In this paper, Aitken’s extrapolation normally applied to convergent fixed point iteration is extended to extrapolate the solution of a divergent iteration. In addition, higher order Aitken extrapolation is introduce...In this paper, Aitken’s extrapolation normally applied to convergent fixed point iteration is extended to extrapolate the solution of a divergent iteration. In addition, higher order Aitken extrapolation is introduced that enables successive decomposition of high Eigen values of the iteration matrix to enable convergence. While extrapolation of a convergent fixed point iteration using a geometric series sum is a known form of Aitken acceleration, it is shown that in this paper, the same formula can be used to estimate the solution of sets of linear equations from diverging Gauss-Seidel iterations. In both convergent and divergent iterations, the ratios of differences among the consecutive values of iteration eventually form a convergent (divergent) series with a factor equal to the largest Eigen value of the iteration matrix. Higher order Aitken extrapolation is shown to eliminate the influence of dominant Eigen values of the iteration matrix in successive order until the iteration is determined by the lowest possible Eigen values. For the convergent part of the Gauss-Seidel iteration, further acceleration is made possible by coupling of the extrapolation technique with the successive over relaxation (SOR) method. Application examples from both convergent and divergent iterations have been provided. Coupling of the extrapolation with the SOR technique is also illustrated for a steady state two dimensional heat flow problem which was solved using MATLAB programming.展开更多
The frequency domain analysis of systems is an important topic in control theory. Powerful graphical tools exist in classic control, such as the Nyquist plot, Bode plots, and Nichols chart. These methods have been wid...The frequency domain analysis of systems is an important topic in control theory. Powerful graphical tools exist in classic control, such as the Nyquist plot, Bode plots, and Nichols chart. These methods have been widely used to evaluate the frequency domain behavior of system. A literature survey shows that various approaches are available for the computation of the frequency response of control systems under different types of parametric dependencies, such as affine, multi-linear, polynomial, etc. However, there is a lack of tools in the literature to construct the Bode envelopes for the general nonlinear type of parametric dependencies. In this paper, we address the problem of computation of the envelope of Bode frequency response of a non-rational transfer function with nonlinear parametric uncertainties varying over a box. We propose two techniques to compute the Bode envelopes：first, based on the natural interval extensions （NIE） combined with uniform subdivision and second, based on the existing Taylor model combined with subdivision strategy. We also propose the algorithms to further speed up both methods through extrapolation techniques.展开更多
The extrapolation technique has been proved to be very powerful in improving the performance of the approximate methods by large time whether engineering or scientific problems that are handled on computers. In this p...The extrapolation technique has been proved to be very powerful in improving the performance of the approximate methods by large time whether engineering or scientific problems that are handled on computers. In this paper, we investigate the efficiency of extrapolation of explicit general linear methods with Inherent Runge-Kutta stability in solving the non-stiff problems. The numerical experiments are shown for Van der Pol and Brusselator test problems to determine the efficiency of the explicit general linear methods with extrapolation technique. The numerical results showed that method with extrapolation is efficient than method without extrapolation.展开更多
The purpose of this research is to investigate the effciency of explicit diagonally implicit multi-stage integration methods with extrapolation. The author gave detailed explanation of explicit diagonally implicit mul...The purpose of this research is to investigate the effciency of explicit diagonally implicit multi-stage integration methods with extrapolation. The author gave detailed explanation of explicit diagonally implicit multi-stage integration method and compared the base method with a technique known as extrapolation to improve the effciency. Extrapolation for symmetric Runge-Kutta method is proven to improve the accuracy since with extrapolation the solutions exhibit asymptotic error expansion, however for General linear methods, it is not known whether extrapolation can improve the effciency or not. Therefore this research focuses on the numerical experimental results of the explicit diagonally implicit multistage integration with and without extrapolation for solving some ordinary differential equations. The numerical results showed that the base method with extrapolation is more effcient than the method without extrapolation.展开更多
文摘In this paper, Aitken’s extrapolation normally applied to convergent fixed point iteration is extended to extrapolate the solution of a divergent iteration. In addition, higher order Aitken extrapolation is introduced that enables successive decomposition of high Eigen values of the iteration matrix to enable convergence. While extrapolation of a convergent fixed point iteration using a geometric series sum is a known form of Aitken acceleration, it is shown that in this paper, the same formula can be used to estimate the solution of sets of linear equations from diverging Gauss-Seidel iterations. In both convergent and divergent iterations, the ratios of differences among the consecutive values of iteration eventually form a convergent (divergent) series with a factor equal to the largest Eigen value of the iteration matrix. Higher order Aitken extrapolation is shown to eliminate the influence of dominant Eigen values of the iteration matrix in successive order until the iteration is determined by the lowest possible Eigen values. For the convergent part of the Gauss-Seidel iteration, further acceleration is made possible by coupling of the extrapolation technique with the successive over relaxation (SOR) method. Application examples from both convergent and divergent iterations have been provided. Coupling of the extrapolation with the SOR technique is also illustrated for a steady state two dimensional heat flow problem which was solved using MATLAB programming.
文摘The frequency domain analysis of systems is an important topic in control theory. Powerful graphical tools exist in classic control, such as the Nyquist plot, Bode plots, and Nichols chart. These methods have been widely used to evaluate the frequency domain behavior of system. A literature survey shows that various approaches are available for the computation of the frequency response of control systems under different types of parametric dependencies, such as affine, multi-linear, polynomial, etc. However, there is a lack of tools in the literature to construct the Bode envelopes for the general nonlinear type of parametric dependencies. In this paper, we address the problem of computation of the envelope of Bode frequency response of a non-rational transfer function with nonlinear parametric uncertainties varying over a box. We propose two techniques to compute the Bode envelopes：first, based on the natural interval extensions （NIE） combined with uniform subdivision and second, based on the existing Taylor model combined with subdivision strategy. We also propose the algorithms to further speed up both methods through extrapolation techniques.
文摘The extrapolation technique has been proved to be very powerful in improving the performance of the approximate methods by large time whether engineering or scientific problems that are handled on computers. In this paper, we investigate the efficiency of extrapolation of explicit general linear methods with Inherent Runge-Kutta stability in solving the non-stiff problems. The numerical experiments are shown for Van der Pol and Brusselator test problems to determine the efficiency of the explicit general linear methods with extrapolation technique. The numerical results showed that method with extrapolation is efficient than method without extrapolation.
文摘The purpose of this research is to investigate the effciency of explicit diagonally implicit multi-stage integration methods with extrapolation. The author gave detailed explanation of explicit diagonally implicit multi-stage integration method and compared the base method with a technique known as extrapolation to improve the effciency. Extrapolation for symmetric Runge-Kutta method is proven to improve the accuracy since with extrapolation the solutions exhibit asymptotic error expansion, however for General linear methods, it is not known whether extrapolation can improve the effciency or not. Therefore this research focuses on the numerical experimental results of the explicit diagonally implicit multistage integration with and without extrapolation for solving some ordinary differential equations. The numerical results showed that the base method with extrapolation is more effcient than the method without extrapolation.
