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Book Reviews Introduction, 565 Featured Reviews: Pattern Formation and Dynamics in Nonequilibrium Systems (Michael Cross and Henry Greenside), Edgar Knobloch, 567 Analytic Combinatorics (Philippe Flajolet and Robert Sedgewick), Robin Pemantle, 572 Numerical Methods for Roots of Polynomials. Part I (J. M. McNamee), Götz Alefeld, 576 Pearls of Discrete Mathematics (Martin Erickson), Robert A. Beezer, 577 Mathematical Optimization and Economic Analysis (Mikulas Luptáčik), Bruce D. Craven, 578 Variational Methods in Imaging (O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen), Heinz W. Engl, 580 Real Analysis and Applications. Theory in Practice (Kenneth R. Davidson and Allan P. Donsig), Palle Jorgensen, 581 Maple and Mathematica: A Problem Solving Approach for Mathematics. Second Edition (Inna Shingareva and Carlos Lizárraga-Celaya), Grant Keady, 582 A Primer on Scientific Programming with Python (Hans Petter Langtangen), Jaan Kiusalaas, 585 Introduction to Applied Algebraic Systems (Norman R. Reilly), William M. Mcgovern, 585 The Pleasures of Statistics: The Autobiography of Frederick Mosteller (Frederick Mosteller, Stephen E. Weinberg, David C. Hoaglin, and Judith M. Tanur, editors), Robert E. O’Malley, Jr., 586 Differential Equations and Mathematical Biology. Second Edition (D. S. Jones, M. J. Plank, and B. D. Sleeman), Robert E. O’Malley, Jr., 586 Perspectives in Computation (Robert Geroch), Arthur Pittenger, 587 Matrix Methods in Data Mining and Pattern Recognition (Lars Eldén), David S. Watkins, 588


This issue has two featured reviews, one on Cross and Greenside’s Pattern For-mation and Dynamics for Nonequilibrium Systems by Edgar Knobloch of the University ofCalifornia at Berkeley, and another on Flajolet and Sedgewick’s Analytical Combinatoricsby Robin Pemantle of the University of Pennsylvania. These topics are important onesand you’ll find the reviews to be articulate and informative.

The other reviews span a wide variety of subjects. Some of the reviews seem quitedemanding. But this is appropriate. Even though personal and institutional libraries aregaining huge amounts of shelf space, due to the purchase of e-books rather than paperversions, readers still simply don’t have time for less satisfactory publications.

The valuable opinions of the reviewers are very much appreciated.

Bob O’MalleySection Editor

[email protected]

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SIAM REVIEW c© 2010 Society for Industrial and Applied MathematicsVol. 52, No. 3, pp. 567–590

Book Reviews

Edited by Robert E. O’Malley, Jr.

Featured Reviews: Pattern Formation and Dynamics in Nonequilibrium Systems andAnalytic Combinatorics.

Pattern Formation and Dynamics in Nonequilibrium Systems. By Michael Cross andHenry Greenside. Cambridge University Press, Cambridge, UK, 2009. $90.00. xvi+535 pp.,hardcover. ISBN 978-0-521-77050-7.

Pattern formation is ubiquitous in nature and occurs on all scales from the microscopicto that of the universe, in which galaxies are organized into filamentary structuresseparated by partial voids. The word pattern can be used to describe all departuresfrom spatial homogeneity. Patterns can be extended, filling available space, or spa-tially localized and embedded in a homogeneous background. Extended patterns maybe regular or irregular and may be time-dependent; this is also true for localized struc-tures. As a result, pattern formation is of interest in all sciences, and indeed in thehumanities too, where patterns in woven fabrics, on ceramic tiles, or on architecturalcolumns have fascinated man since ancient times.

The language of patterns is that of mathematics, specifically that of group theory.Group theory is invaluable in describing and classifying patterns that are regular, andtherefore repeat. However, not all patterns are regular, and some, like the quasipatterns constructed by Penrose, may look locally periodic without being periodicglobally. The description of disordered or chaotic patterns is harder and remainsincomplete. Existing measures of “chaos” or “disorder” in terms of correlations, etc.,provide a crude description but leave out much of what is of interest.

In many branches of science we need more than a description and classification ofpatterns—we need to understand why and when patterns form. In other words, weneed a theory of patterns. Physical systems are generally described by partial differ-ential equations (PDEs) and are dissipative. Patterns in such systems are dissipativestructures, that is, they are maintained against dissipation by forcing. While suchstructures may be produced in response to inhomogeneous forcing, the essence of pat-tern formation is captured by the spontaneous development of a symmetry-breakinginstability, that is, an instability that breaks the translation or rotation invarianceof the homogeneous state. That such instabilities arise in response to homogeneousforcing is familiar. The instability discovered by Turing that now bears his name is anexample of a steady state instability that evolves into steady regular patterns, such asa pattern of stripes or hexagons. The instability arises in systems with local activationand long range inhibition, and has an intrinsic scale that is determined by the diffusioncoefficients of the activator and inhibitor species. In contrast, in many systems arisingin fluid mechanics, such as a layer of liquid heated from below (Rayleigh–Benard con-vection), or the flow between coaxial cylinders in which the inner cylinder is rotating

Publishers are invited to send books for review to Book Reviews Editor, SIAM, 3600 Market St.,6th Floor, Philadelphia, PA 19104-2688.

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568 BOOK REVIEWS

(Couette–Taylor system), the length scale of a symmetry-breaking instability is set byan externally imposed length scale—in convection, this may be the distance betweenthe top and bottom plates; in the Couette–Taylor system, this will be the width of thefluid-filled annulus. Worse yet, in most physical systems the presence of boundariesis inevitable, and one seeks to discover whether the boundaries play an essential rolein the observed patterns or whether the pattern is intrinsic and only modified by theboundaries in their vicinity.

In these systems the patterns that are observed are selected by nonlinear termsin the equations: at the threshold of the instability the marginal mode generally hasa unique wavenumber but its orientation is arbitrary. Moreover, for slightly super-critical parameter values there is a range of wavenumbers centered on the criticalwavenumber that are all weakly unstable. The nonlinear terms come into play asthese modes grow and suppress some terms at the expense of others. For periodicpatterns this process can be studied rigorously using equivariant bifurcation theory.In this approach one selects a spatial lattice and identifies the critical wavevectors onthe corresponding reciprocal (or Bravais) lattice. This procedure determines the de-generacy of the bifurcation problem (i.e., the number of independent modes involvedin the bifurcation) as well as the action of the symmetries of the lattice (symmetry ofthe unit cell together with translations) on the amplitudes of the critical wavevectors.This action provides a representation of the symmetry group of the lattice. At thispoint one can employ group theory, in particular the equivariant branching lemma, toidentify the symmetries of solutions that must bifurcate from the homogeneous state.The beauty of this approach is that it is entirely equation-free and so applies to anyproblem on the same lattice. Of course, one typically desires additional information.For this it is necessary to construct the Hilbert basis of both invariant functions andequivariant vector fields. These are then used to construct the most general equationsdetermining the evolution of the critical modes subject to the restriction that theequations respect the symmetry of the lattice. This process is algorithmic and canbe performed to any desired order. In some cases, particularly for time-dependentpatterns created in a symmetry-breaking Hopf bifurcation, normal form transforma-tions are used to simplify the resulting equations further. These are performed at thecritical parameter value and may introduce additional “normal form symmetries” intothe problem; departures from the critical parameter value are then included througha process called unfolding. The resulting equations can be used in two ways. First,they can be used to compute the stability of the primary patterns established group-theoretically with respect to all perturbations on the lattice. The required eigenvaluecomputations can be dramatically simplified by taking advantage of the high degreeof symmetry that the primary solutions inevitably possess. Second, one can explic-itly solve the equations near the critical parameter value. In addition to the solutionsguaranteed group-theoretically, one often finds additional primary solutions with lowersymmetry. Once identified, the stability of these states can also be computed.

This approach, summarized by Golubitsky, Stewart, and Schaeffer in [5], providesa complete discussion of pattern selection on periodic lattices. A nontrivial applica-tion to Turing structures in three dimensions is described in [2]. The results, suchas the direction of branching or the stability calculations, depend on the arbitrarycoefficients in the equivariant vector field. For applications these coefficients mustbe computed from the governing PDEs in terms of the physical parameters of thesystem. These computations require that one performs explicit center manifold re-duction of the PDEs. These computations can sometimes be performed analyticallyalthough they are typically done using symbolic computation, or directly numerically.


BOOK REVIEWS 569

The abstract theory tells one exactly which quantities are required and which are not;moreover, the coefficients of the nonlinear terms can be computed at criticality sub-ject to nondegeneracy conditions identified in the abstract treatment. In some casesthe coefficients can be determined empirically by fitting the results of direct numericalsimulations of patterns with different imposed symmetries. The imposed symmetrymay render the pattern stable, and the computation then reveals the coefficient thatdetermines its direction of branching.

Of course, this approach leaves out a great deal since it is unable to include modesthat are nearly critical. Such modes are responsible for spatial modulation of pat-terns on scales large compared to the critical wavelength. In addition, the presence ofsuch modes triggers different types of instabilities that delimit the stability balloon ofeach periodic pattern. The two best-known instabilities of this type are the Eckhausinstability and the zigzag instability. Both are long wave instabilities that serve torestrict the wavenumber range within which a stripe pattern is stable. Often theseinstabilities trigger interesting dynamics. For example, the Eckhaus instability of astripe pattern whose wavelength is too short leads to the formation of a defect inthe pattern at which a pair of stripes is eliminated, thereby shifting its wavenumberinto the stability region. Defects of this type are associated with a singularity in thespatial phase of the patterns and are called topological since their presence can bedetected by computing the phase change along any closed contour surrounding thedefect. In general, defects move, and it is through their motion that the wavenumberof the pattern is adjusted or the pattern becomes disorganized. The study of thisaspect of pattern formation has been advanced mainly by applied mathematiciansand theoretical physicists, and has led to a number of canonical or prototypical en-velope or phase equations derived by formal asymptotics (complex Ginzburg–Landauequation, Kuramoto–Sivashinsky equation, and many others) or constructed as usefulmodels on the basis of intuition (Swift–Hohenberg equation and others). These haveprovided several generations of applied mathematicians and physicists with fascinat-ing examples of complex dynamics, and in turn have motivated pure mathematiciansto try to establish some of their basic properties (existence, well-posedness, estimatesof attractor dimension, etc.).

