BOOK REVIEWS

G.E. Andrews, Partitions : Yesterday and Today, The New Zealand Mathematical

Society, 1979, 56 pages ($5.00).

This very readable booklet contains three excellent essays in partition

theory, featuring applications of the Durfee square technique. The spot­

light focuses in turn on three mathematicians: J.J. Sylvester, S. Ramanujan

and L.J. Rogers, each essay illuminating the relevant portion of the life

and work of one of these mathematical giants, and the continuing topicality

of his discoveries. The essays, though independent of each other, have

been so cleverly put together that they blend beautifully into a unified

whole. The booklet contains enough previously unpublished material and

pertinent comment to make it compelling reading for the specialist. On

the other hand, its masterly exposition makes it self-contained and easily

accessible to the general reader.

As its title suggests, the booklet deals with historical aspects as

well as modern developments. On reading it, one gets a strong sense of

history coming alive and reaching into the present. The author teaches

a valuable object lesson here, of how rich a reward may be reaped from

studying the history of a mathematical topic and from consulting original

source material of even the relatively distant past.

The main theme of the first essay is the work of Sylvester and his

group at Johns Hopkins University in partition theory, culminating in

Franklin's purely combinatorial proof of Euler's celebrated Pentagonal

Number Theorem. A brief, but skilful, introduction to the fundamentals

allows the newcomer to partition theory to absorb the prerequisites:

notation, Ferrers graph, generating functions and the interplay between

combinatorics and analysis. With the exception of a very recent

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application of successive Durfee squares, the topics in the first essay

will be familiar to any student of number theory, but even the expert will

find something of interest here: The author invents a hypothetical

sequence of questions that Sylvester may have raised with his students,

and that leads almost inevitably to Franklin's proof. Fanciful perhaps,

but good pedagogy and intellectually more satisfying than the deus ex

maohina approach.

The second essay begins with a brief biographical sketch of

Ramanujan, and an account of the author's recent discovery of Ramanujan's

'lost' notebook. A natural generalisation of the methods explained in

the first essay then leads to a combinatorial proof of one of Ramanujan's

earlier formulae, which in turn facilitates the proof of three rather

unwieldy identities that appear in the 'lost' notebook. These latter

identities are rather surprising generalisations of formulae that occur

in G.N. Watson's work on mock-theta functions. Thus there are combina­

torial aspects of modular equations that were previously unsuspected.

The author comments on this in some detail, and also mentions new avenues

of research in combinatorics arising in this context.

In the third essay, the author gives an appreciation of the work of

L.J. Rogers. Rogers contributed many fine results to many areas of

mathematics but, apart from the Rogers-Ramanujan identities for which he

is justly famous, the true worth and significance of his achievements

went largely unrecognised during his lifetime. The author gives a number

of instances of the continuing rediscovery of Rogers, modern advances in

partition theory being just one case in point. There are several examples

of combinatoric interpretations of Rogers' 'false theta function'

identities. Using the concept of successive Durfee squares, a theorem is

formulated whose proof is seen to lead to a 'false theta function'

identity.

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As one would expect from an author of Professor Andrews' stature,

who is an acknowledged expert in partition theory and a master of exposi­

tion, the essays are well written, entertaining and informative. The

material is meticulously researched and well supported by comprehensive

lists of references. It is therefore with considerable dismay and

embarrassment that I now have to mention some less pleasing aspects of

this publication.

There are numerous misprints in the text. Most of these are easily

spotted and corrected, and thus are merely minor irritations. Two mis­

prints in formula (3.3.1) can be put right by reference to the quoted

source (which unfortunately I did not have to hand): I believe that the

(aq/bk) just after the = in (3.3.1) should be (a q / b k , and that

the exponent of (b^ ̂ c^ j) at the end of (3.3.1) should be

m l + m2 + • • + 2 ' diagram on Page 23 will puzzle the reader who

does not realise that the brackets should stop short of the last row and

the last column of dots. There appears to be a genuine mistake in the

proof of theorem 2.4 and in formula (2.3.3) which is asserted by this

theorem. The exponent of a in the sum on the far right of (2.3.3)

should be -(2n+ 1) . The mistake in the proof occurs in the first line

of page 33, where N should be replaced by -N , with consequential

changes to get the corrected (2.3.3). It would, of course, be easier to-N

start the proof of theorem 2.4 by considering the coefficient of a

Formula (2.3.10) does follow, by the method indicated, from the correct

(2.3.3). Curiously enough, there is also a misprint in Watson's paper

where (2.3.10) was first proved, but stated incorrectly.

