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Analysis of numerical methods
BOOK REVIEWS E. ISAACSON and H. B. KELLER.
Analysis
John Wiley and Sons, London-New
of numerical methods.
ix + 541 pp.
York, 1966.
THE present book is based on long experience of the authors as teachers of courses of lectures at New York University.
The authors assume that the reader
is familiar with the usual course of mathematical analysis and matrix algebra. They avoid applying many other sections of mathematical analysis, for example, the theory of functions of a complex variable and functional analysis.
Some of the simplest concepts of functional analysis such as the norms of vectors and matrices are explained in chapter 1. The book deals with the following problems: the solution of systems of linear algebraic and non-linear algebraic and transcendental equations, the calculation of the eigenvalues and eigenvectors of matrices, the elements of the theory of approximation of functions by algebraic and trigonometric polynomials, interpolation, numerical integration, and the numerical solution of ordinary differential equations and partial differential equations. The authors do not attempt to give a survey and comparison of the different methods of solving the problems indicated. They select the most promising from their point of view and analyze these methods in detail. The analysis includes the correctness of the problem formulated, the errors of the method and calculations,
and the stability of the algorithm.
We give more details of the individual chapters.
In chapters 2, 3 and 4 most
attention is given to iteration methods of solving algebraic and transcendental equations. A great deal of attention is paid to accelerated convergence. In chapter 5 theoretical problems of the uniform and mean square approximation of functions of one variable by algebraic and trigonometric polynomials are considered.
Algorithms for the construction of best approximation polynomials
are not presented. -*Zh.
uFchisl.
Mat. mat. Fiz. 9, 1, 252-255, 349
1969.
Book reviews
350
Chapter 6 and 7 are devoted to interpolation and its applications in numerical differentiation and integration.
Other approaches for obtaining numerical integration
formulas are also briefly indicated.
Problems of integration on classes
of functions
are not considered. Chapter 8 deals with the numerical solution of ordinary differential equations. Most attention is given to the solution of the Cauchy problem. Finally, the last chapter, chapter 9, is devoted to the solution of partial differential equations. On the whole all the reasoning is based on the example of classical representatives of these equations: Poisson’s equations, the oscillation of a string, and the one-dimensional heat conduction equation. The theory of stability and its connection with convergence is briefly explained. The book contains sufficient examples and exercises illustrate but also extend its contents.
which not only
A bibliography of foreign authors and
of translations from Russian is given. The book is written clearly and accurately. The authors have succeeded in efficiently formulating and presenting to the reader the main ideas in numerical methods.
The book may be useful to a wide circle of mathematicians
and engineers concerned with the solution of applied problems. translated into Russian.
It should be
N. P. Zhidkov
Translated by J. Berry
P. II. E. MEIJER. Quantum statistical
mechanics.
Breach Sci. Publs., New York-London-Paris,
viii + 172~~.
Gordon and
1967.
THIS book is one of a series under the general title of Documents on modern physics (the editors are Montroll, Vineyard and Levy), including specially written reviews, reproductions of lectures, conference material, series of papers on a definite subject etc. The book combines lectures given by a number of authors at a summer school which was organized by the Catholic University of America (Washington). Chapter I (pp. l-40, Meyer) is devoted to the density matrix. The most general questions concerning the pure and mixed quantum mechanical states of a system are considered, the statistical operator is introduced, and also its
Analysis
John Wiley and Sons, London-New
of numerical methods.
ix + 541 pp.
York, 1966.
THE present book is based on long experience of the authors as teachers of courses of lectures at New York University.
The authors assume that the reader
is familiar with the usual course of mathematical analysis and matrix algebra. They avoid applying many other sections of mathematical analysis, for example, the theory of functions of a complex variable and functional analysis.
Some of the simplest concepts of functional analysis such as the norms of vectors and matrices are explained in chapter 1. The book deals with the following problems: the solution of systems of linear algebraic and non-linear algebraic and transcendental equations, the calculation of the eigenvalues and eigenvectors of matrices, the elements of the theory of approximation of functions by algebraic and trigonometric polynomials, interpolation, numerical integration, and the numerical solution of ordinary differential equations and partial differential equations. The authors do not attempt to give a survey and comparison of the different methods of solving the problems indicated. They select the most promising from their point of view and analyze these methods in detail. The analysis includes the correctness of the problem formulated, the errors of the method and calculations,
and the stability of the algorithm.
We give more details of the individual chapters.
In chapters 2, 3 and 4 most
attention is given to iteration methods of solving algebraic and transcendental equations. A great deal of attention is paid to accelerated convergence. In chapter 5 theoretical problems of the uniform and mean square approximation of functions of one variable by algebraic and trigonometric polynomials are considered.
Algorithms for the construction of best approximation polynomials
are not presented. -*Zh.
uFchisl.
Mat. mat. Fiz. 9, 1, 252-255, 349
1969.
Book reviews
350
Chapter 6 and 7 are devoted to interpolation and its applications in numerical differentiation and integration.
Other approaches for obtaining numerical integration
formulas are also briefly indicated.
Problems of integration on classes
of functions
are not considered. Chapter 8 deals with the numerical solution of ordinary differential equations. Most attention is given to the solution of the Cauchy problem. Finally, the last chapter, chapter 9, is devoted to the solution of partial differential equations. On the whole all the reasoning is based on the example of classical representatives of these equations: Poisson’s equations, the oscillation of a string, and the one-dimensional heat conduction equation. The theory of stability and its connection with convergence is briefly explained. The book contains sufficient examples and exercises illustrate but also extend its contents.
which not only
A bibliography of foreign authors and
of translations from Russian is given. The book is written clearly and accurately. The authors have succeeded in efficiently formulating and presenting to the reader the main ideas in numerical methods.
The book may be useful to a wide circle of mathematicians
and engineers concerned with the solution of applied problems. translated into Russian.
It should be
N. P. Zhidkov
Translated by J. Berry
P. II. E. MEIJER. Quantum statistical
mechanics.
Breach Sci. Publs., New York-London-Paris,
viii + 172~~.
Gordon and
1967.
THIS book is one of a series under the general title of Documents on modern physics (the editors are Montroll, Vineyard and Levy), including specially written reviews, reproductions of lectures, conference material, series of papers on a definite subject etc. The book combines lectures given by a number of authors at a summer school which was organized by the Catholic University of America (Washington). Chapter I (pp. l-40, Meyer) is devoted to the density matrix. The most general questions concerning the pure and mixed quantum mechanical states of a system are considered, the statistical operator is introduced, and also its