Numerical methods for equations and its applications.

*(English)*Zbl 1254.65068
Boca Raton, FL: CRC Press (ISBN 978-1-57808-753-2/hbk; 978-1-46-651711-0/ebook). viii, 465 p. (2012).

The first goal of the monograph is to give a review of modern results for finding the solutions to nonlinear equations and variational inequalities, the second one is the study of their applications in various fields of engineering, physical problems optimization, economics and control theory with emphasis on the following problems: 1. successful choice of initial data, guaranteeing that an iteration sequence is well defined and iterations will remain in the domain of operators; 2. various convergence problems; 3. economy of the numerical procedure connected with its convergence rate, the best choice of an algorithm or software program for solving a specific type of the problem.

After the introductory Chapter 1 describing the aims of the book, Chapter 2 is devoted to the most popular Newton iterative process to approximate the solutions of nonlinear equations. Here the convergence of Newton method (NM) is discussed under various conditions and space setting. First, new results for the local convergence of NM in Banach space settings are presented using only the center-Lipschitz conditions and then at the usage both Lipschitz and center-Lipschitz conditions on the Fréchet derivative of the nonlinear operator. Further, semilocal results for the convergence of NM follow at hypotheses on the second Fréchet derivative and new idea about recurrent functions. The advantages of such conditions over earlier ones consist in finer bounds on the distances involved with better information on the location of the solution. Then new sufficient semilocal convergence results are presented for NM for not necessarily bounded domains, where as example a Chandrasekhar-type nonlinear integral equation is solved cannot be solved by NM perilously. Next, local convergence results for continuous analog of NM are provided also in a Banach space setting, where the convergence radius is larger, the error bounds are tighter under the same or weaker assumptions than before. Further, the improved Newton-Kantorovich theorem is presented with application to interior point techniques; the concept of regular smoothness is used for the approximation of a locally unique solution of nonlinear equation \(F(x) = 0\), where \(F\) is a Fréchet differentiable operator defined on an open convex subset \(D\) of a Banach space \(X\) with values in a Banach space \(Y\), with more exact results on error bounds on the distances involved and the location of the solution. Chapter 2 contains also new sufficient convergence conditions for NM using convex major ants and recurrent functions, upper bounds on the limit points of majorizing sequences for NM.

The secant method (SM) as an alternative to NM is studied in Chapter 3 with the aims of its new convergence results under various conditions and space setting (nondiscrete induction and SM; nondiscrete induction and a double step SM; directional S-methods; efficient three step S-methods). The other alternative methods to NM are the Steffensen method presented in Chapter 4 with results on its convergence under Lipschitz-type conditions, and Gauss-Newton method in Chapter 5 also with convergence results under Lipschitz and average Lipschitz-type conditions.

Chapter 6 exhibits Newton-type methods with convergence analysis under Lipschitz-type conditions (new semilocal convergence results for Newton-like method using outer inverses but no Lipschitz conditions; convergence of a Moser-type method; convergence with a slantly differentiable operator; an intermediate Newton method).

The NM has however some disadvantages such as the exact computation of the Jacobians involved, computational cost is greater for large systems. Chapter 7 gives an answer of these problems using inexact methods. Here the local convergence analysis of inexact N-type methods (INTM) is given at the usage of a new type of residual control at the introduction the center Hölder condition. Four local convergence results are presented for two variants of INTM under average Lipschitz-type condition. A new concept of recurrent functions is used in Chapter 7 for the approximation of the locally unique solution by INTM two-step algorithm with tighter error bounds and weaker sufficient convergence conditions. In the conclusive Sec.4 the Zabrejko-Zincenko-type conditions [A. I. Zinchenko, “Some approximate methods for solving equations with non-differentiable operators”, Dopovidi Akad. Nauk Ukrain RSR, 156–161 (1963)] are used to provide a semilocal convergence analysis for Newton-type methods on the base of new type recurrent functions introduction. In Chapter 8, convergence analysis for Werner’s method and second Fréchet derivatives is given on the base of a new concept of recurrent functions. In Chapter 9, local convergence analysis is provided for cubically convergent Halley’s method to approximate a locally unique zero \(x^*\) of the nonlinear operator \(F\) defined on an open convex subset \(D\) of a Banach space \(X\) with values in a Banach space \(Y\).

Chapter 10 contains convergence results for variational inequalities under Lipschitz-type conditions on the derivative and Lipschitz-like assumption on set-valued maps in Banach space setting. A new approach of Chebyshev-type iterative method for variational inclusions with locally superquadratic convergence is presented under Lipschitz, Hölder or center-Hölder type conditions with the usage of \(\omega\)-type-conditioned second order Fréchet derivative. For nondifferentiable variational inclusions in a Banach space local convergence of the Newton-like method to a unique solution is proved at the usage of Lipschitz-type property of set-valued mappings. For the Newton-Josephy the method Kantrovich-type semilocal convergence analysis is given. Chapter 11 contains the convergence analysis for two classes of two-point Newton-type methods for solving of nonlinear equations in a Banach space. In the conclusive Chapter 11, the convergence analysis is presented for successive methods of fixed point problems solving. The contraction mapping principle is used guaranteeing the existence of a unique solution, however with more weakened sufficient convergence conditions comparatively with the results of E. Cătina [“Sufficient convergence conditions for certain accelerated successive approximations”, International Series of Numerical Mathematics 151, 71–75 (2005; Zbl 1081.65522)].

