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**Numerical initial value problems in ordinary differential equations.**
*(English)*
Zbl 1145.65316

Englewood Cliffs, NJ: Prentice-Hall. xvii, 253 p. (1971).

The author presents a thorough and well-balanced treatment of discrete methods for the numerical integration of initial value problems associated with ordinary differential equations. Both mathematical and computer implementation aspects are considered, and three complete programs are included.

The reader should have a background in advanced calculus (including a basic knowledge of ordinary differential equations), matrix theory, introductory numerical methods, and Fortran programming. However, the theoretical chapters can be omitted if the reader wishes to concentrate only on the development of the methods, since theoretical results are quoted again whenever needed in the chapters on development. Fortran programming is needed only to read the three programs.

The introductory chapter includes a detailed treatment of Euler’s method. This is followed by chapters on Higher-order one-step methods; Systems of equations and equations of order greater than one; Convergence, error bounds, and error estimates for one-step methods; The choice of step size and order; Extrapolation methods; Multivalue or multistep methods—Introduction; General multistep methods, order, and stability; Multivalue methods; Existence, convergence, and error estimates for multivalue methods; Special methods for special problems; Choosing a method.

The chapter on special methods contains sections on stiff systems, algebraic and singular equations, and parameter estimation. One program is based on the standard fourth-order Runge-Kutta formulas, another is based on the extrapolation method of Bulirsch and Stoer, and the third is the author’s popular multivalue method, known as DIFSUB, for handling either stiff or non-stiff systems. Many problems and examples are given. It is altogether an extremely useful text.

The reader should have a background in advanced calculus (including a basic knowledge of ordinary differential equations), matrix theory, introductory numerical methods, and Fortran programming. However, the theoretical chapters can be omitted if the reader wishes to concentrate only on the development of the methods, since theoretical results are quoted again whenever needed in the chapters on development. Fortran programming is needed only to read the three programs.

The introductory chapter includes a detailed treatment of Euler’s method. This is followed by chapters on Higher-order one-step methods; Systems of equations and equations of order greater than one; Convergence, error bounds, and error estimates for one-step methods; The choice of step size and order; Extrapolation methods; Multivalue or multistep methods—Introduction; General multistep methods, order, and stability; Multivalue methods; Existence, convergence, and error estimates for multivalue methods; Special methods for special problems; Choosing a method.

The chapter on special methods contains sections on stiff systems, algebraic and singular equations, and parameter estimation. One program is based on the standard fourth-order Runge-Kutta formulas, another is based on the extrapolation method of Bulirsch and Stoer, and the third is the author’s popular multivalue method, known as DIFSUB, for handling either stiff or non-stiff systems. Many problems and examples are given. It is altogether an extremely useful text.

Reviewer: T. E. Hall (MR 47 #4447)