In fact, the connection between these two quite distinct approaches remains un-clear. The rigorous approach can establish the existence of spatially periodic patternsbut cannot explain all of their observed properties. For example, selection betweensquares and hexagonal patterns cannot be studied using equivariant bifurcation theorysince the two patterns live on different lattices. In contrast, the scaling requirementsof multiple scale approaches are sometimes too restrictive to capture all the behaviornear a bifurcation, which may require the retention of formally higher-order terms.And, of course, envelope equations cannot be constructed by the algorithmic approachthat works so well for spatially periodic structures.

Pattern formation has been the subject of several books in the recent past. Theseinclude Hoyle [6], Desai and Kapral [4], and Pismen [9], all dealing with pattern forma-tion in general, although with different emphases. The book under review, PatternFormation and Dynamics in Nonequilibrium Systems by Michael Cross and HenryGreenside, is the latest addition to this body of literature and is the most accessible.

Michael Cross is responsible for writing, with Pierre Hohenberg, an immenselyinfluential review article [3] that has already been cited 3,361 times. This article isa compendium of results about pattern formation and a very thorough survey of thesubject as of 1993. Despite its length, the article is dense, with insufficient detailprovided to make it a useful learning tool. Many of us, when we learned that Michael


570 BOOK REVIEWS

Cross was writing a book on pattern formation, hoped that the book would fill inthe gaps in the article, turning it into a classic monograph. However, the book underreview is not that kind of a book.

The authors approach pattern formation from a physics point of view. They takegreat effort, largely successful, to develop an intuitive understanding of the physicalprocesses involved in pattern formation. These involve the growth rate of an insta-bility, saturation of the instability by nonlinear terms, and the competition amongdegenerate modes. This topic takes up a third of the book and is written at an elemen-tary level. The authors employ several physical examples of pattern forming systems,focusing on Rayleigh–Benard convection, the Couette–Taylor system, and the Turinginstability. They explain the physics of the observed instabilities very well, and il-lustrate the phenomena with experimental observations and measurements from themany beautiful experiments by Guenter Ahlers, Eberhard Bodenschatz, Jerry Gollub,Harry Swinney, and their colleagues. However, the Navier–Stokes equation does notmake an appearance, as if to underline the authors’ view that universal properties ofpatterns can be understood without knowledge of the field equations. Instead, theauthors use the planar Swift–Hohenberg equation for all illustrative examples. This isnot an unreasonable approach for introducing the basic ideas, but gives a misleadingimpression of both the subject of pattern formation and the calculations that have tobe done, for example, to show that convection rolls bifurcate supercritically, in otherwords, to make predictions for real systems. In addition, the authors’ approach of“constructing” rather than deriving amplitude equations must be viewed with suspi-cion. In the simplest cases this approach does indeed give the correct answer, but Iwould argue that we know this because the problem has in fact already been donesystematically. In the field of envelope equations there are simply too many pitfallsto rely on a simplistic albeit pedagogically useful approach. For example, envelopeequations may turn out to be nonlocal, or there may be states one does not alreadyknow about and would like to predict.

The book is intended for graduate students “in biology, chemistry, engineering,mathematics, physics, and other fields.” In an attempt to satisfy such a wide rangeof potential readers the authors go easy on the math. Thus, on p. 63, where the(one-dimensional) Swift–Hohenberg equation

(1) ∂tu = ru − (1 + ∂2x)

2u − u3

is introduced as the first equation in the book, the derivatives are described as in-timidating. It takes the authors many more pages to determine the growth rate ofinfinitesimal perturbations of the homogeneous state u = 0 as a function of the pertur-bation wavenumber, even though students who have taken any undergraduate physicsor engineering class should be thoroughly familiar with the notion of a dispersion re-lation, certainly in the context of electromagnetic waves. But this connection is notmentioned. And on p. 132 we are told that AA∗ can be written as |A|2. So it doesnot come as a surprise that group theory is not discussed and that the discussionof bifurcation theory is really quite superficial. Global bifurcations or homoclinicorbits are not mentioned, despite their importance in studies of localized patternsand traveling pulses. Quasi patterns are introduced but the fundamental mathemat-ical issues illustrated, for example, by the recent work of Iooss and Rucklidge [7] onsmall divisors that would interest the readers of SIAM Review, are not mentioned.The multiple scales technique is only introduced in an appendix, and even then theFredholm alternative is described as a “mouthful” and illustrated on symmetric(!)


BOOK REVIEWS 571

matrices. Yet physics and engineering students should be familiar with Green’s iden-tity from an undergraduate class on electromagnetism, and the notion of an adjointoperator could surely be introduced via an appeal to this identity. And it does nothelp that the application of the multiple scales procedure to the Swift–Hohenbergequation in section A2.3.2 has numerous typographical errors (in equations (A2.42)–(A2.48)). Another potentially confusing error is on p. 371 where equations (10.36a)and (10.42) are in conflict.

The authors are reluctant to quote mathematical results, or even to provide ref-erences to original papers. Thus we are not told about Aronson and Weinberger’swork on front propagation in the nonlinear diffusion equation in one dimension. In-stead, Cross and Greenside prefer to present the (linear) stationary phase derivation ofthe critical velocity, attributing the “pinch-point analysis” to Lifshitz and Pittaevskii(sic)—I was always under the impression that this was developed in 1963 by A. Bersand R. J. Briggs. Incidentally, other names are also consistently misspelled, includingWesfreid, van der Pol, Bonhoeffer, Garfinkel, and even Snell, as in Snell’s law. In viewof the authors’ advocacy of Galerkin truncation I would also have liked to have seen aderivation of the Lorenz equations, used briefly in the text for calculational purposes,or at least a reference to E. Lorenz’s seminal paper to indicate why these equationsare in fact interesting.

I found Chapters 8–11 of greatest interest. These focus on “Defects and Fronts”(Ch. 8), “Patterns Far from Threshold” (Ch. 9), “Oscillatory Patterns” (Ch. 10), and“Excitable Media” (Ch. 11). There is also a useful chapter on “Numerical Methods”(Ch. 12), although numerical continuation that has proved so valuable in studies ofpattern formation is not discussed. Although these go over much the same materialas the other (more advanced) books on patterns cited above, the presentation here isclear and accessible, particularly in Chapter 8. A helpful glossary of technical termsis also included. But there are also missed opportunities, for example, to explainthat local bifurcations on the real line can create spatially localized structures andto discuss their remarkable properties away from onset. And yes, this too can all bedone within the Swift–Hohenberg equation [1, 8].

Pattern Formation and Dynamics in Nonequilibrium Systems will be valuable tostudents and researchers approaching the subject of pattern formation for the firsttime. It succeeds in conveying the richness and beauty of the subject and the basicphysical concepts behind pattern formation. A nice feature of the book is the Etudes,that is, worked problems, that are sprinkled throughout the book and designed toillustrate theoretical concepts. While these could have been pushed further, theyare an essential part of the book. This is also true of the exercises at the end ofeach chapter—in contrast to the body of the text, which is quite elementary, theseare decidedly not and the student who avoids the exercises will miss out on a veryvaluable learning experience. But overall I found the book too wordy and feel thata shorter work covering the same material would ultimately have worked out better(and probably cheaper).

REFERENCES

[1] J. Burke and E. Knobloch, Homoclinic snaking: Structure and stability, Chaos, 17 (2007),article 037102.

[2] T. K. Callahan and E. Knobloch, Symmetry-breaking bifurcations on cubic lattices, Nonlin-earity, 10 (1997), pp. 1179–1216.

[3] M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. ModernPhys., 65 (1993), pp. 851–1112.


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[4] R. C. Desai amd R. Kapral, Dynamics of Self-Organized and Self-Assembled Structures, Cam-bridge University Press, Cambridge, UK, 2009.

[5] M. Golubitsky, I. Stewart, and D. G. Schaeffer, Singularities and Groups in BifurcationTheory, Vol. II, Springer, New York, 1988.

[6] R. Hoyle, Pattern Formation, Cambridge University Press, Cambridge, UK, 2006.[7] G. Iooss and A. M. Rucklidge, On the existence of quasipattern solutions of the Swift-

Hohenberg equation, J. Nonlinear Sci., 20 (2010), pp. 361–394.[8] D. J. B. Lloyd, B. Sandstede, D. Avitabile, and A. R. Champneys, Localized hexagon

patterns of the planar Swift–Hohenberg equation, SIAM J. Appl. Dynam. Syst., 7 (2008),pp. 1049–1100.

[9] L. Pismen, Patterns and Interfaces in Dissipative Dynamics, Springer, Berlin, 2006.

EDGAR KNOBLOCH

University of California at Berkeley

Analytic Combinatorics. By Philippe Flajolet and Robert Sedgewick. Cambridge UniversityPress, Cambridge, UK, 2009. $81.00. xiv+810 pp., hardcover. ISBN 978-0-521-89806-5.

This book has been a long time in the making. Drafts have been circulating for over tenyears. Courses at many universities have been taught from the manuscript versions.For several years, this text has been the syllabus for one of the the qualifying examoptions for math graduate students at this reviewer’s institution. Among other things,this review will explain why so many of us have rushed to order a hardcover copy ofa book that has been, and still is, available for free at http://algo.inria.fr/flajolet/Publications/AnaCombi/book.pdf.

This is one of those books that marks the emergence of a subfield. The “FlajoletSchool” of combinatorics emerged in the 1980s and subsequent decades from a groupof researchers at the Institut National de Recherche en Informatique et en Automa-tique (INRIA) in Rocquencourt, south of Paris. One of their principal aims was themathematical analysis of algorithms. Most of these algorithms were already in use,sans rigorous analysis, by computer scientists. While the bulk of this group’s researchis published in mathematics journals, it is the intimate connection with computerscience that has given this research purpose and depth.

