Relaxed Newton's Method Research Articles

The Solution of Non-Linear Equations System Containing Interpolation Functions by Relaxing the Newton Method

Many world phenomena lead to nonlinear equations systems. For some applications, the non-linear equations which construct the non-linear equations system are interpolation functions. However, the interpolation functions are usually not represented as mathematical expressions but as computer programs in specific programming languages. The research proposed using the relaxed Newton method to solve the non-linear equations system that contained interpolation functions. The interpolation functions were represented in the R programming language. Then, the experiment used the Spline interpolation function to construct a two-dimensional non-linear equations system. Eleven initial guesses, maximum of ten-time iterations, and 10-7 precision were applied. The solution of the non-linear equations system and the iteration needed on each initial guess were observed. The experiment shows that the proposed method works well for solving the non-linear equations system constructed by Spline interpolation functions. By observing the initial guesses used in the experiment, there are four possible results: true solution, spurious solution, false solution, and no solution. Applying 11 initial guesses have five initial guesses resulting in true solutions, one initial guess in spurious solution, three initial guesses in false solutions, and one initial guess in no solution. The discussions imply that this method can be generalized to the three-dimensional non-linear equations system or higher dimensions.

Convergence of relaxed Newton method for order-convex matrix equations

In this paper, we suggest singular monotone Newton theorem and a relaxed Newton method for a nonlinear matrix equation $$F(X) = 0$$ where F is an order-convex matrix function. At first, we introduce monotone Newton theorem from Ortega and Rheinboldt (SIAM, 2000), and we show singular monotone Newton theorem which shows that the Newton sequence converges to a solution of $$F(X) = 0$$ even if the Frechet derivative at the solution is singular. At last, we provide a relaxed Newton method for $$F(X) = 0$$ which is better than the pure Newton method and we give some numerical experiments for the relaxed Newton method.

Waveform relaxation–Newton method to determine steady state of an electromagnetic structure

PurposeThe purpose of this study is to determine the steady state of an electromagnetic structure using the finite element method (FEM) without calculation of the transient state. The proposed method permits to reduce the computation time if the transient state is important.Design/methodology/approachIn the case of coupling magnetic and electric circuit equations to obtain the steady state with periodic conditions, an approach can be to discretise the time with periodic conditions and to solve the equation system. Unfortunately, the computation time can be prohibitive. In this paper, the authors proposed to use the waveform relaxation method associated with the Newton method to accelerate the convergence.FindingsThe obtained results show that the proposed approach is efficient if the transient state is important. On the contrary, if the transient state is very low, it is preferable to use the classical approach, namely, the time-stepping FEM.Research limitations/implicationsThe main limitation of the proposed approach is the necessity to evaluate or to know the time constant and consequently the duration of the transient state. Moreover the method requires some important memory resources.Practical/implicationsIn the context of the use of the time-stepping FEM, one of the problems is the computation time which can be important to obtain the steady state. The proposed method permits avoidance of this difficulty and directly gives the steady state.Social/implicationsThe proposed approach will permit to model and study the electromagnetic systems in the steady state, and particularly the transformers. Because of the gain in computing time, the use of optimisation techniques will be facilitated.Originality/valueThe novelty of this study is the proposal of the waveform relaxation–Newton method to directly obtain the steady state when applied to the three-phase transformer.

Connections among several chaos feedback control approaches and chaotic vibration control of mechanical systems

This study reveals the essential connections among several popular chaos feedback control approaches, such as delayed feedback control (DFC), stability transformation method (STM), adaptive adjustment method (AAM), parameter adjustment method, relaxed Newton method, and speed feedback control method (SFCM), etc. Meanwhile, the generality and practical applicability of these approaches are evaluated and compared. It is shown that for discrete chaotic maps, STM can be regarded as a kind of predictive feedback control, and AAM is actually a special case of STM which is merely effective for a particular dynamical system. The parameter adjustment method is only a different expression of the relaxed Newton method, and both of them represent just one search direction of STM, i.e., the gradient direction. Moreover, the intrinsic relation between the STM and SFCM for controlling the equilibrium of continuous autonomous systems is investigated, indicating that STM can be viewed as a special form of the SFCM. Finally, both the STM and SFCM are extended to control the chaotic vibrations of non-autonomous mechanical systems effectively.

