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Nonlinear Mathematics

Thomas L. Saaty and Joseph Bram
Dover Publications
Publication Date: 
Number of Pages: 
International Series in Pure and Applied Mathematics
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a textbook and a manifesto on the mathematics of non-linear models, originally published in 1982. The revolution it called for has happened, but in a way the authors did not anticipate, through computers and statistics. The book covers only a small portion of present-day work in nonlinear mathematics. It contains much still-useful classical material; conspicuous omissions include chaos theory, dynamical systems, and statistical methods in optimal control and filtering. The present work is an unaltered reprint of the 1964 McGraw-Hill edition.

This is the first of a two-volume work on the subject, and covers solution of equations (algebraic and transcendental), ordinary differential equations (including both solution methods and stability), optimization, and automatic control and filtering. The second volume is Saaty’s Modern Nonlinear Equations (McGraw-Hill, 1967) that covers more esoteric types of equations, such as nonlinear difference equations, differential-difference equations, and integral equations. Between them, the two books covered everything in nonlinear mathematics at the time of their publication except for nonlinear partial differential equations.

The book is primarily organized as a reference, and is composed of six largely-independent chapters. There are a large number of exercises scattered through the book; many of these ask the reader to fill in some missing details in the exposition, but many ask for the methods to be applied to particular problems. The book is applied mathematics, as it is aimed at problems for which linear approximations are not adequate, but for the most part it treats the matter abstractly, assuming we already have a (nonlinear) mathematical model and wish to get useful and accurate information from it. The material is slanted toward mathematical analysis, with an especially large section on nonlinear ordinary differential equations, but also includes nonlinear programming. Despite the manifesto, there is a lot of linear mathematics involved, including linear functionals and Newton’s method of root-finding.

Bottom line: reference work that covers only a portion of a now-large field, but still valuable for the parts it does cover.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

1. Linear and Nonlinear Transformations
1.1 Introduction
1.2 Vector Spaces; Linear Transformations
1.3 Eigenvalues; Eigenvectors
1.4 Inner-product Spaces
1.5 Self-adjoint Transformations
1.6 The Infinite-dimensional Case
1.7 Operators
1.8 Applications
1.9 Banach Spaces and Linear Functionals
1.10 Fixed-point Theorems and Applications
1.11 Lebesgue Integration, a Survey
2. Nonlinear Algebraic and Transcendental Equations
2.1 Introduction
2.2 The Newton-Raphson Method
2.3 The Method of Steepest Descent
2.4 Saddle-point Method or Steepest-descent Method of Complex Integration
3. Nonlinear Optimization; Nonlinear Programming and Systems of Inequalities
3.1 Introduction
3.2 Maxima, Minima, Quadratic Forms, and Convex Functions
3.3 Nonlinear Programming
3.4 Linear Programming
3.5 Characterization of the Optimum on the Boundary; Saddle Points; Duality
3.6 Construction of Solutions
3.7 Optimization Problems with Infinitely Many Constraints
4. Nonlinear Ordinary Differential Equations
4.1 Introduction
4.2 Some Nonlinear Equations
4.3 Existence and Uniqueness for First-order Systems
4.4 Linear Equations--Oscillatory Motion, Stability
4.5 Nonlinear Equations--Perturbation Method
4.6 Phase-plane Analysis--Stability Behavior in the Small (Systems of Two Equations)
4.7 Limit Cycles--Stability Behavior in the Large (Systems of Two Equations)
4.8 Topological Considerations: Indices and the Existence of Limit Cycles (Systems of Two Equations)
4.9 Periodic Solutions of Systems with Periodic Coefficients
4.10 Periodic Solutions of Nonlinear Systems with Periodic Coefficients
4.11 Lyapunov Stability
4.12 General Methods of Solution
5. Introduction to Automatic Control and the Pontryagin Principle
5.1 Introduction
5.2 Stability and a Class of Control Equations
5.3 Pontryagin's Maximum Principle
5.4 Functional Analysis and Optimum Control
6. Linear and Nonlinear Prediction Theory
6.1 Introduction
6.2 The Discrete Stationary Case
6.3 Construction of the Discrete-case Estimate
6.4 The Discrete Prediction Problem
6.5 The Prediction Error
6.6 The Special Case of Rational Densities
6.7 The Continuous Stationary Case
6.8 Construction of the Continuous-case Estimate
6.9 The Continuous Prediction Problem
6.10 Examples
6.11 Conditional Expectation
6.12 The General Estimation Problem
6.13 Polynomial Estimation
6.14 The Karhunen-Loeve Expa
6.15 Dynamical Systems with Control Variables
Appendix; Index