As with any field, the roots go back farther than one might imagine. For example,a result of Hipparchus of Rhodes (190–120 BC, published in the first century AD byPlutarch) is an enumeration of certain propositional clauses. Imagine having to derivethe numbers of positive and negative compound statements on ten Boolean variables(103,049 and 310,952, respectively) with the Roman numeration system and no zero!Inclusion of this tidbit (see page 69) and others is one of the delightful features ofAnalytic Combinatorics.

Part A of the book comprises three chapters, which give an overview of modernenumerative combinatorics. A central theme is the connection between recursive con-structions of combinatorial families and the derivation of generating functions thatcount these families. The modern viewpoint stresses the automatic nature of thisconnection, with notations for recursive combinatorial operations such as SET andSEQ which practically build the generating function as they describe the combina-torial class. Historically, the use of generating functions dates back to Euler, butthe ancestry of the modern view probably begins with Polya [Pol37], who linkedthe recursions satisfied by a combinatorial class to analytic properties of its generat-ing function. The formal recursion operators emerged (under different names) fromSchutzenberger’s work on formal languages [CS63]. If one is interested only in combi-natorial enumeration, many texts already exist, ranging from the encyclopedic work

http://algo.inria.fr/flajolet/Publications/AnaCombi/book.pdf

http://algo.inria.fr/flajolet/Publications/AnaCombi/book.pdf


BOOK REVIEWS 573

of Goulden and Jackson [GJ83] to the two volumes of Stanley [Sta97, Sta99] thatare teeming with problems and new directions. Still, viewed on its own, Part A ofAnalytic Combinatorics is a significant contribution. The segments that touch onprobability theory are reminiscent of Feller’s seminal work [Fel68]. For example, theygive a combinatorial analysis of the classical birthday problem and the coupon col-lector’s problem, not found in most other combinatorics texts and differing from theprobabilistic analysis found in probability texts. Probably the greatest intellectualdebt of Part A is to Donald Knuth, but it goes without saying that any considerationof combinatorial structures in theoretical computer science begins in the shadow castby [Knu06].

The heart of the book is Part B. This concerns the use of complex contour integralsto approximate coefficients of a generating function. The point is that, using themethods of Part A, one may obtain a generating function

f(z) :=∞∑

n=0

anzn,

whose coefficients an count the objects of size n in a given combinatorial class. Arecursive description of the class leads to a recursive but not an explicit understandingof {an}. Suppose now that f has a positive radius of convergence, representing ananalytic function in some (complex) neighborhood of zero. Cauchy’s integral formulatells us how to recover an from f :

(1) an =12πi

∮z−n−1 f(z) dz,

where the integral is over any contour that encircles the origin and stays within thedomain of analyticity of f . This identity is exact. Unless f is a very special function(say, a rational function or (1−x)α), however, there is usually no way to evaluate (1)exactly. However, if f has a finite radius of convergence, R, it follows immediatelyfrom (1) that lim supn→∞ |an|1/n = 1/R. Deriving more precise estimates is a scienceknown as asymptotic analysis, for which an excellent introduction may be found in[dB81]. In the present case, better asymptotics may be derived once the behaviorof f near the dominant singularity—the singularity of least modulus—is understood.A range of analytic techniques is employed, which may be found in classic appliedcomplex analysis texts such as [Hen91]. The use of these techniques to convert infor-mation about f near its singularities to asymptotic information about {an} is knownas singularity analysis and is the mainstay of analytic combinatorics.

The use of complex analytic methods to extract asymptotics from a generat-ing function is not a new trick. Hardy and Ramanujan [HR17] used this methodnearly a century ago to convert the generating function for partitions, which is aninfinite product of reciprocals of binomials, to a rather intricate asymptotic estimateeπ√

2n/3/(4√3n). The methodical use of contour integrals to extract coefficients of

generating functions dates back at least to [Hay56], who proved a general theorem onapplicability of the saddle point method for a class of functions he termed admissible,which includes most entire functions. When f has singularities, different deformationssuch as Henkel contours are required. On Flajolet’s webpage, the division betweenthe history of analytic combinatorics and its prehistory occurs in 1990, when [FO90]was published. This paper synthesizes known complex variable techniques to give anautomatic conversion between asymptotic behaviors of f near its dominant singularityand asymptotics of the coefficients {an}.


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Chapter IV is the workhorse chapter in which the basic theory underlying sin-gularity analysis is developed. If you need to know what singularity analysis is allabout and can read only one chapter, read this one. As it claims in the introduction,this book is indeed user-friendly. There are plenty of examples. My favorite examplein Chapter IV is Polya’s determination of the number an of chemical isomers of thealcohol CnH2n+1OH without asymmetric carbon atoms. The generating function isderived on pages 283–284, and turns out to be implicitly described as the solutionto a functional equation; strictly speaking, the derivation is Part A material, but itworks here. Next, it is shown how to turn the functional equation into an expansion.Finally, meromorphic singularity analysis is used to derive asymptotics of an.

Chapter V is a compendium of examples. It includes large classes such as lan-guages, path counting, and the transfer matrix method, as well as some strikingisolated cases such as compositions into primes. Chapter VI contains the body of re-sults that together can be called singularity analysis. Here various cases are presentedthat have been worked out over the past decades. These include the Flajolet–Odlyzkotransfer theorems, Darboux’ method, and some Tauberian theory, as well as the me-thodical application of singularity analysis to compositions, inverse functions, and soforth. Chapter VII contains another wealth of applications. One section is devoted toalgebraic generating functions. These arise often in recursively generated structures,the most well known being the Catalan numbers, which count binary trees as well asa host of other things. Toward the end of the chapter, the topic is broached of gen-erating functions that solve linear ODEs over the rational functions. These so-calledholonomic functions are usually not representable in terms of elementary functions,but for ODEs that are regular in the sense of Frobenius, one may find the singularpoints and obtain asymptotics of the solutions near these points by applying symbolicalgebraic methods to the ODE.

The last chapter of Part B, Chapter VIII, is on the saddle point method. AsFlajolet and Sedgewick say,

Saddle point method = Choice of contour + Laplace’s method.

Saddle point techniques deserve their own chapter because their scope is huge. Itmay be argued that saddle point techniques underlie the more specialized integrals intransfer theorems. Understanding saddle point integrals requires some sophisticationin complex analysis, and because the book is intended for a readership that doesnot specialize in this, Appendix B gives a 40-page summary of the salient theory.Their use in evaluating the Cauchy integral (1) is nicely introduced by an example(page 555) in which Stirling’s formula is derived. Hayman’s results from the 1950sare reviewed, along with later extensions such as [HS68]. The topic of analytic de-Poissonization, popularized in [JS98], is largely beyond the scope of the book, but itis introduced and a basic result is proved (pages 572–573). Another worked exampleis the leading term in the Hardy–Ramanujan result on integer partitions. The saddlepoint method is particularly easy to apply when the integrand is equal or nearlyequal to a large power of a known function. This leads to distributional limit lawsfor many combinatorial classes. The case of large powers and the application toprobability limit laws are developed, respectively, in the eighth and ninth sections ofChapter VIII. The last section of Chapter VIII is devoted to the trickiest class ofintegrals in the book, where the contour must pass between two coalescing saddles.The resulting limit behavior involves Airy functions. A well-known application of thismethod is to planar maps, which are a class of planar graphs. I find it surprisingthat analytic combinatorial methods are able to get at a problem that is inherently


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geometric, and inspiring that the answer involves a distributional limit in the shapeof an Airy function. Only the most basic computations are done in the book, withthe reader being referred to [BBMD+02] for more detailed analysis.

Part C of the book is a single chapter on multivariate asymptotics. Here, thecombinatorial classes being counted are indexed by a vector r = (r1, . . . , rd) of integersinstead of a single integer, and the generating functions are d-variate power series inthe variables z = (z1, . . . , zd):

F (z) =∑r

arzr =∑r

arzr11 · · · zrd

d .

The chapter is largely concerned with the bivariate case (d = 2). There are two knownapproaches to multivariate asymptotics. One approach is to replace the univariateCauchy integral (1) by the corresponding multivariate formula

ar =(

12πi

)d ∫T

F (z)dz1

zr1+11

· · · dzd

zrd+1d

.

The second approach is to consider a d-variate generating function as a sequence of(d − 1)-variate functions. This approach is most natural when one of the parame-ters is a size parameter: then F (z) =

∑∞n=0 Fn(z1, . . . , zd−1)zn

d represents a naturaldecomposition in which the generating function Fn describes the ensemble of size n.The natural outcome of this viewpoint is limit theorems, describing the behavior ofthe nth ensemble, suitably rescaled, as n → ∞.

The first of these two approaches is quite powerful but requires more sophisticatedmultidimensional complex analytic techniques. These include multivariate residues,deformation of multivariate contours via Morse theory, and some complex algebraicgeometry that is needed to examine singularities in the multivariate case. Inclusionof these would expand the size of Appendix B (complex analysis background) by afactor of perhaps six. The authors judiciously stop short of this and refer the readerinterested in this approach to the work of this reviewer (see, e.g., [PW08, Pem10]).

The second approach was pioneered in the 1970s and 1980s, beginning with Bender[Ben73]. The idea is that Fn will be a quasi power, meaning that it is well approxi-mated by cnh(z)g(z)n for some functions g and h and constants cn. The second ideais that coefficients of quasi powers obey a local central limit theorem. This theorem,stated as Theorem IX.8, was proved by [BR83] and later extended by others such as R.Canfield, Z. Gao, and H. K. Hwang to wider classes of functions. In a chapter of over100 pages, Flajolet and Sedgewick take the reader through the development of thismathematics, amply illustrated by examples concerning random walks, trees, strings,and even a run time analysis of the Euclidean algorithm. The first several sectionsdo a good job, explaining the analytic underpinnings, namely, a perturbation analysisof F (z, u) as u increases to 1. The last several sections go outside the central limitregime to large deviations, a final word on the Airy phenomenon, and the case d ≥ 3.

