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Numerical analysis : theory and experiments
Sutton B., SIAM-Society for Industrial and Applied Mathematics, Philadelphia, PA, 2019. 431 pp. Type: Book (978-1-611975-69-7)
Date Reviewed: May 17 2021

In the depressingly similar world of numerical analysis textbooks, Sutton’s book is a welcome relief. It is based on MATLAB, but differently to most. Many of the included methods are those not covered in other books (although there’s nothing to stop such books from including them). This book is also aimed at the learner, assuming only a background of basic calculus and linear algebra.

There are 34 chapters divided among seven sections: “Computation,” “Interpolation,” “Integration,” “Systems of Linear Equations,” “Linear Differential Equations,” “Zero Finding,” and “Non-Linear Differential Equations.” These sections vary greatly in size, for example, the shortest section, “Zero Finding,” has only two chapters (25 pages).

The book makes very heavy use of Chebyshev approximations, and the use of interpolation using Chebyshev nodes: here defined as the values of cos(2 pi i/n) for i between zero and n inclusive, providing unequally spaced points between -1 and 1. Runge’s function 1/(1+ 25 x2) illustrates that polynomials interpolated over equally spaced ordinates tend to oscillate wildly. Such oscillation can be greatly reduced by using ordinates that are more clustered toward the ends of the interval, which is precisely the case with Chebyshev nodes. Thus, interpolating over points defined at Chebyshev notes generally provides a far tighter fit to the function. As is well known, such polynomials can be shown to have the closest fit of any polynomials of equal degree.

Integrating such a polynomial will thus be closer to the area under the curve than obtained by a high-order Newton-Cotes rule, and integrating an interpolating polynomial defined using Chebyshev nodes is called Clenshaw-Curtis integration. This technique is rarely mentioned in most texts, yet it is conceptually simple, can be extremely accurate, and in fact compares very favorably to Gauss-Legendre quadratures. Clenshaw-Curtis integration is often based on nodes, not including the endpoints: cos(2 pi (2i+1)/2n) for i between zero and n-1 inclusive. This allows the integration of a function not defined at an endpoint, such as sin(t)/t between zero and one, without any piecewise definitions. Some authors, such as Trefethen, only use these nodes; however, Sutton only uses the nodes that include the endpoints.

Lagrange interpolation is discussed at length, with the Lagrange interpolation polynomial defined using the barycentric weights, which can be computed beforehand from the ordinates alone. These weights are the reciprocals of the fractions that form the Lagrange polynomial:

Given these precomputed weights, the Lagrange polynomial has a particularly simple form and can be easily extended to include new points. It’s a pity that the author’s proof of the Lagrange polynomial, in terms of its weights, is messy and unclear. Berrut and Trefethen give a neater and far simpler, and more instructive and intuitive, proof [1]. Possibly the notation could be simplified: a friendlier and less symbol-heavy notation would add clarity, especially as this is a beginner’s text.

The sections on differential equations are both interesting and somewhat unsatisfying. Given the importance of adaptive Runge-Kutta methods for individual and systems of ordinary differential equations (ODEs), I think any beginner text owes it to the student to discuss them at least briefly. However, in this text, they are ignored completely. Instead, DEs are solved by collocation methods, which in some ways seems a little ad hoc. If Runge-Kutta methods had been introduced, then collocation methods could be introduced as implicit RK methods; this is more elegant theoretically. As Arieh Iserles points out in A first course in the numerical analysis of differential equations:

The collocation method sounds eminently plausible. Yet you will search for it in vain in most expositions of ODE methods. The reason is that ... collocation is nothing but a Runge-Kutta method in disguise. [2]

The book uses MATLAB exclusively, which makes the text unusable for readers without access to it. Given today’s range of high quality, robust, fast, and open-source software for numerical computations, it’s a pity the author didn’t make use of one. MATLAB certainly remains popular--especially with engineers--but Python, Julia, GNU Octave (to name just three open-source systems) can easily perform any of the computations in this text.

Would I use this text in my own courses? Probably not, as I think there’s still a strong case to be made for a more classical treatment (but using more approachable software). That being said, I like the idea of this text, and much of the material is very good indeed; I would just prefer it as a reference rather than a textbook.

Reviewer:  Alasdair McAndrew Review #: CR147266 (2108-0196)
1) Berrut, J.-P.; Trefethen, L. N. Barycentric Lagrange interpolation. SIAM Review 46, 3(2004), 501–517.
2) Iserles, A. A first course in the numerical analysis of differential equations. Cambridge University Press, Cambridge, UK, 2009.
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