Most nontrivial data in engineering and science is composed of vectors. Linear algebra and multivariate statistics provide the well-known tools for the analysis of vector data, but an often overlooked problem is the estimation of limit values of vector sequences--as needed for prediction, solving systems of equations, and sensitivity analysis. This book treats this topic well. The author presents a fundamental formulation of the problem as follows: find the limiting value of an infinite sequence of vectors. The brute-force solution of calculating each term and hoping for convergence is unsuitable for two reasons: often the series does not actually converge (but useful analysis can still be done), and even if it does, convergence is too slow to allow for reasonable processing times.

Chapter 0 consists of a brief review of the relevant linear algebra, along with a discussion of iterative methods for linear systems that will be needed in the later chapters. The first type of extrapolation described in detail is polynomial extrapolation, covered in chapter 1. It is demonstrated that the limit (or anti-limit) of the series can be calculated by solving an inconveniently large set of linear equations. Minimum polynomial estimation (MPE) approximates the limit of the vector series by solving a much smaller set of equations based on the minimum polynomial of the original data with respect to the difference(s) between successive vectors in the series. This MPE method is the basis for other related methods, such as the modified minimum polynomial extrapolation (MMPE), reduced rank estimation (RRE), and SVD-MPD (where singular value decomposition is applied prior to the determination of the minimum polynomial). After presenting the math behind the methods, the author concisely presents effective algorithms for MPE and RRE in chapter 2. This chapter enables readers to make use of the methods; many authors would have omitted this material and would have written a much less effective book.

Two important areas are discussed in detail in later chapters. Since the slow convergence of brute-force methods is a prime motivation for the entire discussion, it’s vital to assess the convergence performance of these supposedly better techniques. In addition, the error performance of these approximations is addressed. An entire section of the book is devoted to extrapolation methods based on Krylov subspace methods, applicable to sequences defined by multiplication by an unknown matrix, while the MPE-related methods operate on the observed sequence of vectors.

The book addresses an interesting set of problems in prediction and data analysis. The treatment is extremely well supported mathematically--perhaps too much so (at times it was hard to glean the overarching concept from the proofs), but this is an easily forgivable flaw. Vector extrapolation methods with applications is carefully written, complete, and a useful addition to the literature.