A review is supposed to be balanced. For the most part, I have nothing but praisefor this book. Perhaps I should mention that there are typos, such as an incorrectconstant in Proposition IX.7 (missing some factors of ρ); however, there is a well-maintained online errata list at http://algo.inria.fr/flajolet/Publications/AnaCombi/errata.pdf.

The book is somewhat daunting, precisely because it is comprehensive, containingmost known examples of the method. If this and a very reasonable $81.00 price do notalarm you, and you have bothered to read this review, then Analytic Combinatoricsshould probably join your collection.

http://algo.inria.fr/flajolet/Publications/AnaCombi/errata.pdf

http://algo.inria.fr/flajolet/Publications/AnaCombi/errata.pdf


576 BOOK REVIEWS

REFERENCES

[BBMD+02] C. Banderier, M. Bousquet-Melou, A. Denise, P. Flajolet, D. Gardy, and

D. Gouyou-Beauchamps, Generating functions for generating trees, DiscreteMath., 246 (2002), pp. 29–55.

[Ben73] E. A. Bender, Central and local limit theorems applied to asymptotic enumeration, J.Combin. Theory Ser. A, 15 (1973), pp. 91–111.

[BR83] E. A. Bender and L. B. Richmond, Central and local limit theorems applied to asymp-totic enumeration, II: Multivariate generating functions, J. Combin. Theory Ser.A, 34 (1983), pp. 255–265.

[CS63] N. Chomsky and M. Schutzenberger, The Algebraic Theory of Context-Free Lan-guages, North–Holland, Amsterdam, The Netherlands, 1963.

[dB81] N. de Bruijn, Asymptotic Methods in Analysis, 3rd ed., Dover, New York, 1981.[Fel68] W. Feller, An Introduction to Probability Theory and its Applications, Vol. I, 3rd

ed., John Wiley & Sons, New York, 1968.[FO90] P. Flajolet and A. Odlyzko, Singularity analysis of generating functions, SIAM J.

Discrete Math., 3 (1990), pp. 216–240.[GJ83] I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley & Sons,

New York, 1983.[Hay56] W. Hayman, A generalisation of Stirling’s formula, J. Reine Angew. Math., 196 (1956),

pp. 67–95.[Hen91] P. Henrici, Applied and Computational Complex Analysis. Vol. 2: Special Functions—

Integral Transforms—Asymptotics—Continued Fractions, John Wiley & Sons,New York, 1991.

[HR17] G. H. Hardy and S. Ramanujan, Asymptotic formulae for the distribution of integersof various types, Proc. London Math. Soc., 16 (1917), pp. 112–132.

[HS68] B. Harris and L. Schoenfeld, Asymptotic expansions for the coefficients of analyticfunctions, Illinois J. Math., 12 (1968), pp. 264–277.

[JS98] P. Jacquet and W. Szpankowski, Analytical de-Poissonization and its applications,Theoret. Comput. Sci., 201 (1998), pp. 1–62.

[Knu06] D. Knuth, The Art of Computer Programming, Vols. I–IV, Addison–Wesley, UpperSaddle River, NJ, 2006.

[Pem10] R. Pemantle, Analytic combinatorics in d variables: An overview, in Proceedings ofthe AMS Special Sessions on Algorithmic Probability and Combinatorics, AMS,Providence, RI, 2010, pp. 195–220.

[Pol37] G. Polya, Kombinatorische Anzahlbestimmungen fur Gruppen, Graphen und chemis-che Verbindungen, Acta Math., 68 (1937), pp. 145–254.

[PW08] R. Pemantle and M. C. Wilson, Twenty combinatorial examples of asymptoticsderived from multivariate generating functions, SIAM Rev., 50 (2008), pp. 199–272.

[Sta97] R. P. Stanley, Enumerative Combinatorics. Vol. 1, Cambridge University Press, Cam-bridge, UK, 1997.

[Sta99] R. P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge University Press, Cam-bridge, UK, 1999.

ROBIN PEMANTLE

University of Pennsylvania

Numerical Methods for Roots of Polyno-mials. Part I. By J. M. McNamee. Elsevier,Amsterdam, 2007. $170.00. xx+333 pp., hard-cover. ISBN 978-0-444-52729-5.

The book under review constitutes the firstpart of two volumes describing methods forfinding roots of polynomials. Besides an in-troduction it contains six chapters (“Eval-

uation, Convergence, Bounds”; “Sturm Se-quences and Greatest Common Divisors”;“Real Roots by Continued Fractions”; “Si-multaneous Methods”; “Newton’s and Re-lated Methods”; “Matrix Methods”).Unfortunately, the presentation of the

material is not in the shape one might ex-pect. Even the first formulation in the in-troduction is very unusual: “A polynomial


BOOK REVIEWS 577

is an expression of the form

p(x) = cnxn + · · ·+ c0.”

Although mathematical strength is usuallynot necessary in the introduction of a book,it would be more precise to define p as afunction of x, stating first from which setthe coefficients c0, c1, . . . , cn are taken andwhat the domain of p is. Throughout thisbook no period is printed after a formula atthe end of a sentence. This is very unusual.What is even worse is the fact that formu-las are numbered as usual within parenthe-ses, e.g., (10); however, they are referredto without parentheses, e.g., 10. Further-more, it is very annoying that in formulasand in the text variables are denoted usingdifferent typefaces. Also, the arrangementof the formulas is very often not carefullydone; see, for example, page xiv, equations(4) and (5). Repeatedly, the notation ischanged without warning. For example, onpage xvi, in formula (17) the letter i de-notes an iteration index. In formula (19),following only a couple of lines later, theletter i can take on only the integers be-tween 1 and n and denotes the ith zero ofan nth-degree polynomial. Also repeatedly,notations are used without any definitions.These and several other little flaws meanthat the book is not a pleasure to read.On the other hand, it contains a collection

of material which cannot be found in anyother book. In the introduction, the authorsays that he has compiled a bibliographycontaining over 8000 entries. Reading thebook, one gets the impression that the au-thor is more or less evaluating this bibliog-raphy. Many methods and ideas are merelymentioned without explanation. Also, thebehavior of many methods is only reportedfrom the original literature. In rare casesproofs are performed or indicated. There-fore, it is not surprising that occasionallyresults are not cited correctly or are in-complete. Take, for example, the relation(5.228), which only holds under additionalassumptions on the derivative of the under-lying function (which always hold for poly-nomials). As already mentioned, the bookcontains a lot of material. Therefore, every-one interested in computing approximationsto the zeros of polynomials could use it toget an overview of existing methods. How-

ever, if one really needs an algorithm forcomputing zeros of polynomials (which isin my experience very seldom the case), itis not easy to find the appropriate method.Therefore, I look forward to what materialis treated in Part II, only a few examples ofwhich are mentioned in the introduction ofPart I.

GOTZ ALEFELD

Karlsruhe Institute of TechnologyGermany

Pearls of Discrete Mathematics. By MartinErickson. CRC Press, Boca Raton, FL, 2009.$59.95. x+270 pp., softcover. ISBN 978-1-4398-1616-5.

If you were first attracted to mathemat-ics by puzzles, games, elementary numbertheory, geometry, or basic graph theory (in-stead of, say, second-order semilinear par-tial differential equations), then this bookwill remind you why you think mathematicsis fun. The twenty-four chapters are dividedevenly between the eight parts. Each is rel-atively short and relatively disjoint fromthe others. No attempt is made to be sys-tematic, and, in some cases, no attempt ismade to be complete or rigorous. This isnot criticism; it is the nature of the collec-tion. Hit a new topic quickly, do somethinginteresting, and move on. While suggestedas appropriate for a topics course, I thinkthe real strengths of this text would be asself-study for the motivated student (whocan consult more careful references) or asource of interesting examples for a moretraditional course in various areas of dis-crete mathematics. It might also serve asan enticing expose of the many facets ofmathematics captured under the umbrellaterm “discrete mathematics.”Each chapter begins with one or two lead-

ing questions or startling results, which areeventually addressed in the chapter. For ex-ample, the number of ways to make changefor a million dollars is

88,265,881,340,710,786,348,934,950,201,

250,975,072,332,541,120,001

by using standard U.S. currency from apenny up to a $100 note. Or, in any group


578 BOOK REVIEWS

of six people there is either a group ofthree, all of whom knows each other, or agroup of three, none of whom knows eachother. The experienced discrete mathe-matician will recognize these as teasers forgenerating functions and Ramsey theory,respectively. As mentioned, some of thechapters are shorter than the seven-pageaverage. For example, random tournamentstakes three pages to explain and includesone page of questions. The first chapteris one page of narrative and one equallylong page of exercises. So some topics arepresented without all the details, given thespace allowed, or a chapter may be longerbut still contain many more new ideas thanthe space allows for careful explanation.For example, Chapter 10 covers discreteprobability spaces, introducing randomvariables, expectation, variance, derange-ments, inclusion-exclusion, and a limit lawin eleven pages. The next section includesa one-page description of the eigenvaluesof a matrix and their role in diagonalizingthe matrix, though there is no discussionof when this might be possible for thetransition matrix of a Markov chain. Animprovement to the book would be a clearerindication for the student of where to go tolearn more about a topic, either to fill indetails or to continue their study.A small selection of the topics addressed,

one from each of the eight parts, will givean indication of the range: Pascal’s trian-gle, Fibonacci numbers, rook paths, Markovchains, partitions of integers, coding theory,higher-dimensional tic-tac-toe, and listingpermutations in lexicographic order. Eachchapter contains roughly ten to fifteen ex-ercises, with some marked as “particularlydifficult,” or requiring a computing device,or “of theoretical importance.” These arecollectively another strength of the book,especially given the hints and/or solutionsfor every problem, which together comprise58 of the 270 pages.Motivated students will learn much new

mathematics if they work through this bookcarefully. For a student curious about ex-actly which topics make up the field of“discrete mathematics,” a more cursory tripthrough the book will do a good job of an-swering the question. For the library witha collection in recreational mathematics,this book will serve as a nice bridge to the

“more serious” associated areas of math-ematics. And finally, for the professionalmathematician, using this book as bedtimereading just might remind you of why youfound math fun in the first place.

