scientific programming 12 (2004) 197–198 ios press book...

Scientific Programming 12 (2004) 197–198 197IOS Press

Book Review

The Lanczos Method: Evolution and Application,by Louis Komzsik, Society for Industrial and AppliedMathematics, Philadelphia, PA, USA, 2003. ISBN: 089871 537 7

Louis Komzsik is an expert on structural analysiswho has worked for 20 years as the chief numericalanalyst at MSC Software. He was responsible for thenumerical methods in the NASTRAN finite elementsolver. This book is an introduction to the Lanczosmethod and its applications in finite element analysis.The Lanczos method has obviously been very impor-tant in Kmozsik’s work on NASTRAN. The book isalso a homage to Kmozsik’s Hungarian countrymanCornelius Lanczos.

The first half of the book is a brief (33 pages) in-troduction to the Lanczos method. In Chapter one,Komzsik begins with a discussion of the connectionbetween the eigenvalues of a symmetric matrix A andthe ellipsoid given by the equation xTAx = 1. Thissection is marred by a serious misstatement of the rela-tionship between A’s eigenvalues and the lengths of theaxes of the ellipsoid. The author continues by givinga fairly standard derivation of the Lanczos iteration asa way to compute the characteristic polynomial of asymmetric matrix.

The method is extended to unsymmetric matrices andthen to the computation of a tridiagonal matrix T that issimilar to A. In Chapter 2, the author discusses how theeigenvalues and eigenvectors of T can be obtained. InChapter 3 a version of the algorithm for finite precisionarithmetic is developed. In fi- nite precision, loss of or-thogonality of the Lanczos vectors is a serious problem.Komzsik discusses the full orthogonalization strategyfor recovering from this loss of orthogonality. Blockmethods for symmetric and unsymmetric matrices areintroduced in Chapters four and five.

The presentation is made unnecessarily complicatedby several changes in notation. The discussion of re-orthogonalization strategies other than full orthogonal-ization is particularly brief. The lack of any discussionof the Arnoldi iteration is notable. There is also no dis-cussion of available software that implements the Lanc-

zos method. The introduction to the Lanczos method inthe first half of the book contains standard material thatis already present in many textbooks on numerical lin-ear algebra [2,3,5]. Given the lack of exercises in thisbook, most students would be better served by read-ing about the Lanczos iteration in one of the standardtextbooks.

The second half of the book discusses applicationsof the Lanczos method in finite element analysis. InChapter six, the spectral transformation is intro- ducedas a way to spread out closely spaced eigenvalues. Theauthor discusses frequency domain decomposition andgeometric domain decomposition as ways to parallelizethe computation of the eigenvalues of large matricesarising from finite element analysis. Applications ofthe Lanczos method to free undamped vibrations, freedamped vibrations, and forced vibrations are the sub-ject of chapters seven through nine. Examples of theapplication of these approaches to engineering designproblems are given. In Chapter ten, the method ofconjugate gradients is derived as an application of theLanczos iteration.

The discussion of applications of the Lanczos iter-ation to finite element analysis is the most interestingaspect of the book. This half of the book might be ofinterest to mathematicians who are comfortable withnumerical linear algebra but not familiar with finiteelement analysis. However, these chapters would bemuch more interesting if readers could actually tinkerwith the data sets and software. Furthermore, the ap-plication of the Lanczos method to structural analysisproblems is commonly covered in textbooks on finiteelement analysis [1,4].

References

[1] R.D. Cook, D.S. Malkus, M.E. Plesha and R.J. Witt, Conceptsand Applications of Finite Element Analysis, John Wiley &Sons, New York, fourth edition, 2001.

[2] J.W. Demmel, Applied Numerical Linear Algebra, SIAM,Philadelphia, 1997.

[3] G.H. Golub and C.F. van Loan, Matrix Computations, JohnsHopkins University Press, Baltimore, third edition, 1996.

ISSN 1058-9244/04/$17.00 2004 – IOS Press and the authors. All rights reserved

198 Book Review

[4] T.J.R. Hughes, The Finite Element Method: Linear Static andDynamic Finite Element Analysis, Dover Publications, Mine-ola, NY, 2000.

[5] L.N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM,Philadelphia, 1997.

Brian BorchersDepartment of Mathematics

New Mexico TechSocorro, NM, USA

January 21, 2004

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