Scientific Programming 12 (2004) 197–198 IOS Press Book...
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Scientific Programming 12 (2004) 197–198 197 IOS Press Book Review The Lanczos Method: Evolution and Application, by Louis Komzsik, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2003. ISBN: 0 89871 537 7 Louis Komzsik is an expert on structural analysis who has worked for 20 years as the chief numerical analyst at MSC Software. He was responsible for the numerical methods in the NASTRAN finite element solver. This book is an introduction to the Lanczos method and its applications in finite element analysis. The Lanczos method has obviously been very impor- tant in Kmozsik’s work on NASTRAN. The book is also a homage to Kmozsik’s Hungarian countryman Cornelius 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 connection between the eigenvalues of a symmetric matrix A and the ellipsoid given by the equation xT Ax =1. This section is marred by a serious misstatement of the rela- tionship between A’s eigenvalues and the lengths of the axes of the ellipsoid. The author continues by giving a fairly standard derivation of the Lanczos iteration as a way to compute the characteristic polynomial of a symmetric matrix. The method is extended to unsymmetric matrices and then to the computation of a tridiagonal matrix T that is similar to A. In Chapter 2, the author discusses how the eigenvalues and eigenvectors of T can be obtained. In Chapter 3 a version of the algorithm for finite precision arithmetic is developed. In fi- nite precision, loss of or- thogonality of the Lanczos vectors is a serious problem. Komzsik discusses the full orthogonalization strategy for recovering from this loss of orthogonality. Block methods for symmetric and unsymmetric matrices are introduced in Chapters four and five. The presentation is made unnecessarily complicated by several changes in notation. The discussion of re- orthogonalization strategies other than full orthogonal- ization is particularly brief. The lack of any discussion of 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 in the first half of the book contains standard material that is already present in many textbooks on numerical lin- ear algebra [2,3,5]. Given the lack of exercises in this book, most students would be better served by read- ing about the Lanczos iteration in one of the standard textbooks. The second half of the book discusses applications of the Lanczos method in finite element analysis. In Chapter six, the spectral transformation is intro- duced as a way to spread out closely spaced eigenvalues. The author discusses frequency domain decomposition and geometric domain decomposition as ways to parallelize the computation of the eigenvalues of large matrices arising from finite element analysis. Applications of the Lanczos method to free undamped vibrations, free damped vibrations, and forced vibrations are the sub- ject of chapters seven through nine. Examples of the application of these approaches to engineering design problems are given. In Chapter ten, the method of conjugate gradients is derived as an application of the Lanczos iteration. The discussion of applications of the Lanczos iter- ation to finite element analysis is the most interesting aspect of the book. This half of the book might be of interest to mathematicians who are comfortable with numerical linear algebra but not familiar with finite element analysis. However, these chapters would be much more interesting if readers could actually tinker with the data sets and software. Furthermore, the ap- plication of the Lanczos method to structural analysis problems is commonly covered in textbooks on finite element analysis [1,4]. References [1] R.D. Cook, D.S. Malkus, M.E. Plesha and R.J. Witt, Concepts and 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, Johns Hopkins University Press, Baltimore, third edition, 1996. ISSN 1058-9244/04/$17.00 2004 – IOS Press and the authors. All rights reserved
Transcript of 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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