ROBERT A. BEEZER

University of Puget Sound

Mathematical Optimization and Eco-nomic Analysis. By Mikulas Luptacik. Springer,New York, 2010. $99.00. xiv+294 pp., hard-cover. ISBN 978-0-387-89551-2.

Some other books on mathematical opti-mization (or programming) and its applica-tions present the mathematical theory andcomputational methods in detail, but in-clude only fairly simple illustrative applica-tions. In contrast, Luptacik’s book, which isreviewed here, places the emphasis on theeconomic applications. The mathematicaltheory is presented, though in less detailthan a mathematician might desire, andwith more attention paid to the rationalethan the proofs. This reviewer, being amathematician rather than an economist,cannot judge the depth of the economic pre-sentation, but it appears clearly presentedand relates to a number of significant eco-nomic questions.By comparison, an older book by M. In-

triligator (Mathematical Optimization andEconomic Theory, Prentice-Hall, 1971;reprinted by SIAM in 2002) presents themathematics in more detail, including othertopics such as dynamic programming. Eco-nomics topics, such as theory of the firm,are analyzed with more emphasis on themathematics.Most mathematical optimization in eco-

nomics is built on linear programming. Thisfirst became practicable in the 1950s withDantzig’s simplex method (although lineareconomic models were known much earlier).The calculations were first done by hand,for small models, and then for much big-ger models when computers became avail-able. (The word “computer” had previouslymeant a human being.) It is important toremember that systems in economics, ormanagement planning, are not automati-cally linear. A cost, or profit, function isonly linear over a limited range, outside


BOOK REVIEWS 579

of which the slope changes, due to suchthings as setup costs and the need for moremachines when some capacity is exceeded.Moreover, not all cost components are rele-vant to optimization. For example, a wagescomponent of cost may be fixed by a unioncontract, so may not change when produc-tion level changes. So there is then no suchthing as “the cost,” as commonly supposedin textbooks. These matters are importantto an operations researcher, though pos-sibly less so to an economic theorist whois looking at the “big picture,” but theyare relevant to how far a piece of eco-nomic theory can describe a real-world sit-uation.The book under review discusses a long

list of economic models that may be opti-mized. Included are questions of compar-ative advantage (for international trade),the Giffen paradox (where an optimum canbe quite unintuitive), a portfolio selectionmodel, input-output models (leading to theLeotief pollution model), a model for en-vironmental control (this one is nonlinear),the behavior of a firm under regulatory con-straint, models for welfare economics, andthe optimal behavior of a monopolist.The Kuhn–Tucker conditions are pre-

sented with some economic examples,though without detailing constraint qual-ifications. This is followed by a discussionof convex programming. (Of course, a lin-ear program is also convex.) One shouldnote here that if a minimization problem isconvex, then any local minimum is also aglobal minimum, and there are various effi-cient ways to compute it. This extends toproblems satisfying some suitable “general-ized convex” property. But problems thatare far from convex are less tractable; theycommonly have several local minima, andcomputation of a global minimum is muchmore difficult. The emphasis on linear, orconvex, problems in the book under reviewthus directs attention to a class of problemswith certain tractable properties.A convex program, and in particular a

linear program, has an associated “dualprogram,” with close relations between itand the given “primal” program. The op-timal dual prices are interpreted as shadowprices and also as equilibrium prices, andslack variables and complementary slack-ness have economic interpretations. In lin-

ear programming, the change in profit re-sulting from a small change in a requirementis measured by the corresponding shadowprice. But, in fact, the shadow price isthe initial slope of a piecewise-linear curve,whose slope may have many jumps, andmore computation is needed to find whathappens for larger changes. This aspectdoes not seem to be discussed in manytextbooks (including this one).The simplex method for linear program-

ming is described, with its various stepsgiven economic, rather than mathematical,interpretations.A chapter is devoted to data envelopment

analysis. This is concerned with efficiencyof production, considered as a ratio of out-put to input. When there are several inputsand several outputs, this efficiency is mod-eled as the ratio of a linear expression foroutput to a linear expression for input, withweights to be optimized. This “linear frac-tional program” is transformed to an equiv-alent linear program. The chapter gives aclear and detailed presentation, includingexamples with numerical data. There areseveral kinds of efficiency, including “ecoef-ficiency.”Geometric programming is a kind of opti-

mization technique, where the functions are“posynomials,” sums of products of pow-ers of positive variables, with positive co-efficients. There is some association withCobb–Douglas production functions in eco-nomics. A logarithmic transformation ofthe variables turns this nonconvex programinto a convex program to which duality the-ory may be applied. In various instances,the dual problem allows an optimum tobe simply calculated. Various problems ineconomics fit this pattern. These includean open input-output model with continu-ous substitution between primary factors.A multiobjective version of geometric pro-gramming is also given.It is very common for an optimization

problem to have several objectives whoserequirements conflict. At a “Pareto mini-mum” (or maximum), one cannot improveany objective without making another ob-jective worse. At such points, Kuhn–Tuckerconditions are satisfied. But there are manyPareto minima, and how should one bechosen among them? Often, the several ob-jectives are combined into a single utility


580 BOOK REVIEWS

function, but many choices are possible.Often, a decision maker is given informa-tion on the results of several choices andasked to express preferences. An exampleof the Zionts–Wallenius iterative method forsuch calculations is presented. An alterna-tive approach, which is not discussed, wouldbe to optimize one main objective, subjectto additional constraints that the other ob-jectives must not be allowed to be too bad.Cited economic examples of multiobjec-

tive problems include models for welfareeconomics, monetary policy, behavior of amonopolist, and Leontief’s pollution model.However, some of the economic ideas mighthave been explained in more detail. Dual-ity for multiobjective programming is ap-proached by considering a weighted sum ofthe several objectives. The weights arise asmultipliers in the Kuhn–Tucker conditions,and may be considered as parametrizing thedifferent Pareto optimal points. For multi-objective linear programming, a dual prob-lem is presented, but the details are noteasy to follow. The economic meaning ofthe dual variables, considered as shadowprices, is discussed with a numerical ex-ample, but why is the dual problem itselfsignificant here? A vector analog of “weakduality” is not given.This book gives a clear and careful ac-

count of optimization theory as applied toa substantial collection of economic mod-els. The multiobjective section is, however,more difficult, and would have benefittedfrom some more detail. The author is notconcerned with the sort of detailed planningthat an operations researcher might need todo, nor is he greatly concerned with comput-ing algorithms or programs. The detailedcriticisms made in this review relate mostlyto aspects where the current theory doesnot give an adequate description, and somecritical comment might have been useful.The author intends the book for quanti-

tative economists (from student to profes-sional levels), and this is very appropriate.Operations researchers might also read thebook to discover how their discipline relatesto ideas in economics.

BRUCE D. CRAVEN

University of MelbourneAustralia

Variational Methods in Imaging. By O.Scherzer, M. Grasmair, H. Grossauer, M. Halt-meier, and F. Lenzen. Springer, New York, 2009.$79.95. xiv+320 pp., hardcover. ISBN 978-0-387-30931-6.

The following tasks in imaging are consid-ered as prototype models for the methodsdiscussed in this book:

• Denoising, where spurious noise is tobe removed with the aim of making theimage look nicer or, more importantly,as a preprocessing step for image anal-ysis and feature extraction.

• Removing noise in observations of thesky with ground-based telescopes by“chopping and nodding,” i.e., tiltingthe secondary mirror and the wholetelescope to remove background andresidual noise, respectively.

• Image inpainting, i.e., filling in missingor occluded data.

• X-ray and photoacoustic tomography.

• Schlieren tomography, the reconstruc-tion of pressure waves in a fluid fromdiffracted light.

Mathematical models for these problemsare carefully derived.Next, noise models typical for imaging

problems are discussed: additive, multi-plicative, Poisson noise, and sampling er-rors. For discrete images, MAP (maximuma posteriori) estimates based on Bayes’ for-mula are discussed, where different priorsare used for the three different types of im-ages used throughout the book: a naturalimage of mountains (close to Innsbruck), anartificial image of playing cards, and a med-ical ultrasound image from one of the au-thor’s industrial cooperations. These MAPestimators result in minimizing functionalslike in Tikhonov regularization as used insolving inverse problems and, in the caseof sampling error, involve a nonlinear func-tional containing a discrete gradient (thatreappears later in the book in continuousform as nonconvex BV-regularization).This is the starting point for consider-

ing imaging problems in the framework ofregularization theory for inverse problems;there, the regularization functional to be


BOOK REVIEWS 581

minimized contains two terms, one mod-eling the closeness to the data, the otherone enforcing stability by penalizing, e.g.,oscillations. The form of the second terminvolves a priori assumptions, e.g., aboutsmoothness, for the unknown solution. Thesame is the case when using such a func-tional in imaging, where the penalty termdepends on the type of image. One of theearliest such methods used in imaging wasthe TV-regularization method by Rudin,Osher, and Fatemi, which uses the L1-normof the gradient (not, as is usual in inverseproblems, the L2-norm) in the penalty termto avoid the smearing out of edges. Thisshows that, from the outset, a Hilbert spacetheory for regularization methods, as wasthe usual standard in inverse problems untilsome time ago, is insufficient for variationalmethods in imaging.Hence, the core of the book starts by

developing a variational theory of regular-ization in a quite general setting: nonlinearand in Banach (and more general) spaces.In this framework, the usual questions ofexistence of minimizers, stability, and con-vergence (rates) are considered, as well asappropriate choices for regularization pa-rameters. In addition to classical terms like(semi)norms, the Bregman distance appearsin the penalty term. The Bregman distance(based on a given convex function) betweentwo points can be interpreted as the differ-ence in ordinate between the convex func-tion and the tangent, drawn in the firstpoint, measured at the second point. Itplays a key role in variational and iterativemethods in imaging, as shown in this book.Some results about regularization are also

of interest to the inverse problems commu-nity, e.g., concerning convergence rates forsparsity-enforcing regularization.Different concrete methods based on

these considerations are applied to the threeprototype images mentioned above (with-out concrete reference to numerical and im-plementation issues).The next two chapters concern con-

vex and nonconvex regularization meth-ods; minimizers are characterized (and,hence, alternative versions of methods de-rived) via tools from convex analysis. Asspecial cases, classical methods like TV-regularization (and its connection to the

taut string algorithm), Mumford–Shah reg-ularization, and cartoon-texture-noise de-composition are discussed.The final chapters contain scale space and

inverse scale space methods, a frameworkfor diffusion filtering, backwards diffusion,and Bregman iteration and its continuousversion. One step of Bregman iteration canbe seen as TV denoising, enhancement ofthe filtered data, and adding back the tex-ture.The mathematical prerequisites for this

book are quite high. But since all resultsfrom functional analysis, convex analysis,calculus of variations, and measure and in-tegration theory used are collected and ref-erenced in an extensive appendix, the bookcan serve as a quite self-contained reference.It can certainly be recommended as a textfor a graduate course or seminar if the aim isto describe the mathematical theory behindimaging algorithms to an audience which al-ready has some familiarity with imaging ona more elementary level. Also, the bookshould be of great interest to the inverseproblems community.The book is dedicated to Zuhair Nashed

on the occasion of his 70th birthday, whocollaborated with two of the authors of thisbook (and also with the reviewer, who wantsto join them in congratulating Zuhair on hisbirthday and thanking him for influencingthree generations of Austrian mathemati-cians in the early stages of their careers).

