A Stochastic Newton Method for Nonlinear Equations

Incomplete Jacobian Newton method for nonlinear equations

Problem statement: The major weaknesses of Newton method for nonlinear equations entail computation of Jacobian matrix and solving systems of n linear equations in each of the iterations. Approach: In some extent function derivatives are quit costly and Jacobian is computationally expensive which requires evaluation (storage) of n×n matrix in every iteration. Results: This storage requirement became unrealistic when n becomes large. We proposed a new method that approximates Jacobian into diagonal matrix which aims at reducing the storage requirement, computational cost and CPU time, as well as avoiding solving n linear equations in each iterations. Conclusion/Recommendations: The proposed method is significantly cheaper than Newton’s method and very much faster than fixed Newton’s method also suitable for small, medium and large scale nonlinear systems with dense or sparse Jacobian. Numerical experiments were carried out which shows that, the proposed method is very encouraging.

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Preface 1. Introduction. Problems to be considered Characteristics of 'real-world' problems Finite-precision arithmetic and measurement of error Exercises 2. Nonlinear Problems in One Variable. What is not possible Newton's method for solving one equation in one unknown Convergence of sequences of real numbers Convergence of Newton's method Globally convergent methods for solving one equation in one uknown Methods when derivatives are unavailable Minimization of a function of one variable Exercises 3. Numerical Linear Algebra Background. Vector and matrix norms and orthogonality Solving systems of linear equations-matrix factorizations Errors in solving linear systems Updating matrix factorizations Eigenvalues and positive definiteness Linear least squares Exercises 4. Multivariable Calculus Background Derivatives and multivariable models Multivariable finite-difference derivatives Necessary and sufficient conditions for unconstrained minimization Exercises 5. Newton's Method for Nonlinear Equations and Unconstrained Minimization. Newton's method for systems of nonlinear equations Local convergence of Newton's method The Kantorovich and contractive mapping theorems Finite-difference derivative methods for systems of nonlinear equations Newton's method for unconstrained minimization Finite difference derivative methods for unconstrained minimization Exercises 6. Globally Convergent Modifications of Newton's Method. The quasi-Newton framework Descent directions Line searches The model-trust region approach Global methods for systems of nonlinear equations Exercises 7. Stopping, Scaling, and Testing. Scaling Stopping criteria Testing Exercises 8. Secant Methods for Systems of Nonlinear Equations. Broyden's method Local convergence analysis of Broyden's method Implementation of quasi-Newton algorithms using Broyden's update Other secant updates for nonlinear equations Exercises 9. Secant Methods for Unconstrained Minimization. The symmetric secant update of Powell Symmetric positive definite secant updates Local convergence of positive definite secant methods Implementation of quasi-Newton algorithms using the positive definite secant update Another convergence result for the positive definite secant method Other secant updates for unconstrained minimization Exercises 10. Nonlinear Least Squares. The nonlinear least-squares problem Gauss-Newton-type methods Full Newton-type methods Other considerations in solving nonlinear least-squares problems Exercises 11. Methods for Problems with Special Structure. The sparse finite-difference Newton method Sparse secant methods Deriving least-change secant updates Analyzing least-change secant methods Exercises Appendix A. A Modular System of Algorithms for Unconstrained Minimization and Nonlinear Equations (by Robert Schnabel) Appendix B. Test Problems (by Robert Schnabel) References Author Index Subject Index.

To minimize the average of a set of log-convex functions, the stochastic Newton method iteratively updates its estimate using subsampled versions of the full objective’s gradient and Hessian. We contextualize this optimization problem as sequential Bayesian inference on a latent state-space model with a discriminatively-specified observation process. Applying Bayesian filtering then yields a novel optimization algorithm that considers the entire history of gradients and Hessians when forming an update. We establish matrix-based conditions under which the effect of older observations diminishes over time, in a manner analogous to Polyak’s heavy ball momentum. We illustrate various aspects of our approach with an example and review other relevant innovations for the stochastic Newton method.

