http://inis.jinr.ru/sl/m_mathematics/mn_numerical methods/mnd_numerical calculus/allgower e.l., georg k. introduction to numerical continuation methods (colorado, 1990)(397s).pdf

Numerical Continuation Methods

From the Publisher:
Introduction to Numerical Continuation Methods continues to be useful for researchers and graduate students in mathematics, sciences, engineering, economics, and business looking for an introduction to computational methods for solving a large variety of nonlinear systems of equations. A background in elementary analysis and linear algebra is adequate preparation for reading this book; some knowledge from a first course in numerical analysis may also be helpful.

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Introduction to Numerical Continuation Methods

The classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present-day computing. We briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms. We end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost.

Solving a Polynomial Equation: Some History and Recent Progress

Geometric constructions of iterative functions to solve nonlinear equations

A survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non-Hermitian matrices is presented. In addition, a modification of one method based on a better normalizing condition is proposed. Methods are classified as Direct or Adjoint and are evaluated for efficiency. Operation counts are presented in terms of matrix size, number of design variables and number of eigenvalues and eigenvectors of interest. The effect of the sparsity of the matrix and its derivatives is also considered, and typical solution times are given. General guidelines are established for the selection of the most efficient method.

Derivatives of eigenvalues and eigenvectors of a general complex matrix

A new iterative method for solving nonlinear equations is presented which is shown to converge locally withR-order of convergence $$1 + \sqrt 2 $$ at least under suitable differentiability assumptions. The method needs as many function evaluations per step as the classical Newton method.

Über ein Verfahren der Ordnung $$1 + \sqrt 2 $$ zur Nullstellenbestimmung

On the simultaneous determination of polynomial roots

The celebrated Banach fixed point theorem provides conditions which assure that the method of successive substitution is convergent; the convergence, however, may take place very slowly so that it may be desirable to use a Newton-like method for the computation of the fixed point. If Newton's method itself is applied one ignores the additional information that the problem arises from a frixed point problem with a contraction mapping. In the present note some variants of Newton's method are discussed which make use of this contraction information; it turns out that the convergence of Newton's method can be accelerated without any relevant additional computational labour.

Newton-like methods for the computation of fixed points

A simple but efficient method to obtain accurate solutions of a system of nonlinear equations with a singular Jacobian at the solution is presented. This is achieved by enlarging the system to a higher dimensional one whose solution in question is isolated. Thus it can be computed e. g. by Newton's method, which is locally at least quadratically convergent and selfcorrecting, so that high accuracy is attainable.

On the accurate determination of nonisolated solutions of nonlinear equations

Recently an iterative method for the solution of systems of nonlinear equations having at leastR-order 1+ $$\sqrt 2 $$ for simple roots has been investigated by the author [7]; this method uses as many function evaluations per step as the classical Newton method. In the present note we deal with several properties of the method such as monotone convergence, asymptotic inclusion of the solution and convergence in the case of multiple roots.

Wilhelm Werner

Papers

Über ein Verfahren der Ordnung $$1 + \sqrt 2 $$ zur Nullstellenbestimmung

On the simultaneous determination of polynomial roots

Newton-like methods for the computation of fixed points

On the accurate determination of nonisolated solutions of nonlinear equations

Some supplementary results on the 1+ $$\sqrt 2 $$ order method for the solution of nonlinear equations