scispace - formally typeset
Search or ask a question
Proceedings ArticleDOI

Globally convergent homotopy methods: a tutorial

01 May 1989-Applied Mathematics and Computation (ACM)-Vol. 31, pp 369-396
TL;DR: Homotopy algorithms for solving nonlinear systems of (algebraic) equations, which are convergent for almost all choices of starting point, are globally convergent with probability one and exhibit a large amount of coarse grain parallelism.
About: This article is published in Applied Mathematics and Computation.The article was published on 1989-05-01 and is currently open access. It has received 123 citations till now. The article focuses on the topics: n-connected & Homotopy analysis method.
Citations
More filters
BookDOI
01 Jan 1990

1,149 citations

Book
01 Jan 1987
TL;DR: The Numerical Continuation Methods for Nonlinear Systems of Equations (NCME) as discussed by the authors is an excellent introduction to numerical continuuation methods for solving nonlinear systems of equations.
Abstract: 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.

889 citations

Journal ArticleDOI
TL;DR: Changes to HOMPACK include numerous minor improvements, simpler and more elegant interfaces, use of modules, new end games, support for several sparse matrix data structures, and new iterative algorithms for large sparse Jacobian matrices.
Abstract: HOMPACK90 is a Fortran 90 version of the Fortran 77 package HOMPACK (Algorithm 652), a collection of codes for finding zeros or fixed points of nonlinear systems using globally convergent probability-one homotopy algorithms. Three qualitatively different algorithms— ordinary differential equation based, normal flow, quasi-Newton augmented Jacobian matrix—are provided for tracking homotopy zero curves, as well as separate routines for dense and sparse Jacobian matrices. A high level driver for the special case of polynomial systems is also provided. Changes to HOMPACK include numerous minor improvements, simpler and more elegant interfaces, use of modules, new end games, support for several sparse matrix data structures, and new iterative algorithms for large sparse Jacobian matrices.

167 citations

Journal ArticleDOI
TL;DR: In this paper, an analytical technique, namely the homotopy analysis method (HAM), is applied to obtain an approximate analytical solution of the Burgers-Huxley equation.
Abstract: In this paper, an analytical technique, namely the homotopy analysis method (HAM) is applied to obtain an approximate analytical solution of the Burgers–Huxley equation. This paper introduces the two theorems which provide us with a simple and convenient way to apply the HAM to the nonlinear PDEs with the power-law nonlinearity. The homotopy analysis method contains the auxiliary parameter ħ , which provides us with a simple way to adjust and control the convergence region of solution series.

149 citations

Journal ArticleDOI
TL;DR: In this article, the authors used the homotopy analysis method (HAM) to evaluate the analytical approximate solutions and efficiency of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient.

131 citations

References
More filters
Book
01 Feb 1996
TL;DR: In this paper, Schnabel proposed a modular system of algorithms for unconstrained minimization and nonlinear equations, based on Newton's method for solving one equation in one unknown convergence of sequences of real numbers.
Abstract: 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.

6,831 citations

Book
01 Mar 1983
TL;DR: Newton's Method for Nonlinear Equations and Unconstrained Minimization and methods for solving nonlinear least-squares problems with Special Structure.
Abstract: 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.

6,217 citations

Journal ArticleDOI
TL;DR: In this paper, an incremental approach to the solution of buckling and snapping problems is explored, where the authors use the length of the equilibrium path as a control parameter, together with the second order iteration method of Newton.

1,821 citations