Different modifications in the Newton's method with cubic convergence have become popular iterative methods to find the roots of non-linear equations. In this paper, a modified Newton's method for solving a single nonlinear equation is proposed. This method uses harmonic mean while using Simpson's integration rule, thus replacing �� ' �� in the classical Newton's method. The convergence of the proposed method is found to be order three. Numerical examples are provided to compare e the efficiency of the method with few other cubic convergent methods. In this paper,we propose amodification in the method of(3) including Harmonic mean instead of Arithmetic meanwhile using the Simpson's 1/3 rule of integration for the equation (3). The proposed new method has the advantage of evaluating only the firstderivativeand less number of iterations to achieve third order accuracy.In Section II, we present some definitions related to our study. In Section III, some known variants of Newton's method are discussed. Section IV presents the new method and its analysis of convergence. Finally, Section V gives numerical results and a discussion is carried out in section VI.

In the recent past, different variants of Newton's method with cubic convergence have become popular iterative methods to find the roots of non-linear equations. In this paper, a new class of Newton's method for solving a single nonlinear equation is proposed. This method is the generalization of Simpson's integration rule applied on the Newton' theorem. Some of the existing methods become special cases of this method. Third order convergence of the proposed methods is established. Numerical examples are provided. A comparison study is done to show the efficiency of this method for different parameters in the method. Keywords: Iterative Method, Newton's Method, Non-linear equation, Third order convergence, Simpson integration method.

A Newton-type method is proposed for the recovery of the unknown coefficient function in the canonical Sturm-Liouville differential equation from two spectral data. Specifically, the two spectral data will be used to produce two Cauchy data, which in turn will serve as the input in a non-linear equation whose unknown is the coefficient function in the canonical Sturm-Liouville differential equation. This non-linear equation is to be solved numerically by the Newton method. Each Newton iterate requires that a Goursat-Cauchy boundary value problem be solved numerically. The Fréchet differentiability of the non-linear map is also discussed in the present paper. The numerical implementation of the Newton method for this inverse Sturm-Liouville problem is illustrated with examples, and it will be compared with a quasi-Newton method, and with a variational method discussed in previous literature. The numerical examples show that the Newton method needs fewer iterates to recover the true coefficient function than the quasi-Newton method. The Newton method is comparable with the variational method in terms of accuracy and number of iterates when the boundary parameters are given, and it requires a much smaller number of iterates than the variational method, when the boundary parameters are reconstructed from the available spectra.

Location, separation and approximation of solutions of nonlinear Hammerstein-type integral equations

Newton’s method is frequently employed as a means to solve nonlinear equations because of its rapid local convergence properties; however, the behaviour of Newton's method is very much dependent upon how close the initial guess is to the true solution. An inappropriate or poor initial guess can severely hinder the convergence of Newton’s method. A novel hybrid numerical/data driven approach to estimating the initial guess that can be used in conjunction with Newton's method is proposed, whereby a machine learning model is trained to make predictions about the location of the root of a specified function from sampled function values taken from a bounded interval. To develop this approach, a synthetic dataset of nonlinear equations was produced and the ML algorithm was trained to predict the location of the root, based on these training examples, that would be subsequently used as the initial guess for the numerical solver. Experimental results indicate that the proposed hybrid numerical/data-driven approach will increase the success rate of obtaining a solution with Newton's method from roughly 97% to approximately 99.75% and decrease the average number of iterations required to obtain a solution by over 30% when compared to standard methods of initialization (midpoint or random guesses). Thus, the results presented in this paper indicate that data driven initial guessing can improve both the robustness and the efficiency of classical numerical methods, without changing the classical mathematical constructs by which they operate.

Finding the approximate solution of linear or nonlinear equations is a mathematical problem, which is often encountered in engineering. As it is known, Galois E. had proven that over five orders algebraic equation was no root-finding formula. The solutions of high-order equation are obtained by some of iterative algorithms, e.g. Binary method and Newton method. To Binary method and Newton method, the function must be continuous, monotone and differentiable. The function must be only simple root in solution space, too. A few of solution subspace have been obtained by dividing solution space according to Binary method's idea. In each subspace, real root will be searched for by algorithm proposed. If subspace has simple root or multi roots, the subspace will be reserved. If no, the subspace will be discarded. The subspace that has solution will be divided again. With increasing of iterations, solution space will be compressed, and converge at the solution of equations. Solution space compression will be finished until all real roots are found. The proposed algorithm is applicable to solve equation's simple root and multi roots problem. As such, it is fit for linear equations and nonlinear equations. Eventually, some test cases illustrate this approach is very available.