Lectures on Numerical Methods

ISBN-13: 9789401174855 / Angielski / Miękka / 2012 / 344 str.

I. P. Mysovskih

Opis

The course of lectures on numerical methods (part I) given by the author to students in the numerical third of the course of the mathematics mechanics department of Leningrad State University is set down in this volume. Only the topics which, in the opinion of the author, are of the greatest value for numerical methods are considered in this book. This permits making the book comparatively small in size, and, the author hopes, accessible to a sufficiently wide circle of readers. The book may be used not only by students in daily classes, but also by students taking correspondence courses and persons connected with practical computa tion who desire to improve their theoretical background. The author is deeply grateful to V. I. Krylov, the organizer ofthe course on numerical methods (part I) at Leningrad State University, for his considerable assistance and constant interest in the work on this book, and also for his attentive review of the manuscript. The author is very grateful to G. P. Akilov and I. K. Daugavet for a series of valuable suggestions and observations. The Author Chapter I NUMERICAL SOLUTION OF EQUATIONS In this chapter, methods for the numerical solution of equations of the form P(x) = 0, will be considered, where P(x) is in general a complex-valued function."

Spis treści

I. Numerical solution of equations.- 1. Finding an initial approximation.- 2. The secant method.- 3. The method of iterations.- 4. The method of iterations for systems of equations.- 5. Numerical evaluation of polynomials and their derivatives.- 6. Newton’s method.- 7. Theorems on the convergence of Newton’s method.- 8. Remarks on the practical application of Newton’s method.- 9. Lobacevskii’s method.- 10. Factorization methods.- Exercises for Chapter I.- II. Algebraic interpolation.- 1. Introduction.- 2. Finite differences.- 3. Divided differences.- 4. The general problem of interpolation.- 5. Interpolation of function values.- 6. The remainder term in interpolation.- 7. Interpolation at equidistant points. Newton’s formulas for interpolation at the beginning and end of tables.- 8. Interpolation at equidistant points. The formulas of Gauss, Stirling, and Bessel.- 9. Inverse interpolation. Interpolation without differences.- 10. Hermite interpolation.- 11. Numerical differentiation.- Exercises for Chapter II.- III. Approximate calculation of integrals.- 1. Interpolation quadrature formulas.- 2. The simplest interpolation quadrature formulas.- 3. Numerical integration of periodic functions and the rectangular quadrature formula.- 4. Gaussian type quadrature formulas.- 5. Legendre polynomials and the Gauss formula.- 6. Other special cases of quadrature formula of the Gaussian type.- 7. A. A. Markov’s quadrature formulas.- 8. ?ebyšev’s quadrature formula.- 9. Bernoulli numbers and polynomials.- 10. Representation of functions by means of Bernoulli polynomials.- 11. The Euler-Maclaurin formula.- 12. Concluding remarks.- Exercises for Chapter III.- IV. The numerical solution of the Cauchy problem for ordinary differential equations.- 1. Introduction.- 2. The Runge-Kutta method.- 3. On difference methods for the solution of the Cauchy problem.- 4. The Adams extrapolation method.- 5. The construction of the beginning of the table.- 6. The Adams interpolation method.- 7. Methods of Cowell type.- 8. Numerical integration of systems of equations of the first order.- 9. Störmer’s extrapolation method.- 10. Störmer’s interpolation method.- 11. Cowell’s method.- 12. On the estimate of error of the Adams method.- Exercises for Chapter IV.