CONVERGENCE OF THE RELAXED NEWTON'S METHOD

In this work we study the local and semilocal convergence of the relaxed Newton's method, that is Newton's method with a relaxation parameter 0 < � < 2. We give a Kantorovich-like theorem that can be applied for operators defined between two Banach spaces. In fact, we obtain the recurrent sequence that majorizes the one given by the method and we characterize its convergence by a result that involves the relaxation parameter �. We use a new technique that allows us on the one hand to generalize and on the other hand to extend the applicability of the result given initially by Kantorovich for � = 1. In many areas related to the applied sciences one confronts the problem of solving a nonlinear equation of the form f(x) = 0. The solutions of these equa- tions can rarely be found in closed form. That is why most solution methods are iterative. There exist lots of iterative methods with different properties that allow us to solve this kind of equations, but the most well-known and used is the Newton's method, which has the following form:

Convergence of pure and relaxed Newton methods for solving a matrix polynomial equation arising in stochastic models

We consider a matrix polynomial equation which has the form of AnXn+An−1Xn−1+⋯+A0=0 where An,An−1,…,A0 and X are square matrices assuming the positivity of coefficients from stochastic models. The monotone convergence of Newtonʼs method for solving the equation is considered and we show that the elementwise minimal nonnegative solution can be found by the method with the zero starting matrix. Moreover, the relaxed Newton method preserving the monotonicity result is provided. Finally, numerical experiments show that our method reduces the number of iterations significantly.

Two‐Step Relaxation Newton Method for Nonsymmetric Algebraic Riccati Equations Arising from Transport Theory

We propose a new idea to construct an effective algorithm to compute the minimal positive solution of the nonsymmetric algebraic Riccati equations arising from transport theory. For a class of these equations, an important feature is that the minimal positive solution can be obtained by computing the minimal positive solution of a couple of fixed‐point equations with vector form. Based on the fixed‐point vector equations, we introduce a new algorithm, namely, two-step relaxation Newton, derived by combining two different relaxation Newton methods to compute the minimal positive solution. The monotone convergence of the solution sequence generated by this new algorithm is established. Numerical results are given to show the advantages of the new algorithm for the nonsymmetric algebraic Riccati equations in vector form.

The relaxed Newton method derivative: Its dynamics and non-linear properties

The dynamic behaviour of the one-dimensional family of maps f ( x ) = c 2 [ ( a − 1 ) x + c 1 ] − λ / ( α − 1 ) is examined, for representative values of the control parameters a , c 1 , c 2 and λ . The maps under consideration are of special interest, since they are solutions of the relaxed Newton method derivative being equal to a constant a . The maps f ( x ) are also proved to be solutions of a non-linear differential equation with outstanding applications in the field of power electronics. The recurrent form of these maps, after excessive iterations, shows, in an x n versus λ plot, an initial exponential decay followed by a bifurcation. The value of λ at which this bifurcation takes place depends on the values of the parameters a , c 1 and c 2 . This corresponds to a switch to an oscillatory behaviour with amplitudes of f ( x ) undergoing a period doubling. For values of a higher than 1 and at higher values of λ a reverse bifurcation occurs. The corresponding branches converge and a bleb is formed for values of the parameter c 1 between 1 and 1.20. This behaviour is confirmed by calculating the corresponding Lyapunov exponents.

FINITENESS OF THE AREA OF BASINS OF ATTRACTION OF RELAXED NEWTON METHOD FOR CERTAIN HOLOMORPHIC FUNCTIONS

For a nonconstant function F and a real number h∈]0, 1] the relaxed Newton's method NF,h of F is an iterative algorithm for finding the zeroes of F. We show that when relaxed Newton's method is applied to complex function F(z)=P(z)eQ(z), where P and Q are polynomials, the basin of attraction of a root of F has finite area if the degree of Q exceeds or equals 3. The key point is that NF,h is a rational map with a parabolic fixed point at infinity.

