branches from the roots that are found as described above. If there are some isolas, we can find all of them, so the remaining roots can be found. This method is applied to four systems of nonlinear equations that have some isolas. In all cases, all isolas are reached and all roots are found. In one example there are several isolas and 119 solutions are found using the new method.