HEINZ W. ENGL

Austrian Academy of Sciences

Real Analysis and Applications. Theoryin Practice. By Kenneth R. Davidson and AllanP. Donsig. Springer, New York, 2010. $79.95.xii+513 pp., hardcover. ISBN 978-0-387-98097-3.

Introduction. The ways in which mathe-matics departments structure and presenttheir undergraduate analysis courses areboth important and varied. Here are a fewvariations: the level of the course, the de-gree of emphasis on proofs, the balance be-tween pure and applied, and the selectionof topics.I believe that the book under review

strikes a good balance on all of these judg-


582 BOOK REVIEWS

ment calls, but it does assume a class ofmotivated students.This book by Davidson and Donsig is

subdivided by first covering the fundamen-tals, and then later the details. The firstchapters cover fundamentals every studentmust master in order to move onto moreadvanced topics. Then the tools are pre-sented, tools that serve a variety of demandsfrom applications and from the areas servedby mathematics, physics, engineering, bi-ology, business, economics and finance, tomention a few.Each year, the books we typically use

in mathematical analysis reflect these com-peting demands. Within the number ofsemesters available, every department andevery instructor must make choices. Thebook by Davidson and Donsig offers an ex-cellent selection, and it strikes a sound bal-ance. It builds on an earlier more condensedversion by the same authors.Most importantly, this new book has a

wider appeal, successfully reaching under-graduates in diverse mathematics depart-ments and allowing for classroom use inboth long and short courses. While it willwork better with ambitious students, itcould be used also when less emphasis isplaced on proofs.

Comparison with Other Books: Rudin[1] and Strichartz [2]. As a reasonablecomparison two books come to mind: [1],a classic, and Strichartz’s [2], of more re-cent vintage. What perhaps sets Davidsonand Donsig apart is that it may be easierfor beginning students, and in addition itcovers a few specialized topics not seen inearlier books at the same level. Also, theexercises are many and excellent, coveringthe range from easy to difficult, with lotsin between.

Level and Students. Davidson and Don-sig’s book is well suited for motivated under-graduates, mainly math majors. The bookmay also be helpful for beginning graduatestudents in math and neighboring areas.As with comparable books at this level,

for instance, [1] and [2], this new book willwork best for motivated students who wantto master proofs. However, I expect thatDavidson and Donsig will help to encourage

weaker students as well. In any case, it hasa wider appeal in its choice of a host of newtopics: wavelets, iterated function systems,and a selection of partial differential equa-tions.

Emphasis on Proofs and Completeness.The authors have adopted the theorem-proof model, and that is in the natureof mathematics. They also include pas-sages for motivation, though, as mentionedabove, students must have some level ofperseverance in order to make headway.

The Exercises. The nature of the exercisesis a real strength of this book; they areexcellent choices, and really helpful in thelearning process.

Traditional Subjects. While the book cov-ers such traditional subjects as the realnumber system, integration and differentia-tion, function spaces, and Fourier methods,it includes new topics not previously cov-ered in comparable undergraduate texts.

New Subjects. Wavelets, differential equa-tions (including some partial differentialequations), fractals, and iterated functionsystems.

Approach to Applications. The authorsinclude a nice selection of applications and,more importantly, they are well integratedwith the more theoretical parts of the book.

REFERENCES

[1] W. Rudin, Principles of MathematicalAnalysis, 2nd ed., McGraw-Hill, NewYork, 1964.

[2] R. S. Strichartz, The Way of Analysis,Jones and Bartlett, Boston, 2000.

PALLE JORGENSEN

The University of Iowa

Maple and Mathematica: A Problem Solv-ing Approach for Mathematics. SecondEdition. By Inna Shingareva and Carlos Lizarraga-Celaya. Springer, New York, 2009. $69.95. xviii+483 pp., softcover. ISBN 978-3-211-99431-3.


BOOK REVIEWS 583

I believe that the material in this bookbelongs in electronic form only: I doubtwhether anyone would actually want toread the print version. Having said that,the content on the CD may be useful to afew people, and CD or Web publication ofthe material—without the book—may beappropriate.John D. Carter, in a review dated March

2008 of the first edition (fac-staff.seattleu.edu/carterj1/web/papers/Shingareva.pdf),began as follows:

The book under review treats thetwo computer algebra systems (CAS)Maple and Mathematica. The purposeof the book is not to establish thateither Maple or Mathematica is “bet-ter” than the other, but is to simplypresent and describe a wide variety ofsample Maple and Mathematica codesside-by-side.

Carter states, and I agree, that “the bookdoes not provide a good introduction to ei-ther Maple or Mathematica for first-timeCAS users.” Carter continues to write thatthere may be some who will find it a useful“introduction to Maple for Mathematicausers and vice-versa,” and with the sec-ond edition providing a CD with the codes,this is truer of it than of the first edition.However, I’m less sure of the value of thematerial even for this purpose. Such read-ers would find equally useful many of thenumerous other books, including references[1] and [27] of the second edition, whichprovide parallel accounts in each of Mapleand of Mathematica.Perhaps it is worth considering the range

of situations in which both CAS togethermight be of interest.

• There is interest in comparing themathematical capabilities of the CAS.Wester’s book [We], reference [52] inthe book, provided test suites in sev-eral CAS. The value of the test suitesdiminishes rapidly with time as dif-ferent versions of the software addressissues shown up by the tests.There are books which discuss algo-rithms used to provide the algebra ca-pabilities of both CAS, e.g., [GKW].

• There are many applied mathemati-cians who make straightforward use ofboth Mathematica and Maple. Some-

times this is done to check the re-sult of one against the other. Some-times one starts with one CAS and, atthe very last step, even though one’sintuition says the step might be do-able, that CAS doesn’t work, but theother one does. In practice, the on-line helps in the CAS combined, if onewishes, with one of the many goodbooks on programming in the CASis enough. And, of course, there aremany places where “one-liner” transla-tions are provided, e.g., http://amath.colorado.edu/computing/mmm/.

• There are mathematical programmerswho code with some sophistication spe-cialized packages and provide these inboth CAS.

• There are people providing tools fortranslation between subsets of theCAS, especially concentrating on themathematical functions. The likelyusers of such tools include the math-ematical programmers mentioned inthe preceding situation.

While I consider the material in the book in-appropriate for the first three groups above,it is just possible that the material on theCD could be developed into a form whichmight be of use for the final group. Moredetail is given later in this review.When switching between the two CAS,

there are many details to consider. Toname but one, which is easy to describe,it happens that EllipticK(x) in Maple (atleast in version 9.5) is almost equivalent toEllipticK[x^2] in Mathematica (at leastin versions up to 7); one might expect toget the same result in both systems, at leastwhile 0 < x < 1. Collecting items of thiskind and other items where the translationis simpler, as is done by the authors, is aworthwhile endeavor.There are continuing efforts at devel-

oping tools to help in the translation ofcode between each CAS. Personally, I thinkthe efforts that are most likely to be suc-cessful, and useful, are those for Maple-to-Mathematica translation, particularly con-centrating on mathematical applications.Mathematica as a language lends itself toother uses, e.g., cellular automata, and someof the programming constructs are not nat-ural to Maple. Some of the translation tools

http://amath.colorado.edu/computing/mmm/

http://amath.colorado.edu/computing/mmm/


584 BOOK REVIEWS

are written in Perl or similar. The othersI mention here are written in the languageof the CAS. Those written in Mathematicainclude the following:

• For the conversion of Mathematicainto Maple syntax: MapleForm byJuergen Schmidt, 1996; Format.m byMark Sofroniou, also from the mid1990s and still available from the li-brary.wolfram.com website.

• For the conversion of Maple 9 work-sheets to Mathematica notebooks:code byYves Papegay, 2004. The workon this seems to have been discontin-ued, and I think it may be too muchto expect that anything other thantranslation of subsets of the languagesmight be achievable.

Maplesoft has provided with recent versionsof Maple (from Maple 11) their MmaTrans-lator, with its usage

with(MmaTranslator);

lprint(FromMma(‘some Mma commands‘));

Again this is, as its online help explains,really a very incomplete translator.It may well be that the authors of such

software find real value in a good, large col-lection of codes with a mathematical cover-age similar to that on the CD of the secondedition. It may be that a small subset ofthe CD posted on the Web, in some sortof wiki allowing different styles, would beworthwhile. The authors could then, whenappropriate, adapt their style in later re-visions of the CD to good ideas submittedthrough the wiki. It is very easy to findfault with the style of the programming inthe present CD.There are many typos in the text, but

not, I think, in the code.I have assessed the CD using Mac OSX.

