Interval Newton Algorithm And Its Application In Chemical Engineering Calculation

The central problem in modeling, optimization and design is for chemical engineering process is how to solve nonlinear equations f(x) = 0 ,where These equations are typically high nonlinear and may have multiple solutions. However, conventional solution methods such as Newton iterative method, quasi-Newton iterative method are initialization dependent and maybe could not reliable converge. What's more, these methods can not find multiples solutions when they exist.A global convergence method called Interval Newton algorithm was developed in this thesis. The method is initialization independent and can provide a mathematical and computational guarantee that all solutions of an nonlinear equation are enclosed.' If Interval Newton algorithm can not find any solution, it can be concluded that there is no solution in the given interval vector. Interval Newton algorithm is used extensively in the calculation of chemical engineering such as the determination of phase stability, reactive azeotropes, robust process simulation, reliable parameter estimation, etc. All of these results show that Interval Newton algorithm is suitable to solve most kinds of nonlinear equation successfully.Based on interval the computation cost of Interval Newton algorithm is larger than the general Newton iteration method. The computation time of Interval Newton algorithm tends to grow exponentially with the dimension. Even for small problems, Interval Newton algorithm often requires enormous computation time if the nonlinearity of the problem is very large or the problem is ill-conditioned.Two new methodologies are presented here for improving the efficiency of Interval Newton algorithm. One is based on the bounded simplex method. At first, nonlinear equations was transferred to linear equations whose feasible region contained all solutions in the given interval vector, then the bounded simplex method was used to determine whether the feasible region was empty or not . If the feasible region was empty, the interval vector could be excluded . The other are a hybrid preconditioning strategy, in which a simple pivoting preconditioner is used in combination with the standard inverse-midpoint method and a new scheme for selecting the real point in formulating Interval Newton equation. These techniques can be implemented with relatively little computational overhead and lead to a large reduction in the number of subintervals that must be tested during the interval Newton procedure. Results show that both iterative times and computation time can be substantially reduced.