|Original Title||Computational Mathematics|
|Author||B. P. Demidovich, I. A. Maron|
|Topics||mathematics, programmes, monte-carlo method, approximate numbers, continued fractions, computations, functionsm, solutions. algebraic equations, transcendental equations, newton’s method, convergence of series, matrix algebra, systems of linear equations, linear vector spaces, eigenvalues, eigenvectors, matrices, non-linear equations, interpolation, approximate differentiation, approximate integration|
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The basic aim of this book is to give as far as possible a systematic and modern presentation of the most important methods and techniques of computational mathematics on the basis of the general course of higher mathematics taught in higher technical schools. The. book has been arranged so. that the basic portion constitutes a manual for the first .cycle of ·studies in approximate computations for higher technical colleges. The text contains supplementary ma- tetial Which goes beyond the scope of the ordinary college course, but the reader can select those sections which interest him and omit any extra material without loss of continuity. The chapters and sections which may be dropped out in a first reading are marked with an asterisk.
This text makes wide use of matrix calcu]us. The concepts of a vector, matrix, inverse matrix, eigenvalue and eigenvector of a matrix, etc. are workaday tools. The use of matrices offers a number of advantages in presenting the subject matter since they greatly facilitate an understanding of the development of many computations. In this sense a particular gain is achieved in the proofs of the convergence theorems of various numerical processes. Also, modern high-speed computers are nicely adapted to the performance of the basic matrix operations.
For a full comprehension of the contents of this. book, the reader should have a background of linear algebra and the theory of linear vector spaces. With the aim of making the text as self-contained as possible, the authors have included all the necessary starting material in these subjects. The appropriate chapter~ are completely independent of the basic text and can be omitted by readers who have already studied these sections.
A few words about the contents of the book. In the main it is devoted to the following problems: operations involving approximate numbers, computation of functions by means of series and iterative processes, approximate and numerical solution of algebraic ·and transcendental equations, computational methods of linear algebra, interpolation of functions, numerical differentiation and integration of functions, and the Monte Carlo method.
A great deal of attention is devoted to methods of error estimation. Nearly all processes are provided with proofs of convergence theorems, and the presentation is such that the proofs may be omitted if one wishes to confine himself to the technical aspects of the matter. In. certain case?, in order to pictorialize and lighten the presentation, the computational techniques are given as simple recipes.
The basic methods are carried to numerical applications that include computational schemes and numerical examples with de- tailed step.s of solution. To facilitate understanding the essence of the matter at hand, most of the problems are stated in simple form and are of an illustrative nature. References are given at the. end of each chapter and the complete list (in alphabetical order) is given at the end of the book.
The present text offers selected methods in computational mathematics and does not include material that involves empirical formulas, quadratic approximation of functions, approximate solutions of differential equations, etc. Likewise, the book does not include material on programming and the technical aspects of solving mathematical problems on computers. The interested reader must consult the special literature on these subjects.