Vilenkin N. Method of Successive Approximations 1979

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Vilenkin N. Method of Successive Approximations 1979

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Textbook in PDF format The main purpose of this book is to present various methods of approximate solution of equations. Their practical value is beyond doubt, but still little attention is paid to them either at school or a col­lege and so someone who has passed a college level higher mathema­tics course usually has difficulty in solving a transcendental equation of the simplest type. Not only engineers need to solve equations, but also mechnicians, production technologists and people in other profes­sions as well. It is also good for high-school students to become acquainted with the methods of approximate solution of equations. Since most approximate solution methods involve the idea of the derivative we were forced to introduce this concept. We did this intui­tively, making use of a geometric interpretation. Hence, a knowledge of secondary school mathematics will be sufficient for anyone wanting to read this book. In writing this book the author made use of a lecture he delivered to 9th. and 10th. form pupils, members of the school mathematics circle at the Lomonosov State University of Moscow. The material contained in this lecture was used by a teacher at the Moscow secondary school No. 425, S. I. Schwartzburd, for extracurri­cular work with the 9th. form pupils. The author expresses his gratitude to S. I. Schwartzburd for supplying problems involving the solu­tion of equations by the method of iteration. These problems were made use of in the writing of the book. Preface to the Second Russian Edition. Preface to the First Russian Edition. Introduction. Successive Approximations. Achilles and the Tortoise. Division On Electronic Computers. Extraction of Square Roots by Method of Successive Approximations. Extraction of Roots with Positive Integer Indices Using Method of Successive Approximations. Method of Iteration. Geometrical Meaning of Method of Iterations. Contraction Mappings (Contractions). Contraction Mappings and Method of Iteration. Method of Chords. Improved Method of Chords. Derivative of Polynomial. Newton’s Method for Approximate Solution of Algebraic Equations. Geometrical Meaning of Derivative. Geometrical Meaning of Newton’s Method. Derivatives of Arbitrary Functions. Computation of Derivatives. Finding the First Approximations. Combined Method of Solving Equations. Convergence Test for Method of Iterations. Rate of Convergence of Iteration Process. Solving Systems of Linear Equations by Method of Successive Approximations. Solving Systems of Non-linear Equations Using Method of Successive Approximations. Modified Distance. Convergence Tests for Process of Successive Approximations for Systems of Linear Equations. Successive Approximations in Geometry. Conclusion. Exercises. Solutions