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A brief introduction to numerical analysis. (English) Zbl 0874.65001

Boston: Birkhäuser. xii, 202 p. (1997).
The title of this book reflects its structure and style with great exactness: the book contains twentyone brief lectures on the numerical methods of linear algebra, analysis, optimization and solution of integral equations; these lectures are divided into very short sections which are, as a rule, less than one page in length.
The beginning Lectures 1 and 2 are introductory. They contain some basic material on functional analysis and matrix theory which will be necessary later on. Lectures 3, 4 and 5 are concerned with the foundations of the perturbation theory and eigenvalue problems. In Lecture 6 floating-point arithmetic and roundoff errors are discussed. Lectures 7, 8 and 9 are dealing with direct methods for the solution of linear systems of equations and eigenvalue problems with special emphasis on QR decomposition.
Lectures 10 and 11 are devoted to the QR algorithm, its generalizations and the question of its convergence. The next two Lectures 12 and 13 are concerned with polynomial interpolation and the problem of convergence of the interpolation process. Lecture 14 provides a basic introduction to the theory of splines, and Lecture 15 includes some aspects of uniform and least squares approximation theory. Methods of numerical integration are considered in Lecture 16. Problems of numerical solution of nonlinear equations are discussed in Lecture 17. Lectures 18, 19 and 20 are dealing with optimization methods including minimization over the subspace, projection methods and the method of conjugate gradients. The final Lecture 21 contains foundations of methods for the numerical solution of integral equations.
The presentation of the material is very clear and supported by properly chosen exercises of good didactic value added to each lecture. The book is oriented to students in mathematical and physical disciplines, applied mathematicians and higher school lecturers.

MSC:

65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
65Fxx Numerical linear algebra
65Dxx Numerical approximation and computational geometry (primarily algorithms)
65Jxx Numerical analysis in abstract spaces
65Hxx Nonlinear algebraic or transcendental equations
65Kxx Numerical methods for mathematical programming, optimization and variational techniques
65Rxx Numerical methods for integral equations, integral transforms
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