OF “ ORTHOGONAL POLYNOMIALS IN MATLAB : EXERCISES AND SOLUTIONS ” BY GAUTSCHI

Walter Gautschi is a leading expert in constructive orthogonal polynomials and special functions. He is well-known to the SIAM readership through his fundamental contributions to numerical analysis that span over 65 years. So, you can imagine my excitement when his new book “Orthogonal Polynomials in MATLAB: Exercises and Solutions” arrived in the mail. I hastily ripped off the plastic packaging and blocked out the following weekend to devour its words of wisdom. Before finishing breakfast that Saturday, I realized that this is not an armchair book for a weekend. A desk chair, MATLAB, and the manuscript [2] are useful accessories. Several months later I picked up the book again with those accessories. I was pleased I did. Gautschi reminds us of some important concepts and computational techniques in constructive orthogonal polynomials. To get the most out of this book, I had to execute some MATLAB codes, regularly consult [2], and carefully work through some of the derivations. I suspect you will need to do the same. Since every set of orthogonal polynomials satisfies a three-term recurrence relationship, Gautschi views the recurrence relation as the definitive characterization of a set of orthogonal polynomials. As he describes, it provides access to Gauss quadrature rules (Chap. 4), evaluation procedures (Sec. 5.2), and the zeros of orthogonal polynomials (Sec. 3.3). Thus, the goal of Chapter 2 (which takes up a third of the book) is to compute the recurrence coefficients for the associated set of orthogonal polynomials when given an alternative description. There is the discretized Stieltjes procedure (Sec. 2.5) for computing the recurrence when the inner-product defining the orthogonality is known, the Chebyshev algorithm when the first 2n moments are given (Sec. 2.4), the modified Chebyshev algorithm when modified moments are prescribed, and a modification algorithm when a known weight is updated by a rational function (Sec. 2.6). These algorithms are sophisticated and state-of-the-art. Most are numerical procedures, but the ill-conditioning inherent in moments means that the Chebyshev algorithm is usually performed in symbolic mode. I appreciated the mastery in Chapter 2 and the distinction between problems that can be solved numerically and those that require symbolic manipulations. Chapter 2 also reminds us why folklore is abound in the subject of constructive orthogonal polynomials. Algorithmic details do matter! As a beginner it is easy to be mislead by quick experiments. After experience, one can successfully identity how to modify a question to a well-conditioned problem and carefully tip-toe around nearby numerical pitfalls. To get that experience, you will need that desk chair. Decades after Gautschi’s original work on the subject in Chapter 2, this material is still important and very welcomed. Chapter 2 pays off in chapter 4 on Gauss quadrature, where the Golub–Welsch algorithm is the main computational tool [3]. While the key relationship between the eigenvalues of a Jacobi matrix and the Gauss quadrature nodes is unceremoniously stated in Exercise 4.3, Gautschi beautifully illustrates the so-called circle theorem, i.e., appropriately scaled Gauss quadrature weights asymptotically tend to a semicircle. He goes on to present many standard and nonstandard Gauss quadrature