Methods for solving systems of nonlinear equations. 2nd ed.

*(English)*Zbl 0906.65051
CBMS-NSF Regional Conference Series in Applied Mathematics. 70. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. ix, 148 p. (1998).

[For the first edition (1974) see Zbl 0325.65022.]

A short description of the content of the 10 chapters of this basic work on methods for solving systems of nonlinear equations:

Chapter 1: Introduction. As an introduction to the topic the author begins with a brief overview of various questions that arise in connection with the computational solution of systems of nonlinear equations.

Chapter 2: Model problems. Systems of finitely many nonlinear equations in several real variables arise in connection with numerous scientific and technical problems and, correspondingly, they differ widely in form and properties. Chapter 2 introduces some typical examples of such systems without attempting to be exhaustive or to enter into details of the underlying problem areas.

Chapter 3: Iterative processes and rates of convergence. Iterative processes for the solution of finite-dimensional nonlinear equations vary almost as widely in form and properties as do the equations themselves. In Chapter 3 an algorithmic characterization of a class of these processes is introduced and some measures of their efficiency and rate of convergence are discussed.

Chapter 4: Methods of Newton type. This chapter introduces several basic types of iterative methods derived by linearizations, including, in particular, the classical Newton method and a few of its modifications.

Chapter 5: Methods of secant type. The discussion of discretized Newton methods begun in Section 4.3 is continued in this chapter.

Chapter 6: Combinations of processes. The chapter addresses the analysis of combined iterative processes where at each step of a primary method a secondary iteration is applied. For example, when linearization methods are used for solving high-dimensional problems it may become necessary to work with a secondary linear iterative process to solve the resulting linear systems at each step. In the study of such combined processes it has to be distinguished between controlling the secondary methods by means of a priori specified conditions or by some adaptive strategies based on the performance of the computation.

Chapter 7: Parametrized systems of equations. Continuation methods, in particular an ordinary differential equations approach, the code PITCON and simplicial approximation of manifolds are contained.

Chapter 8: Unconstrained minimization methods. The problem of solving a system of nonlinear equations may be replaced by a problem of minimizing a nonlinear functional on \(\mathbb{R}^n\). The literature in this area is extensive. The author presents an introduction to some convergence theory for some classes of methods for solving unconstrained minimization problems.

Chapter 9: Nonlinear generalizations of several matrix classes. The author discusses some classes of mappings on \(\mathbb{R}^n\) which represent nonlinear generalizations of several closely related types of matrices arising frequently in applications.

Chapter 10: Outlook at further methods. This chapter presents a brief look at some examples of further classes of methods without going into details, for instance higher-order methods and piecewise linear methods.

A short description of the content of the 10 chapters of this basic work on methods for solving systems of nonlinear equations:

Chapter 1: Introduction. As an introduction to the topic the author begins with a brief overview of various questions that arise in connection with the computational solution of systems of nonlinear equations.

Chapter 2: Model problems. Systems of finitely many nonlinear equations in several real variables arise in connection with numerous scientific and technical problems and, correspondingly, they differ widely in form and properties. Chapter 2 introduces some typical examples of such systems without attempting to be exhaustive or to enter into details of the underlying problem areas.

Chapter 3: Iterative processes and rates of convergence. Iterative processes for the solution of finite-dimensional nonlinear equations vary almost as widely in form and properties as do the equations themselves. In Chapter 3 an algorithmic characterization of a class of these processes is introduced and some measures of their efficiency and rate of convergence are discussed.

Chapter 4: Methods of Newton type. This chapter introduces several basic types of iterative methods derived by linearizations, including, in particular, the classical Newton method and a few of its modifications.

Chapter 5: Methods of secant type. The discussion of discretized Newton methods begun in Section 4.3 is continued in this chapter.

Chapter 6: Combinations of processes. The chapter addresses the analysis of combined iterative processes where at each step of a primary method a secondary iteration is applied. For example, when linearization methods are used for solving high-dimensional problems it may become necessary to work with a secondary linear iterative process to solve the resulting linear systems at each step. In the study of such combined processes it has to be distinguished between controlling the secondary methods by means of a priori specified conditions or by some adaptive strategies based on the performance of the computation.

Chapter 7: Parametrized systems of equations. Continuation methods, in particular an ordinary differential equations approach, the code PITCON and simplicial approximation of manifolds are contained.

Chapter 8: Unconstrained minimization methods. The problem of solving a system of nonlinear equations may be replaced by a problem of minimizing a nonlinear functional on \(\mathbb{R}^n\). The literature in this area is extensive. The author presents an introduction to some convergence theory for some classes of methods for solving unconstrained minimization problems.

Chapter 9: Nonlinear generalizations of several matrix classes. The author discusses some classes of mappings on \(\mathbb{R}^n\) which represent nonlinear generalizations of several closely related types of matrices arising frequently in applications.

Chapter 10: Outlook at further methods. This chapter presents a brief look at some examples of further classes of methods without going into details, for instance higher-order methods and piecewise linear methods.

Reviewer: J.Guddat (Berlin)

##### MSC:

65H10 | Numerical computation of solutions to systems of equations |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65H20 | Global methods, including homotopy approaches to the numerical solution of nonlinear equations |

65K05 | Numerical mathematical programming methods |

90C30 | Nonlinear programming |