I followed the instructions on the CD: in-stall Acrobat Reader (as it has a pdf-to-texttranslator), then load start.pdf. (I lookedaround on the CD for .txt files with thecode, but failed to find any.) One then hasto edit the code into separate files—a filefor each problem in each CAS—to be ableto run it. Better organization of the codeon the CD is possible.I find it disconcerting that the code on

the CD isn’t always identical to that in

the book, e.g., the Mathematica of Problem4.29, where the mult function comes earlierin the print version. I assume that therehas been some editing to save paper. Thecode on the CD should have been “pretty-printed,” put through some sort of codeformatter to make it more humanly read-able. After converting the pdfs to text andediting so that the different languages werein different files, I used vim to highlight, incolor, the reserved words in the files. As vimknows about both Maple and Mathematica,having the code for the same problem inthe two languages in two simultaneouslyvisible terminal windows enables a moreinformative display than just viewing thepdf on the CD. The code, along with tex-tual explanation, should also be providedin Mathematica notebook and Maple doc-ument/worksheet formats, both of whichautomatically “pretty-print” and color re-served words, and so on.Finally, I’m not happy about a market-

ing issue. Both the preface to the bookand the publisher’s pages on it (http://www.springer.com/springerwiennewyork/mathematics/book/978-3-211-99431-3) endwith the statement

Finally, this book is ideal for scientistswho want to corroborate their Mapleand Mathematica work with indepen-dent verification provided by anotherCAS.

They attribute this to J. Carter, SIAMReview, 50 (2008), pp. 149–152. However,that article by Carter was a review ofMath-ematica 6, with no mention of the book.Carter’s review of the first edition book iselsewhere.

REFERENCES

[We] M. J. Wester, Computer AlgebraSystems: A Practical Guide, JohnWiley, Chichester, UK, 1999.

[GKW] J. Grabmeier, E. Kaltofen, and

V. Weispfenning, Computer Al-gebra Handbook: Foundations,Applications, Systems, Springer-Verlag, Berlin, 2003.

GRANT KEADY

University of Western Australia


BOOK REVIEWS 585

A Primer on Scientific Programming withPython. By Hans Petter Langtangen. Springer,Berlin, 2009. $59.95. xxviii+693 pp., hardcover.ISBN 978-3-642-02474-0.

Python is a relatively new programminglanguage (dating from the late 1980s)that was originally designed for scripting.The language is characterized by its clear,clutter-free syntax which makes programseasy to write and read. Since its introduc-tion, Python’s numerical capabilities havegreatly increased with the release of a se-ries of modules which have transformed itinto a powerful tool for scientific computing.Python is still an emerging language, with anew version being released every year or so.Unfortunately, new releases are often some-what incompatible with previous versions.The code in Langtangen’s book requiresPython 2.6, whereas the latest Python re-lease is version 3.1. The two versions arenot entirely compatible.Considering the increasing popularity of

Python in scientific applications, it is sur-prising that until now there have been nobooks on the subject; Langtangen’s offeringmore than makes up for this oversight. Itis an authoritative and almost monumen-tal work that covers most aspects of thePython language and its numerical mod-ules. It definitely has a prominent place onmy bookshelf.The book is almost 700 pages long and

attractively printed in two colors. It is lib-erally illustrated with snippets of computercode, and the price is more than reason-able. The text emphasizes the Python lan-guage; scientific programming is relegatedto the secondary role of illustrating pro-gramming concepts. In other words, this isnot a numerical methods book, but a texton programming.The basics of Python (variables, opera-

tors, basic constructions, data input, arraycomputing, and plotting) are covered in thefirst four chapters, which take up 233 pages.These chapters could be used as a text for anintroductory Python programming course.The rest of the book is devoted to moreadvanced topics, including two chapters onobject-oriented programming.The text is well written, although some-

what verbose for my taste. All the pro-

gramming details are thoroughly explainedwith only occasional lapses. For example,the function enumerate on page 63 is illus-trated with a line of code, but no further in-formation is given. I had to go on Python’swebsite to find out how the function works.Although the book is written in the style

of a textbook, I feel that it would serve bet-ter as a reference book for Python program-mers with some prior experience. With theexception of the first four chapters, thereis simply too much material for an effec-tive course syllabus. But, as a reference,there is a lot to like about this book. Be-ing an intermediate-level Python program-mer, I found the book invaluable in improv-ing my knowledge of Python. The chapterson object-oriented programming alone areworth the price of the book.I consider the index to be inadequate, as

it omits many of the functions introducedin the text. I also did not like extensivereliance on the scitools module, writtenby the author. It would make more senseto use the mainstream modules scipy andmatplotlib, which most Python users havealready installed (matplotlib is a part ofPython download).In summary, this is the book (the only

book) to have if you are an aspiring Pythonprogrammer of scientific applications. Itsfew shortcomings do not detract signifi-cantly from its well-written, comprehensivecoverage of the Python language.

JAAN KIUSALAAS

Pennsylvania State University

Introduction to Applied Algebraic Sys-tems. By Norman R. Reilly. Oxford Univer-sity Press, Oxford, 2010. $80.00. xiv+509 pp.,hardcover. ISBN 978-0-195-36787-4.

This is a text for an introductory course inabstract algebra with an emphasis on recentapplications of the subject, particularly tocryptography and error-correcting codes. Incontrast to more traditional developmentsof the subject, the author covers rings andfields before groups; more specifically, hebegins with some basic number theory andthen introduces general rings, concentrat-ing especially on the modular integers andfinite fields, with applications to the BCH


586 BOOK REVIEWS

and Reed–Solomon error-correcting codes.He introduces groups in Chapter 3, empha-sizing permutation groups and developingthe counting applications of Cauchy, Frobe-nius, and Polya relating orders of permuta-tion groups to those of their orbits andstabilizers. The following chapter treatsnormal subgroups and group homomor-phisms, continuing with the characteriza-tions of finite abelian groups and conjugacyclasses in the symmetric groups togetherwith the Sylow theorems. In the remain-der of the text the author returns to ringtheory, studying polynomials in one andseveral variables in Chapter 5 and introduc-ing ideals, affine varieties, Groebner bases,parameters, and intersection multiplicities.Chapter 6 introduces elliptic curves andtheir applications to cryptography. The lastchapter rounds off the discussion in previ-ous chapters, discussing prime factorizationvia elliptic curves, the link between ellip-tic curves and Fermat’s last theorem, andPell’s equation.The book is noteworthy for its inclusion

of extensive historical material on breakingthe Enigma code in World War II and thetheorems used to accomplish this (the char-acterization of conjugacy classes in the sym-metric group is described as the “theoremthat won World War II”). It also includes anice discussion at the undergraduate levelof the ideas behind Wiles’s proof of Fer-mat’s last theorem and some of the latestwork in cryptography. There are almost 850exercises included, of various shades of diffi-culty. The whole book would be suitable asa text for a year-long course in algebra, orthe first chapter by itself for a short coursein number theory, or the first three chaptersfor a semester-long course in algebra.

WILLIAM M. MCGOVERN

University of Washington

The Pleasures of Statistics: The Autobi-ography of Frederick Mosteller. Edited byFrederick Mosteller, Stephen E. Weinberg, DavidC. Hoaglin, and Judith M. Tanur. Springer, NewYork, 2010. $39.95. xvi+344 pp., softcover.ISBN 978-0-387-77955-3.

Frederick Mosteller (1916–2006) was atremendously productive and influential

statistician. He published 57 books, 360papers, and was president of the Ameri-can Statistical Association, the Institute ofMathematical Statistics, and the AmericanAssociation for the Advancement of Science,among other organizations.He was hired by Harvard’s Department

of Social Relations in 1946, and ultimatelychaired both it and Harvard’s Departmentsof Statistics, Biostatistics, and Health Pol-icy and Management. This followed math-ematics majors at the Carnegie Institute ofTechnology, wartime work on bombing, anda Princeton Ph.D. with Samuel Wilks.The book is based on notes prepared

by Mosteller before 1990 and is a com-panion to his Selected Papers. Its chaptershighlight many of the diverse projects thiswell-organized statistician carried out withover two hundred collaborators. These in-clude a study of why Dewey beat Trumanin the preelection polls of 1948, the use ofBayesian methods and computation to at-tribute authorship of twelve Federalist pa-pers to Madison, joint work with anesthe-siologists, surgeons, and social scientists,teaching “Continental Classroom” on earlymornings on NBC, and chairing influentialgovernment panels.There are valuable commentsmade about

Jimmie Savage, John Tukey, and Persi Dia-conis, among others. Overall, however, themanuscript shows a reluctance to make ob-servances about individuals or policies thatmany readers would naturally hope to hearfrom this experienced professor and publiccitizen.

ROBERT E. O’MALLEY, JR.


Differential Equations and MathematicalBiology. Second Edition. By D. S. Jones, M. J.Plank, and B. D. Sleeman. Chapman & Hall/CRC,Boca Raton, FL, 2010. $79.95. xviii+444 pp.,hardcover. ISBN 978-1-4200-8357-6.

While they were colleagues in Dundee,the distinguished applied analysts DouglasJones and Brian Sleeman published a bookin 1983 with Allen & Unwin with the sametitle as this one. The outline of the newbook is nearly the same, except that a


BOOK REVIEWS 587

chapter on “Catastrophe Theory and Bi-ological Phenomena” has been replaced bysuccessful new chapters on “Bifurcation andChaos” and “Numerical Bifurcation Analy-sis,” while more computational approachesand the use of MATLAB have been addedthroughout. Much progress by these au-thors and others over the past quarter cen-tury in modeling biological and other scien-tific phenomena make this differential equa-tions textbook more valuable and bettermotivated than ever.The intended audience is broad and, in

contrast with many texts on modeling inbiology, the differential equations cover-age is quite sophisticated. For example,it’s pointed out how to solve nonhomoge-neous linear equations that can be factored;how to write the matrix exponential as afinite combination of matrix powers withcoefficients satisfying the D operator ver-sion of the characteristic polynomial andappropriate initial conditions; that char-acteristics remain fundamental in solvingsecond-order linear partial differential equa-tions; and that the Poincare–Bendixsontheorem explains the occurrence of limitcycles.Models of the heartbeat, nerve impulses,

chemical reactions, fishing, pattern forma-tion, and tumor growth are all carefullydescribed. Many of these biological prob-lems lead to systems of differential equationswith a small parameter multiplying deriva-tives. Clever detailed arguments are usedto describe solution behavior, without ex-plicitly introducing boundary layer theory,though averaging is ultimately developed.The writing is clear, though the modelingis not oversimplified.Overall, this book should convince math

majors how demanding math modelingneeds to be and biologists that taking an-other course in differential equations willbe worthwhile. The coauthors deserve con-gratulations as well as course adoptions.

