Algorithms for solving nonlinear equation systems assist students to become better problem solvers

In this paper, we present the distinguishing characteristics of "easy" and "hard" sets of equations and show the difference between simulation and design type problems. Easy equation sets have one unknown variable in each equation such that the equations can be solved sequentially. Hard equation sets have one critical unknown located in many (or all) of the equations in the set such that many (or all) of the equations must be solved simultaneously. We call these equations "coupled" and we teach students how they can analyze equation sets (based on the location of known and unknown variables) to "decouple" the equations and make the solution path easy. We describe the algorithm called the DeCoupler, which is used to identify variable interactions to determine the optimal decoupling variables. We also describe the SmartSwapper algorithm, which uses information from the DeCoupler to simplify the solution path and parametrically iterate to a solution. The algorithms are logic based rather than numerically focused. The algorithms have been implemented into an equation solving software program (SmartSolve2) to demonstrate their effectiveness compared to other commercial strategies. It has been shown that these algorithms can solve nonlinear simultaneous equations without the need for highly refined initial guesses. By teaching the concepts of these algorithms, students are assisted in becoming more effective problem solvers.