# Introduction to Numerical Continuation Methods

@inproceedings{Allgower1987IntroductionTN, title={Introduction to Numerical Continuation Methods}, author={Eugene L. Allgower and Kurt Georg}, year={1987} }

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.

## 826 Citations

### Parametric continuation method with correction and its applications

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- 2009

The article discusses the parametric continuation method for nonlinear equations. A continuation algorithm with correction is proposed, an approximation accuracy theorem is proved, and issues of…

### On homotopy continuation method for computing multiple solutions to the Henon equation

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Motivated by numerical examples in solving semilinear elliptic PDEs for multiple solutions, some properties of Newton homotopy continuation method, such as its continuation on symmetries, the Morse…

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This paper first gives a simple survey on methods and theory for numerically solving multiple solution problems existed in computational mathematics, physics, chemistry, biology, etc. and then…

### A New Method to Solve a Non Linear Differential System

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- 2007

The objective is the analysis of the resolution of non-linear differential systems by combining Newton and Continuation (N-C) method to solve the problem of Partial Differential Equation.

### Convergent regions of Newton homotopy methods for nonlinear systems: theory and computational applications

- Mathematics2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353)
- 2000

It is shown that convergent regions of the Newton homotopy method are equal to the stability regions for the Newton flow x/spl dot/=-adj(DF(x))F(x), which leads to the development of a numerical method to determine the convergent region.

### Numerically stable homotopy methods without an extra dimension

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We give new versions of the global Newton method and the Kellogg & Li & Yorke method for calculating zero points and fixed points of nonlinear maps, which are numerically stable, but do not require…

### HEURISTIC PATH CHOICE TO AVOID ILL-CONDITIONING IN HOMOTOPY CONTINUATION

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Homotopy continuation is a numerical method rooted in numerical linear algebra. When paired with some theory from algebraic geometry, it provides a means for approximating solutions of polynomial…

### Software for numerical algebraic geometry: a paradigm and progress towards its implementation

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- 2008

An enumeration of the operations that an ideal software package in this field would allow is described and the current and upcoming capabilities of Bertini, the most recently released package, are described.

### A homotopy method for nonlinear second-order cone programming

- MathematicsNumerical Algorithms
- 2014

Global convergence of a smooth curve determined by constructed homotopy is proven under mild conditions and the considered algorithm is applicable and efficient.

## References

SHOWING 1-10 OF 673 REFERENCES

### Continuation methods: Theory and applications

- MathematicsIEEE Transactions on Systems, Man, and Cybernetics
- 1983

This paper surveys in a tutorial fashion theoretical and applications aspects of the continuation method for the solution of large scale system engineering problems. The continuation method is…

### Numerical Solutions by the Continuation Method

- Mathematics
- 1973

The continuation method is developed with a special emphasis on its suitability for numerical solutions on fast computers. Four problems are treated in detail : finding roots of a polynomial,…

### A note on continuation methods for the solution of nonlinear equations

- MathematicsThe Journal of the Australian Mathematical Society. Series B. Applied Mathematics
- 1977

Abstract In this note we present a variable order continuation method for the solution of nonlinear equations when only a poor estimate of a solution is known. The method changes continuously from…

### Lectures on Numerical Methods in Bifurcation Problems

- Mathematics
- 1988

These lectures introduce the modern theory and practical numerical methods for continuation of solutions of nonlinear problems depending upon parameters. Bifurcations are one of the many types of…

### Some analytic techniques for parametrized nonlinear equations and their discretizations

- Mathematics
- 1983

Abstract : This paper presents general techniques based on the theory of Fredholm operators for analyzing the solutions of parametrized nonlinear equations and their finite-dimensional…

### Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems

- Computer Science, Mathematics
- 1987

This introduction to polynomial continuation remains a useful starting point for the reader interested in learning how to solve practical problems without advanced mathematics.

### Finite dimensional approximation of nonlinear problems

- Mathematics
- 1981

SummaryWe continue here the study of a general method of approximation of nonlinear equations in a Banach space yet considered in [2]. In this paper, we give fairly general approximation results for…

### Continuation Methods in Computational Fluid Dynamics

- Mathematics
- 1982

Continuation methods are extremely powerful techniques that aid in the numerical solution of nonlinear problems and were laid by mathematicians very much concerned with fluid-dynamical problems and in particular with the construction of existence proofs for the Navier-Stokes equations.

### A fast solver for nonlinear eigenvalue problems

- Mathematics
- 1982

A numerical method recently proposed by the author is shown to be a very efficient and robust method for the solution of a class of discrete nonlinear eigenvalue problems. In particular it is applied…