ROBERT E. O’MALLEY, JR.


Perspectives in Computation. By RobertGeroch. University of Chicago Press, Chicago,2009. $25.00. vi+200 pp., softcover. ISBN 978-0-226-28855-0.

The idea of quantum computation camefrom a confluence of several lines of research.One of these was the study of reversible com-puting (e.g., Bennett, Landauer), and an-other dealt with using quantum systems ascomputers (e.g., Benioff, Feynman). Nextcame some innovative algorithms based onthe use of quantum mechanics (Deutsch-Jozsa, Simon, Shor, Grover) and otherapplications (e.g., Ekert). Significant newfunding quickly developed and provided thecatalyst for a continuing surge of researchin a new, highly interdisciplinary field.For the physicist this has opened up

new vistas as well as provided new per-spectives on existing work. For other dis-ciplines the general strategy has been toadd the novel features of quantum mechan-ics to an existing topic: quantum infor-mation theory, quantum coding, quantumalgorithms, quantum complexity, and evenquantum bits (aka qubits). Thus, someoneinterested in a subspecialty of this subjectneeds to know both the classical results aswell as the perspective introduced by quan-tum mechanics.In the book under review Geroch follows

this paradigm in discussing several subjectsin the theory of computation. His motivat-ing topics are the computability of prob-lems and the computational efficiency ofalgorithms. The foundation for the book isa set of lecture notes for a graduate coursefor physics students, and the author’s inten-tion is to provide a “walking tour throughthe subject” with particular attention towhat he considers the most illuminatingissues.Accordingly, he only assumes an audience

accustomed to proofs and abstract reason-ing, and he concentrates on making the keyproblems and questions accessible. In par-ticular, his focus is on a few open questions.There is no attempt to survey the field;there are only fourteen references, and themodest number of definitions are mostlyembedded in the text.The book is roughly divided into three

parts. The first part deals with the idea ofthe computability of a problem, based onTuring machines, and the appropriate con-text and tools are developed from scratch.The author includes a proof that the haltingproblem is not computable in this context


588 BOOK REVIEWS

and makes a short detour into the idea ofnoncomputable numbers.The second classical topic deals with

computational efficiency and with ways ofmeasuring the difficulty of actually imple-menting an algorithm. The concepts of adifficulty function and equivalence classesof such functions are introduced, and it isshown that in this formulation there doesn’tappear to be a way of defining an intrinsicdifficulty of a problem. To facilitate a com-parable argument when quantummechanicsis added, the author defines a “language forefficiency” rather different from the Turingmachine context used in discussing com-putability.After a section on probabilistic comput-

ing and a short section on finite-dimensionalquantummechanics, Geroch has the tools todiscuss the impact of quantummechanics oncomputability and computational efficiency.He begins with a detailed discussion ofGrover’s algorithm and indulges his originalaudience of graduate students by spending alittle time discussing the theoretical feasibil-ity of implementing the several steps of thatalgorithm. This is a nice summary of thealgorithm, and it is presented in a way thatpermits its use as an example in the last fivechapters that deal with quantum-assistedcomputability and quantum-assisted effi-ciency. It is in those five chapters thathe discusses what is probably his motivat-ing issue: while quantum-assisted compu-tation appears to enhance computationalefficiency, a formal proof is still lacking.Geroch’s book is an accessible introduc-

tion to some of the results and open ques-tions involved in computability and compu-tational efficiency, both with and withoutquantum computation. If one of his goalswas to produce a treatise that would en-courage physics graduate students to readfurther into the subject of quantum-assistedcomputation, he has succeeded admirably.

ARTHUR PITTENGER

University of Maryland

Matrix Methods in Data Mining and Pat-tern Recognition. By Lars Elden. SIAM,Philadelphia, 2007. $69.00. x+224 pp., soft-cover. ISBN 978-0-898716-26-9.

This is the fourth in the growing SIAMbookseries Fundamentals of Algorithms edited byNick Higham. These books are short andfairly narrowly focused. Each presents algo-rithms for solving a few specific problems,together with the background material nec-essary for understanding the algorithms.Elden’s book discusses five application

areas in data mining and pattern recog-nition that are amenable to treatment bymatrix methods. He states in the prefacethat his primary audience is undergradu-ate students who have already had a coursein scientific computing or numerical anal-ysis. I would add that such a prerequisitecourse had better have a substantial andpretty serious component on matrix com-putations.The book has three parts, but the heart

of the book is Part II, which consists of fivechapters (10–14) and about 65 pages. Eachchapter discusses one application.Chapter 10 is on machine classification

of handwritten digits, an important prob-lem for postal services around the world.Each digit is represented as a grayscale im-age of 16 × 16 pixels. Thus, it is a vectorin R

256. Sets of training digits determinewhich regions of R

256 are occupied by whichdigits. The singular value decomposition(SVD) is used as a tool to create a low-rank approximation and a low-dimensionalsubspace that captures the important infor-mation about each digit.The effects of simple transformations,

such as rotation, stretching, or thickeningof a digit, are discussed. The notion of tan-gent distance, a method of measuring thedistance between digits that is insensitiveto small transformations, is introduced.Chapter 11 is on text mining. Given a

large set of documents, how does one de-termine which documents are relevant to agiven query? To this end, a term-documentmatrix A is built with one row for eachterm and one column for each document.The (i, j) element is positive if and only ifthe ith term appears in the jth document.The value of aij can be weighted by theimportance of the term and/or the docu-ment. Thus, each document is representedby a vector, a column of A. Queries arealso documents and can be represented asvectors in the same way. A simple measure


BOOK REVIEWS 589

of the relevance of a document to a givenquery is the cosine of the angle between thequery vector and the document vector.This measure of relevance is only mod-

estly successful. Chapter 11 consists mainlyof ways of improving on it. Each of thesemethods replaces the term-document ma-trix by a low-rank approximation in an at-tempt to capture the important informationand discard the irrelevant details. The firstis latent semantic indexing, which uses theSVD to create the low-rank approximation.Viable alternatives are based on clusteringand on a nonnegative matrix factorization.Lanczos bidiagonalization, which builds aKrylov subspace starting from the queryvector, is also considered. This method cangive good results after a very small numberof steps (very low-rank approximation), butbecomes less effective if too many steps aretaken.Chapter 12, which is about ranking web-

pages, consists mainly of a description ofGoogle’s PageRank algorithm. For a Websearch engine it does not suffice to identifyall webpages that have been judged relevantto a given query. It must also make a deci-sion about which of the many relevant pagesare most important and therefore worthy ofbeing recommended to the user. Googleuses PageRank to help make this decision.The idea of PageRank is that a page is im-portant if there are many links to it fromother important pages. This circular def-inition of importance leads to a giganticeigenvalue problem,

r = Qr,

where Q is the Google matrix, a posi-tive, column-stochastic matrix determinedby the link structure of the Web. Its di-mension is n× n, where n is the number ofpages in the Web. Perron–Frobenius theoryguarantees that the equation r = Qr has anessentially unique positive solution r, whichcan be computed by the power method. Theimportance of the ith webpage is given byri, the ith component of r.Chapter 13 is on automatic key word

and key sentence extraction. Given a doc-ument, how can we decide what are themost important terms and the most impor-tant sentences in that document? Insteadof a term-document matrix, we can build

a term-sentence matrix A, in which eachsentence is treated as a separate document.Each term and each sentence is given asaliency score, which is a measure of its im-portance. A sentence is considered impor-tant if it contains many important terms,and a term is considered important if itis contained in many important sentences.This circular definition leads to anothereigenvalue problem, which is actually a sin-gular value problem for the term-sentencematrix. The saliency values for the termsand sentences are the entries of the dom-inant left and right singular vectors of A,respectively.Just as in Chapter 11, we can improve

performance here by replacing the matrix Aby a low-rank approximation, which can bedone using the SVD, a clustering method,or a nonnegative matrix factorization.Chapter 14 considers the question of fa-

cial recognition. It is difficult for a machineto match two photographs of the same facebecause they can be taken from different an-gles, in different lighting, and with differentfacial expressions. The method describedin the book stores several images of eachof several people as a third-order tensor.A tensor SVD (HOSVD) is used to ana-lyze the data. It is safe to say that thisapplication is the least well developed ofthe five applications discussed in the book.This would also make it the ripest for futuredevelopment.So far we have been talking about Part

II of the book. Part I, which is about 110pages long, is a minicourse in matrix com-putations, including such standard topicsas linear systems and least squares prob-lems, QR decomposition, and SVD. Exam-ples pertaining to data mining and patternrecognition are included. Some nonstan-dard topics that are included are tensordecompositions, clustering, and nonnega-tive matrix factorization. The presentationis too compressed to be readable by a neo-phyte, but it could serve as a useful reviewand reinforcement for a reader who has al-ready had a course in matrix computations.Part III, which consists of a single chapter

about 30 pages long, gives a brief overviewof methods for computing matrix decom-positions. Topics discussed include pertur-bation theory, the power method, reduc-


590 BOOK REVIEWS

tion to tridiagonal form, the QR algorithm,SVD computation, and Arnoldi and Lanc-zos methods. Clearly the treatment is verycondensed.The book has an accompanying website

that has theory questions (exercises), com-puter assignments, and more.

If you are planning to teach a course ondata mining or applied matrix computa-tions, consider using this book as a text ora supplement. Or pick it up and read it foryour own edification, as I did.

DAVID S. WATKINS

Washington State University

book reviewstardis.berkeley.edu/reprints/papers/kn_siamrev52_2010.pdf · introduction, 565 featured...

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