Kurt Georg

Introduction to 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.

Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations

This paper presents a digest of recently developed simplicial and continuation methods for approximating fixed-points or zero-points of nonlinear finite-dimensional mappings. Underlying the methods are algorithms for following curves which are implicitly defined, as for example, in the case of homotopies. The details of several algorithms are outlined sufficiently that they should be easily implemented. Applications of simplicial and continuation methods to nonlinear complementarily, location of critical points, location of multiple solutions and bifurcation are presented.

Continuation and path following

The main ideas of path following by predictor–corrector and piecewise-linear methods, and their application in the direction of homotopy methods and nonlinear eigenvalue problems are reviewed. Further new applications to areas such as polynomial systems of equations, linear eigenvalue problems, interior methods for linear programming, parametric programming and complex bifurcation are surveyed. Complexity issues and available software are also discussed.

Exploiting symmetry in boundary element methods

Linear operator equations

Predictor-Corrector and Simplicial Methods for Approximating Fixed Points and Zero Points of Nonlinear Mappings

\mathcal {L}f = g

Piecewise linear methods for nonlinear equations and optimization

are considered in the context of boundary element methods, where the operator

Conjugate gradient methods for continuation problems II

\mathcal {L}

Matrix-Free Numerical Continuation and Bifurcation

is equivariant, i.e., commutes with the actions of a given finite symmetry group. By introducing a generalization of Reynolds projectors, a decomposition of the identity operator is constructed, which in turn allows the decomposition of the problem

The method of resultants for computing real solutions of polynomial systems

\mathcal {L}f = g

A new exclusion test

into a finite number of symmetric subproblems. The data function g does not need to possess any symmetry properties. It is shown that analogous reductions are possible for discretizations. An explicit construction of the corresponding reduced system matrices is given. This effects a considerable reduction in the computational complexity. For example, in the case of the isometry group of the 3-cube, the computational complexity of a direct linear equation solver for full matrices is reduced by 99.65 percent. Specific decompositions of the identity are given for most of the significant finite isometry group...