Showing papers on "Steffensen's method published in 1988"

Newton’s Method and Gauss-Kronrod Quadrature

Abstract: One of us, jointly with CALIO and MARCHETTI (1986), considered the application of Newton’s method (for large nonlinear systems of equations) in the context of computing Gauss-Kronrod quadrature rules. With the equations set up in an appropriate manner, it was found that, by careful choice of initial approximations and continued monitoring of the iteration process, the method could be made to work for rules with up to 81 nodes (40 Gauss and 41 Kronrod nodes). This was documented for the Legendre weight on [‒1,1] (where in fact formulae with up to 161 nodes were computed) and for weight functions on [0,1] involving logarithmic and algebraic singularities. Further evidence of the feasibility of Newton’s method, also for Kronrod extension of Gauss-Radau and Gauss-Lobatto formulae, is contained in NOTARIS’s thesis (1988). If one attempts, however, to repeat Kronrod extension in the manner of PATTERSON (1968), one discovers that Newton’s method quickly deteriorates and eventually fails to converge. The purpose of this note is to shed some light on the reasons for this failure of Newton’s method. One of these is the excessive magnitude of the inverse Jacobian of the nonlinear system (evaluated at the solution) which comes about because of a peculiar behavior of a certain polynomial responsible for the magnitude of this inverse. Graphical evidence is provided to underscore the phenomenon.

Newton’s Method for the Navier-Stokes Equations

Abstract: The work described in this paper is part of a larger effort whose objective is the development of efficient rapid convergence algorithms for finite difference equations which approximate the steady-state, compressible, Reynolds-averaged, Navier-Stokes equations. The overall program objective has been divided into three parts: (a) Investigate the effects of Jacobian-approximations on the convergence rate of Newton’s method applied to the NS equations. (b) Investigate the usefulness of ‘steady-state solvers’, i.e., those algorithms which are not time accurate, (c) Develop an efficient algorithm for solving finite-difference approximations to the steady-state compressible Reynolds-averaged NS equations.

A newton algorithm for steady‐state analysis of nonlinear oscillatory circuits

Abstract: For the steady-state analysis of nonlinear oscillatory circuits, the Newton algorithm proposed by Aprille and Trick is well known. However, often the conventional algorithm needs too many iterations or may not converge, because of the discretization error of numerical integration. This paper discusses in detail the Newton algorithm for the steady-state analysis, and proposes an efficient algorithm which retains the quadratic convergence in the presence of the error of numerical integration. First, a method of computing the exact Jacobians of the functions obtained by numerical integration is proposed. Then it is shown that the quadratic convergence of the Newton algorithm is retained by suitably controlling the truncation errors of the inner Newton iterations for the corrector equations. The proposed Newton algorithm converges to a solution quadratically even if the step length of the numerical integration is large. Utilizing this property, a mesh refinement strategy is proposed which improves the computational efficiency of the algorithm by determining first an inaccurate solution which is computed by using a large steplength. Then this approximate solution is used for the starting point of the next Newton iteration with a smaller steplength to obtain an accurate numerical solution. Also, it is shown that the foregoing technique can also be applied to the Newton algorithm for computing the bifurcation value of the periodic solutions. The validity and the effectiveness of the proposed method are also verified by numerical examples.