The typography leaves much to be desired. There are uncalled-for

changes of typehead in the main text, sometimes on the same page. The

general lay-out could be improved, as could the spacing and alignment of

symbols in many formulae. Moreover, the formulae would make a better

impact, and would be easier to grasp, if they were rendered in Italics

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as would be the normal custom. On the other hand, the references are

printed entirely in Italics. This is just as irritating as the absence

of Italics from the formulae.

Altogether, one is left with the impression of a sparkling diamond

set in a brass ring. However, the diamond is well worth having, even if

$5 is quite a stiff price for a 56-page booklet.

A. Zulauf

R.J. Hosking, D.C. Joyce and J.C. Turner, First Steps in Numerical

Analysis, Hodder 5 Stoughton, London, 1978, 202 pages ($7.80).

This textbook, which is designed for 7th-form students and for

Advanced Level GCE students in the U.K., is divided into 31 brief

chapters, or "steps".

Steps 1 to 5, on errors, deal with sources of error, approximation

to numbers, floating-point arithmetic and approximation to functions.

Steps 6 to 10 deal with solving a non-linear equation by the bisection

method, false position, simple iteration and the Newton-Raphson iterative

method. Steps 11 to 14 deal with systems of linear equations, solved by

elimination and by the Gauss-Seidel iterative method, with discussion of

errors and ill-conditioning. Steps 15 to 18 deal with finite differences,

including the application to the detection and correction of mistakes in

tables. Steps 19 to 24 deal with interpolation, including the techniques

named after Newton, Lagrange and Aitken. Step 25 treats curve fitting,

and step 26 deals with numerical differentiation. Steps 27 to 30 deal

with numerical integration by the trapezoidal rule, Simpson's rule and

Gauss integration formulae; whilst step 31 introduces numerical methods

for ordinary differential equations.

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An Appendix (pp. 149-158) gives flow charts for 8 algorithms, and it

is followed by an extensive set of Answers to Exercises (pp. 160-196).

Anyone considering adopting this book as a text should consider the

following points. In the account of floating-point numbers, there is no

indication given of the special representation required for zero. In

step 3 the relative error of the product of two approximate numbers is

obtained under the assumption that the product of their relative errors

is "negligible", with no discussion of what that might mean. In step 4

the normalized mantissa of a floating-point number is stated to lie in

the range (1,10) or (%,1) , instead of [ 1,10) or [is, 1) , and the

description of the operation of normalization is oversimplified. In

step 7 on the bisection method, the cases of various numbers of roots

could have been clarified by diagrams. In step 8 on the method of false

position, the root estimate x = a - f(a) — ^(a) *s converte(* to

the form 7pfia) ̂ without any warning that the latter version is

subject to much more round-off error than the former. The discussion of

the speed of convergence of the false position method is confusing. In

step 9, the stated condition for the method of simple iteration to converge

is necessary but not sufficient.

In step 12 on errors, the statement that "Generally, however, the

uncertainty in the solution is greater than in the constants" is incorrect.

In step 13, the recommended criterion for stopping iteration of the Gauss-

Seidel method is hazardous if it is applied uncritically. The treatment

of finite differences is much lengthier and more elaborate than is

required in an introductory textbook. Does a student need to be taught

the obsolete interpolation formulae named after Gauss, Stirling, Bessel

and Everett (in step 21) before learning of Lagrange's simple and general

formula for interpolation (in step 22)? Finite differences are used to

construct the trapezoidal rule and Simpson's rule in an unnecessarily

complicated manner. In step 27, the account of piecewise-polynomial

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functions is very muddled. Step 31 does warn that multi-step methods for

ordinary differential equations need more than one starting value, but no

advice is given on how to find them. In the flowcharts in the Appendix,

the operation of assignment is represented by the symbol for equality.

Accordingly, this book cannot be regarded as fulfilling the long-felt

need for a satisfactory introductory textbook on numerical analysis.

G .J . Tee

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