Broyden and Quasi-Newton Methods for Nonlinear Systems of Algebraic Equations and Extension to Homotopy and Optimization Applications
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Date
2020-03-12
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Addis Ababa University
Abstract
In the field of science and engineering, researches often come across with problems model by system of nonlinear algebraic equations that cannot solve easily. Numerical convergence for many problems, typically solved by the Newton-Raphson algorithm, is sensitive to the initial guess and need computations of Jacobi and its inverse at each iteration. Emphasis in the present work is placed on the alternative approach, such as quasi-Newton, Homotopy Method and optimization method.
The choice of the optimal numerical method, which ensures the best convergence rate with minimum error for the corresponding system of nonlinear equations, is discussed. Effectiveness of the method is demonstrated by comparing the results norm errors of each method for our sample problem implemented using a MATLAB Program.
In order to avoid computation of Jacobian matrix at each iteration that is one problem of Newton method, Broyden method that use an approximation matrix that is updated at each iteration in place of the Jacobian matrix. Moreover, to reduce the large step size that leads to wrong path, we use Safeguarded and Damped methods for Newton as well as Broyden’s method. In addition to this, we use different Modified Broyden’s or Quasi-Newton method to solve SNAE. The modified Broyden’s method gives a better optimal solution than both pure Newton and Broyden’s methods.
In order to avoid failure of convergence of poor starting guess of problem for Newton method as well as Broyden’s method we apply Numerical algorithm like NHAM that combining Newton method and Homotopy analysis method with Euler and Runge-Kutta type to increase the range of initial value and efficiency of convergence. The results from NHAM with RK6 NHDE steps were more accurate than other NHAMs.
Finally, solve our sample SNLAE by globally convergent, Optimization methods in order to get good and sufficient initial guess for other methods. The result from Levenberg-Marquard method, were significantly reliable and more accurate than Steepest Decent, BFGS and DFP methods, however, computation cost is high.
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Broyden, Quasi-Newton Methods, Nonlinear Systems, Algebraic Equations, Extension, Homotopy, Optimization, Applications