Werner C. Rheinboldt

Iterative solution of nonlinear equations in several variables

Preface to the Classics Edition Preface Acknowledgments Glossary of Symbols Introduction Part I. Background Material. 1. Sample Problems 2. Linear Algebra 3. Analysis Part II. Nonconstructive Existence Theorems. 4. Gradient Mappings and Minimization 5. Contractions and the Continuation Property 6. The Degree of a Mapping Part III. Iterative Methods. 7. General Iterative Methods 8. Minimization Methods Part IV. Local Convergence. 9. Rates of Convergence-General 10. One-Step Stationary Methods 11. Multistep Methods and Additional One-Step Methods Part V. Semilocal and Global Convergence. 12. Contractions and Nonlinear Majorants 13. Convergence under Partial Ordering 14. Convergence of Minimization Methods An Annotated List of Basic Reference Books Bibliography Author Index Subject Index.

Curriculum 68: Recommendations for academic programs in computer science: a report of the ACM curriculum committee on computer science

This report contains recommendations on academic programs in computer science which were developed by the ACM Curriculum Committee on Computer Science. A classification of the subject areas contained in computer science is presented and twenty-two courses in these areas are described. Prerequisites, catolog descriptions, detailed outlines, and annotated bibliographies for these courses are included. Specific recommendations which have evolved from the Committees 1965 Preliminary Recommendations are given for undergraduate programs. Graduate programs in computer science are discussed, and some recommendations are presented for the development of masters degree programs. Ways of developing guidelines for doctoral programs are discussed, but no specific recommendations are made. The importance of service courses, minors, and continuing education in computer science is emphasized. Attention is given to the organization, staff requirements, computer resources, and other facilities needed to implement computer science educational programs.

A Unified Convergence Theory for a Class of Iterative Processes

Unified theory for convergence results based on nonlinear estimates for iteration function and on majorizing sequences concept

Differential-algebraic systems as differential equations on manifolds

Based on the theory of differential equations on manifolds, existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in various applications. Both the autonomous and nonautonomous case are considered. Moreover, a class of algebraically incomplete systems is introduced for which existence and uniqueness results only hold on certain lower-dimensional manifolds. This class includes systems for which the application of ODE-solvers is known to lead to difficulties. Finally, some solution approach based on continuation techniques is outlined.

On P- and S-functions and related classes of n-dimensional nonlinear mappings☆

Abstract This paper introduces and analyzes certain classes of mappings on R n which represent nonlinear generalizations of the P - and S -matrices of Fiedler and Ptak, and of several closely related types of matrices. As in the case of the corresponding matrices, these nonlinear P - and S -functions arise frequently in applications. Basic properties of the different functions and of their inverses and subfunctions are established, and then a number of theorems are proved about the interrelationships between the various mappings. In particular, it is shown that the well-known monotone mappings, as well as the M -functions and certain of the strictly diagonally dominant mappings recently analyzed by Rheinboldt and More, respectively,are special cases of the P -functions. In turn, these P -functions and also the inverse isotone mappings are subclasses of the S -functions. In a final section, a series of characterization theorems for the different functions are presented in terms of conditions on their derivatives.

Analysis of Optimal Finite Element Meshes in R1

A theory of a posteriori estimates for the finite element method has been developed. On the basis of this theory, for a two-point boundary-value problem the existence of a unique optimal mesh distribution is proved and its properties are analyzed. This mesh is characterized in terms of certain, easily computable local error indicators which in turn allow for a simple adaptive construction of the mesh and also permit the computation of a very effective a posteriori error bound. While the error estimates are asymptotic in nature, numerical experiments show the results to be excellent already for 10% accuracy. The approaches are not restricted to the model problem considered here only for clarity; in fact, they allow for rather straightforward extensions to more general problems in one dimension as well as to higher-order elements. 11 tables.

Monotone Iterations for Nonlinear Equations with Application to Gauss-Seidel Methods

Monotone iterations for nonlinear elliptic differential equations in boundary-value problems applied to Gauss-Seidel methods

On the computation of multi-dimensional solution manifolds of parametrized equations

SummaryA new algorithm is presented for computing vertices of a simplicial triangulation of thep-dimensional solution manifold of a parametrized equationF(x)=0, whereF is a nonlinear mapping fromRn toRm,p=n−m>1. An essential part of the method is a constructive algorithm for computing moving frames on the manifold; that is, of orthonormal bases of the tangent spaces that vary smoothly with their points of contact. The triangulation algorithm uses these bases, together with a chord form of the Gauss-Newton process as corrector, to compute the desired vertices. The Jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local “triangulation patches”. Several numerical examples show that the method is very efficient in computing triangulations, even around singularities such as limit points and bifurcation points. This opens up new possibilities for determining the form and special features of such solution manifolds.

On Steplength Algorithms for a Class of Continuation Methods

The continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form

A Posteriori Error Analysis of Finite Element Solutions for One-Dimensional Problems

Fx = b,F:D \subset R^{n + 1} \to R^n