A Theoretical Introduction to Numerical Analysis

This book introduces the reader to basic ideas of numerical analysis and it covers the main topics in the field. The book is divided into four parts. The introductory part gives the reader fundamental objects of numerical analysis such as errors and basic methods of computation. The first part of the book is devoted to an overview of interpolation (algebraic interpolation and trigonometric interpolation) and quadrature of functions. The second part studies the problem of solving systems of scalar equations (systems of linear algebraic equations; direct methods, iterative methods for solving linear systems and overdetermined linear systems; the method of least squares, numerical solution of nonlinear equations and their systems). The third part studies the method of finite differences for numerical solution of ordinary differential equations, finite difference schemes for partial differential equations, discontinuous solutions and methods of their computation and discrete methods for elliptic problems. The fourth part contains a discussion of the method of boundary equations for the numerical solution of boundary value problems and boundary integral equations and the method of boundary elements, boundary equations with projections and the method of difference potentials.

The basic concepts (discretization, errors, efficiency, complexity, numerical stability, consistency, convergence, etc.) are explained and illustrated in different parts of the book. The prospective reader would be a graduate or senior undergraduate student in mathematics, science or engineering.

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