On infinite area for complex exponential function

Abstract This paper shows via a reduced family of examples, the relaxed Newton's method is applied to complex exponential function F ( z )= z e z and F ( z )= z e z 2 , the basin of roots has infinite area. In addition, we examined their computer pictures which are fractals for the relaxed Newton's basin. In fact, computer experiments F ( z )= P ( z )e z and F ( z )= P ( z )e z 2 , indicate this to hold for arbitrary non-constant polynomial P ( z ).

Holomorphic motions in the parameter space for the relaxed Newton's method.

Holomorphic motions in the parameter space for the relaxed Newton's method.

Parallel-in-time relaxed Newton method for transient stability analysis

The paper presents a new parallel method for transient stability analysis of power systems. For simplicity, the classical power system model is adopted. The implicit trapezoidal rule is used to discretise the set of differential equations which describes the transient stability problem. As is well known, these discretised nonlinear algebraic equations are almost invariably solved by a Newton procedure. A parallel-in-time relaxed Newton method is proposed to solve the overall set of discretised equations concurrently on all the time steps. The proposed method has been tested on Cray-T3D by the use of PVM system. The test results show that the proposed method presents a good compromise between parallelism-in-time and convergence. Some important aspects for parallel transient stability analysis have been clarified.

Convergence of a relaxed Newton's method for cubic equations

This paper presents new results on the behavior of Newton's method when it is used to search for the roots of a cubic equation. When the equation has one real root and two real singular points, Newton's method can exhibit chaotic or periodic behavior. Asymptotic methods are used to derive results for the period p solutions of the iterative map for large p. The effect of introducing a relaxation parameter on the chaotic and periodic behavior is also discussed. The applicability of the results to more general nonlinear functions and to the solution of equations of state is discussed.

Newton's method and a class of meromorphic functions without wandering domains

AbstractLetNbe the class of meromorphic functionsfwith the following properties:fhas finitely many poles;f′ has finitely many multiple zeros; the superattracting fixed points offare zeros off′ and vice versa, with finitely many exceptions;fhas finite order. It is proved that iff∈N, thenfdoes not have wandering domains. Moreover, iff∈Nand if ∞ is among the limit functions offnin a cycle of periodic domains, then this cycle contains a singularity off−1. (Herefndenotes thenth iterate off) These results are applied to study Newton's method for entire functionsgof the formwherepandqare polynomials and wherecis a constant. In this case, the Newton iteration functionf(z) =z−g(z)/g′(z) is inN. It follows thatfn(z) converges to zeros ofgfor allzin the Fatou set off, if this is the case for all zeroszofg″. Some of the results can be extended to the relaxed Newton method.

Surgery for relaxed newton's method

In this paper we consider Newton's method with the help of quasi conformal surgery. Using holomorphic motions the existence of attractive periodic points of relaxed Newton's method for cubic polynomials and any h ∊]0,1[ is established.

The relaxed newton-iteration for rational functions: The limiting case

In this paper we study the approximation of the phase-portrait of the Newton differential equation u′ = –R/R′ o u by means of its Euler discretization NR,h Here NR,h stands for the relaxed Newton-method for finding a zero of a rational function R with disaetization parameter h. For h suficiently small we prove that each basin of a zero of R corresponding to the differential equation is approximated in the sense of Carathedory by the immediate hasin of attraction with respect to the iteration of NR,h. From that we derive conditions. which ensure the convergence of the Julia set J(NR,h) in the sense of the Hausdorff metric, to the union of all boundaries of basins corresponding to the differential equation and the zeros of R, for h tending to zero. In particular the derived conditions apply to all nonconstant polynomials.

The complex dynamics of Newton's method for a double root

We show that the relaxed Newton's method for finding the roots of a cubic with one double root is conjugate, by a linear fractional transformation on the Riemann sphere, to the iterations of the quadratic p( z) = z 2 − 0.75.

A combined relaxation-newton method as a new global approach to the computation of thermal separation processes

Two basically different methods, formerly used sequentially to solve multicomponent thermal separation problems, are now allowed to function simultaneously within a single algorithm to ensure a rapid yet stable convergence to the solution. By combining the stability of the Rose-Sweeney-Ball method with the rapidity of the Newton-Raphson method, the solution is easily reached without encountering the difficulties of employing these methods separately.