%0 Journal Article
%T Interval Arithmetic for Nonlinear Problem Solving
%A Benito A. Stradi-Granados
%J International Journal of Engineering Mathematics
%D 2013
%R 10.1155/2013/768474
%X Implementation of interval arithmetic in complex problems has been hampered by the tedious programming exercise needed to develop a particular implementation. In order to improve productivity, the use of interval mathematics is demonstrated using the computing platform INTLAB that allows for the development of interval-arithmetic-based programs more efficiently than with previous interval-arithmetic libraries. An interval-Newton Generalized-Bisection (IN/GB) method is developed in this platform and applied to determine the solutions of selected nonlinear problems. Cases 1 and 2 demonstrate the effectiveness of the implementation applied to traditional polynomial problems. Case 3 demonstrates the robustness of the implementation in the case of multiple specific volume solutions. Case 4 exemplifies the robustness and effectiveness of the implementation in the determination of multiple critical points for a mixture of methane and hydrogen sulfide. The examples demonstrate the effectiveness of the method by finding all existing roots with mathematical certainty. 1. Introduction There are a large number of problems that require the computation of stationary points and roots of equations. These are found in the fields of optimization [1], economics and finance [2, 3], thermodynamics [4, 5], applied mathematics [6], and similar contributions over the last thirty years. The application of interval arithmetic involves from algebraic equations with known solutions to more complicated systems representing physical phenomena. Among the earlier problems, second- and third-order polynomial problems serve to illustrate the effectiveness of the implementation. Similarly, more complicated multidimensional problems serve to illustrate stability and robustness of the implementation. In particular, the determination of critical points is of interest. Critical point computations are a well-known highly nonlinear problem that has been studied for a long time. The literature is endowed with some worthy analyses of the problem. Michelsen and Heidemann [7] discuss the calculation of critical points from cubic two-constant equations of state. They numerically determine the critical points of mixtures solving the highly nonlinear critical point equations. Hoteit et al. [8] claim an efficient algorithm based on bisection, secant, and inverse quadratic programming methods where their method is apparently faster but incapable of handling the presence of multiple critical points without restarting the program after each critical point determination. Nichita and Gomez [9] discuss the
%U http://www.hindawi.com/journals/ijem/2013/768474/