Nonlinear algebraic equations Newton algorithm

A very large number of methods of solving systems of nonlinear algebraic equations has been devised (Ortega and Rheinbolt, 1970). However, just two methods are employed in the algorithms presented in this book repeated substitution and Newton s method we review these methods below. 

This represents an problem in which C nonlinear algebraic equations are to be solved for C unknowns T and y. In general, these equations must be solved by trial, and the standard method of attack is the Newton-Raphson scheme [9]. However, in particular problems, we hope to find alternative algorithms, for the Newton-Raphson method is computationally expensive and slow to converge. 

Although MATLAB contains a built-in rontine, fsolve, for solving systems of multiple nonlinear algebraic equations (using the trnst-region Newton method discussed above), it is part of the Optimization Tooikit and so is not available in every installation of MATLAB. reduced Newton.m implements the rednced-step Newton algorithm, and can be used if... 

The MESH equations constitute a nonlinear and strongly coupled system of algebraic equations since the equilibrium ratios Ki j and the enthalpies and are complex functions of temperature and concentrations. The system (5.2-71) is numerically solved by the iterative Newton-Raphson algorithm. Commercial software packages (e.g., ASPEN, HYSYS, CHEMCAD) contain both the mathematical solver and the required system properties, such as vapor liquid equilibria and enthalpies. 

---