G. Golub, J. Ortega  Scientific Computing and Differential Equations
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Название: 
Scientific Computing and Differential Equations 
Автор: 
G. Golub, J. Ortega 
Категория: 
Математика

Тип: 
Книга 
Дата: 
04.01.2009 12:08:32 
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98 
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Описание: 
This book is a revision of Introduction to Numerical Methods for Differential Equations by J. M. Ortega and W. G. Poole, Jr., published by Pitman Publishing, Inc., in 1981.
As discussed in Chapter 1, a large part of scientific computing is concerned with the solution of differential equations and, thus, differential equations is an appropriate focus for an introduction to scientific computing. The need to solve differential equations was one of the original and primary motivations for the development of both analog and digital computers, and the numerical solution of such problems still requires a substantial fraction of all available computing time. It is our goal in this book to introduce numerical methods for both ordinary and partial differential equations with concentration on ordinary differential equations, especially boundaryvalue problems. Although there are many existing packages for such problems, or at least for the main subproblems such as the solution of linear systems of equations, we believe that it is important for users of such packages to understand the underlying principles of the numerical methods. Moreover, it is even more important to understand the limitation of numerical methods: "Black Boxes" can't solve all problems. Indeed, it may be that one has several excellent black boxes for solving classes of problems, but the combination of such boxes may yield less than optimal results.
We treat initialvalue problems for ordinary differential equations in Chapter 2 and introduce finite difference methods for linear boundary value problems in Chapter 3. The latter problems lead to the solution of systems of linear algebraic equations; Chapter 4 is devoted to the general treatment of direct methods for this problem, independently of any connection to differential equations. This chapter also considers the important problem of leastsquares approximation. In Chapter 5 we return to boundary value problems, but now they are nonlinear. This motivates the treatment of the solution of nonlinear algebraic equations in the remainder of the chapter. The tool for discretizing differential equations to this point has been finite difference methods; in Chapter 6 we introduce Galerkin and collocation methods as alternatives. These methods lead to the important subtopics of numerical integration and spline approx 
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