Every section of the monograph is complemented with a set of remarks, where literature citations are given, other related results are discussed and various possible extensions of the results are indicated.

After the introductory Chapter 1 describing the aims of the book, Chapter 2 is devoted to the most popular Newton iterative process to approximate the solutions of nonlinear equations. Here the convergence of Newton method (NM) is discussed under various conditions and space setting. First, new results for the local convergence of NM in Banach space settings are presented using only the center-Lipschitz conditions and then at the usage both Lipschitz and center-Lipschitz conditions on the Fréchet derivative of the nonlinear operator. Further, semilocal results for the convergence of NM follow at hypotheses on the second Fréchet derivative and new idea about recurrent functions. The advantages of such conditions over earlier ones consist in finer bounds on the distances involved with better information on the location of the solution. Then new sufficient semilocal convergence results are presented for NM for not necessarily bounded domains, where as example a Chandrasekhar-type nonlinear integral equation is solved cannot be solved by NM perilously. Next, local convergence results for continuous analog of NM are provided also in a Banach space setting, where the convergence radius is larger, the error bounds are tighter under the same or weaker assumptions than before. Further, the improved Newton-Kantorovich theorem is presented with application to interior point techniques; the concept of regular smoothness is used for the approximation of a locally unique solution of nonlinear equation \(F(x) = 0\), where \(F\) is a Fréchet differentiable operator defined on an open convex subset \(D\) of a Banach space \(X\) with values in a Banach space \(Y\), with more exact results on error bounds on the distances involved and the location of the solution. Chapter 2 contains also new sufficient convergence conditions for NM using convex major ants and recurrent functions, upper bounds on the limit points of majorizing sequences for NM.

The secant method (SM) as an alternative to NM is studied in Chapter 3 with the aims of its new convergence results under various conditions and space setting (nondiscrete induction and SM; nondiscrete induction and a double step SM; directional S-methods; efficient three step S-methods). The other alternative methods to NM are the Steffensen method presented in Chapter 4 with results on its convergence under Lipschitz-type conditions, and Gauss-Newton method in Chapter 5 also with convergence results under Lipschitz and average Lipschitz-type conditions.

Chapter 6 exhibits Newton-type methods with convergence analysis under Lipschitz-type conditions (new semilocal convergence results for Newton-like method using outer inverses but no Lipschitz conditions; convergence of a Moser-type method; convergence with a slantly differentiable operator; an intermediate Newton method).

The NM has however some disadvantages such as the exact computation of the Jacobians involved, computational cost is greater for large systems. Chapter 7 gives an answer of these problems using inexact methods. Here the local convergence analysis of inexact N-type methods (INTM) is given at the usage of a new type of residual control at the introduction the center Hölder condition. Four local convergence results are presented for two variants of INTM under average Lipschitz-type condition. A new concept of recurrent functions is used in Chapter 7 for the approximation of the locally unique solution by INTM two-step algorithm with tighter error bounds and weaker sufficient convergence conditions. In the conclusive Sec.4 the Zabrejko-Zincenko-type conditions [A. I. Zinchenko, “Some approximate methods for solving equations with non-differentiable operators”, Dopovidi Akad. Nauk Ukrain RSR, 156–161 (1963)] are used to provide a semilocal convergence analysis for Newton-type methods on the base of new type recurrent functions introduction. In Chapter 8, convergence analysis for Werner’s method and second Fréchet derivatives is given on the base of a new concept of recurrent functions. In Chapter 9, local convergence analysis is provided for cubically convergent Halley’s method to approximate a locally unique zero \(x^*\) of the nonlinear operator \(F\) defined on an open convex subset \(D\) of a Banach space \(X\) with values in a Banach space \(Y\).

Chapter 10 contains convergence results for variational inequalities under Lipschitz-type conditions on the derivative and Lipschitz-like assumption on set-valued maps in Banach space setting. A new approach of Chebyshev-type iterative method for variational inclusions with locally superquadratic convergence is presented under Lipschitz, Hölder or center-Hölder type conditions with the usage of \(\omega\)-type-conditioned second order Fréchet derivative. For nondifferentiable variational inclusions in a Banach space local convergence of the Newton-like method to a unique solution is proved at the usage of Lipschitz-type property of set-valued mappings. For the Newton-Josephy the method Kantrovich-type semilocal convergence analysis is given. Chapter 11 contains the convergence analysis for two classes of two-point Newton-type methods for solving of nonlinear equations in a Banach space. In the conclusive Chapter 11, the convergence analysis is presented for successive methods of fixed point problems solving. The contraction mapping principle is used guaranteeing the existence of a unique solution, however with more weakened sufficient convergence conditions comparatively with the results of E. Cătina [“Sufficient convergence conditions for certain accelerated successive approximations”, International Series of Numerical Mathematics 151, 71–75 (2005; Zbl 1081.65522)].

Every section of the monograph is complemented with a set of remarks, where literature citations are given, other related results are discussed and various possible extensions of the results are indicated.

Reviewer: Boris V. Loginov (Ul’yanovsk)

##### MSC:

65J15 | Numerical solutions to equations with nonlinear operators (do not use 65Hxx) |

65K15 | Numerical methods for variational inequalities and related problems |

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

47J25 | Iterative procedures involving nonlinear operators |

49J40 | Variational inequalities |

65R20 | Numerical methods for integral equations |

45G10 | Other nonlinear integral equations |