Numerical methods and optimization. An introduction (Q2836474)

This text is intended for undergraduate or beginning graduate students. The authors hope that students studying optimization methods would like to have numerical methods as well.NEWLINENEWLINEThe book consists of three parts: I Basics, II Numerical methods for standard problems, III Introduction to optimization.NEWLINENEWLINEPart I consists of two chapters in which some definitions and simple facts about vector spaces, matrices, sets and errors are given.NEWLINENEWLINEPart II consists of five chapters: Elements of numerical linear algebra, Solving equations, Polynomial interpolation, Numerical integration, Numerical solution of differential equations.NEWLINENEWLINEPart III consists of seven chapters: Basic concepts, Complexity issues, Introduction to linear programming, The simplex method for linear programming, Duality and sensitivity analysis in linear programming, Unconstrained optimization, Constrained optimization.NEWLINENEWLINEThe book contains a Bibliography (31 entries), an Index, and Exercises at the end of each chapter.NEWLINENEWLINEThe book is useful for students.NEWLINENEWLINEReviewer's remark: Exercise 13.2.e is strange: it is required to find minimum of the function \([(x_1-x_2-2)^2+(x_1-x_2+1)^4]^{-1}\) over the whole plane. Since \(x_1-x_2\) can go to infinity, the minimum is zero. It is not clear what is the aim of this exercise.NEWLINENEWLINEIn Exercise 13.11, on p. 349, it is claimed that for any initial \(x_0\) the Newton method for the function \(f:=x^{4/3}\) diverges.NEWLINENEWLINEHowever, the Newton method in this case converges for any initial approximation \(x_0\). This is clear from the geometrical point of view and can be easily checked analytically. Indeed, \(x_{n+1}=x_n-x_n^{4/3}/(4/3 x_n^{1/3})=1/4 x_n\), so \(x_{n}=(1/4)^nx_0\to 0\). Thus, the author's claim is not correct.NEWLINENEWLINEIt would be useful to say more about ill-posed problems of linear algebra. A lower estimate of the condition number is mentioned, that follows from a trivial inequality \(||A^{-1}||\geq ||Au||/||u||\). However, the error of this lower estimate is not known and can be very large.