An algorithm for finding all solutions of a nonlinear system

Smiley, Michael W.; Chun, Changbum

doi:10.1016/s0377-0427(00)00711-1

Cited by 19 publications

(11 citation statements)

References 15 publications

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“…According to Smiley and Chun (2001) For each subdivision level, the subintervals obtained are tested for the existence of roots in order that only those containing one or more solutions are maintained. The selection criterion is based on calculation of the Euclidean norm of the set of equations:…”

Section: Subdivision Algorithmmentioning

confidence: 99%

“…the Newton-Raphson method) is employed to determine the roots, taking as initial values the middle points of the last subintervals. Since the solutions are confined to the retained subintervals, the convergence of the adopted method becomes more efficient and reliable (Smiley and Chun, 2001).…”

Section: Subdivision Algorithmmentioning

confidence: 99%

“…In this context, the aim of this work is to provide a systematic comparison between the two mentioned strategies for phase stability analysis: Simulated Annealing (Press et al, 1992), wich is a global optimization procedure, and a subdivision method (Smiley and Chun, 2001;Corazza et al, 2007) -henceforth denominated SubDivNL -employed for solution of the stationary points on TPD of liquid mixtures modeled by the NRTL activity coefficient model (Renon and Prausnitz, 1968). Comparison of the approach used in this work with Interval Newton analysis is also provided.…”

Section: Introductionmentioning

confidence: 99%

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Phase stability analysis of liquid-liquid equilibrium with stochastic methods

Nagatani¹,

Ferrari²,

Filho

et al. 2008

Braz. J. Chem. Eng.

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-Minimization of Gibbs free energy using activity coefficient models and nonlinear equation solution techniques is commonly applied to phase stability problems. However, when conventional techniques, such as the Newton-Raphson method, are employed, serious convergence problems may arise. Due to the existence of multiple solutions, several problems can be found in modeling liquid-liquid equilibrium of multicomponent systems, which are highly dependent on the initial guess. In this work phase stability analysis of liquid-liquid equilibrium is investigated using the NRTL model. For this purpose, two distinct stochastic numerical algorithms are employed to minimize the tangent plane distance of Gibbs free energy: a subdivision algorithm that can find all roots of nonlinear equations for liquid-liquid stability analysis and the Simulated Annealing method. Results obtained in this work for the two stochastic algorithms are compared with those of the Interval Newton method from the literature. Several different binary and multicomponent systems from the literature were successfully investigated.

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Section: Subdivision Algorithmmentioning

confidence: 99%

Section: Subdivision Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Phase stability analysis of liquid-liquid equilibrium with stochastic methods

Nagatani¹,

Ferrari²,

Filho

et al. 2008

Braz. J. Chem. Eng.

View full text Add to dashboard Cite

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“…The performance and success of this method in finding all possible solutions (roots) are directly associated with the choice of homotopy type (Wayburn and Seader, 1987). Smiley and Chun (2001) recently presented a subdivision algorithm with exclusion to locate all roots of a nonlinear algebraic system, where the authors present a section proofing the convergence for the subdivision algorithm. In general, subdivision algorithms consist in establishing initial intervals for the system variables and then a systematically subdividing these intervals.…”

Section: Brazilian Journal Of Chemical Engineeringmentioning

confidence: 99%

“…Considering a system of nonlinear algebraic equations, F(x), in accordance with Smiley and Chun (2001), the basic idea of the proposed subdivision algorithm is as follows: given an initial interval for unknown variables ("rectangle") R ∈ ℜ passes the test, it is retained (saved) and will be a new "parent" rectangle, which in turn will divide at i = i + 1. In the present work, the following selection criterion was used to test the existence of solution in a rectangle:…”

Section: Subdivision Algorithmmentioning

confidence: 99%

A subdivision algorithm for phase equilibrium calculations at high pressures

Corazza

Filho

et al. 2007

Braz. J. Chem. Eng.

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-Phase equilibrium calculations at high pressures have been a continuous challenge for scientists and engineers. Traditionally, this task has been performed by solving a system of nonlinear algebraic equations originating from isofugacity equations. The reliability and accuracy of the solutions are strongly dependent on the initial guess, especially due to the fact that the phase equilibrium problems frequently have multiple roots. This work is focused on the application of a subdivision algorithm for thermodynamic calculations at high pressures. The subdivision algorithm consists in the application of successive subdivisions at a given initial interval (rectangle) of variables and a systematic test to verify the existence of roots in each subinterval. If the interval checked passes in the test, then it is retained; otherwise it is discharged. The algorithm was applied for vapor-liquid, solid-fluid and solid-vapor-liquid equilibrium as well as for phase stability calculations for binary and multicomponent systems. The results show that the proposed algorithm was capable of finding all roots of all high-pressure thermodynamic problems investigated, independent of the initial guess used.

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Computation of Morse decompositions for semilinear elliptic PDEs

Smiley

Chun

2001

Numerical Methods Partial

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A numerical method for computing all solutions of an elliptic boundary value problem Au + g [u, λ] = 0 and their Morse indices as steady-states of the parabolic problem ut + Au + g [u, λ] = 0, is presented. Morse decompositions are also determined. The method uses a finite element approach that is based on the method of alternative problems. Error estimates for the finite element approximations are verified and examples are given.

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An algorithm for finding all solutions of a nonlinear system

Cited by 19 publications

References 15 publications

Phase stability analysis of liquid-liquid equilibrium with stochastic methods

Phase stability analysis of liquid-liquid equilibrium with stochastic methods

A subdivision algorithm for phase equilibrium calculations at high pressures

Computation of Morse decompositions for semilinear elliptic PDEs

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