» J. Stoer, R. Bulirsch - Introduction to Numerical Analysis
J. Stoer, R. Bulirsch - Introduction to Numerical Analysis
||Introduction to Numerical Analysis
||J. Stoer, R. Bulirsch
||On the occasion of this new edition, the text was enlarged by several new sections. Two sections on B-splines and their computation were added to the chapter on spline functions: Due to their special properties, their flexibility, and the availability of well-tested programs for their computation, B-splines play an important role in many applications.
Also, the authors followed suggestions by many readers to supplement the chapter on elimination methods with a section dealing with the solution of large sparse systems of linear equations. Even though such systems are usually solved by iterative methods, the realm of elimination methods has been widely extended due to powerful techniques for handling sparse matrices. We will explain some of these techniques in connection with the Cholesky algorithm for solving positive definite linear systems.
The chapter on eigenvalue problems was enlarged by a section on the Lanczos algorithm; the sections on the LR and QR algorithm were rewritten and now contain a description of implicit shift techniques.
In order to some extent take into account the progress in the area of ordinary differential equations, a new section on implicit differential equations and differential-algebraic systems was added, and the section on stiff differential equations was updated by describing further methods to solve such equations.
The last chapter on the iterative solution of linear equations was also improved. The modern view of the conjugate gradient algorithm as an iterative method was stressed by adding an analysis of its convergence rate and a description of some preconditioning techniques. Finally, a new section on multigrid methods was incorporated: It contains a description of their basic ideas in the context of a simple boundary value problem for ordinary differential equations.