In the Little Mathematics Library we now come to Method of Successive Approximations by N. Ya. Vilenkin. As the title suggests the book has to do with approximation methods, but what kind of approximations and for what kind of use one may ask?

The preface of the book reads:

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 technicians, 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.

The book was translated from the Russian by Mark Samokhvalov and was first published by Mir in 1979. All credits to the original uploader.

The Internet Archive Link

Contents of the book are as under:

1. Introduction 9
2. Successive Approximations 12
3- Achilles and the Tortoise 14
4 Division On Electronic Computers 16
5 Extraction of Square Roots by Method of Successive Approximations 19
6. Extraction of Roots with Positive Integer Indices Using Method of Successive Approximations 25
7 Method of Iteration 27
8. Geometrical Meaning of Method of Iterations 30
9 Contraction Mappings (Contractions) 32
10 Contraction Mappings and Method of Iteration 36
11. Method of Chords 43
12. Improved Method of Chords 47
13 Derivative of Polynomial 49
14 Newton’s Method for Approximate Solution of Algebraic Equations 51
15. Geometrical Meaning of Derivative 54
16. Geometrical Meaning of Newton’s Method 57
17 Derivatives of Arbitrary Functions 59
18 Computation of Derivatives 61
19. Finding the First Approximations 63
20 Combined Method of Solving Equations 66
21 Convergence Test for Method of Iterations 68
22. Rate of Convergence of Iteration Process 71
23. Solving Systems of Linear Equations by Method of Successive Approximations 74
24. Solving Systems of Non-linear Equations Using Method of Successive Approximations 79
25. Modified Distance 82
26. Convergence Tests for Process of Successive Approximations for Systems of Linear Equations 85
27. Successive Approximations in Geometry 91
28 Conclusion 94
Exercises 96
Solutions 98