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This IMA Volume in Mathematics and its Applications RECENT ADVANCES
IN ITERATIVE METHODS is based on the proceedings of a workshop that
was an integral part of the 1991-92 IMA program on "Applied Linear
Algebra. " Large systems of matrix equations arise frequently in
applications and they have the prop erty that they are sparse
and/or structured. The purpose of this workshop was to bring
together researchers in numerical analysis and various ap plication
areas to discuss where such problems arise and possible meth ods of
solution. The last two days of the meeting were a celebration
dedicated to Gene Golub on the occasion of his sixtieth birthday,
with the program arranged by Jack Dongarra and Paul van Dooren. We
are grateful to Richard Brualdi, George Cybenko, Alan George, Gene
Golub, Mitchell Luskin, and Paul Van Dooren for planning and
implementing the year-long program. We especially thank Gene Golub,
Anne Greenbaum, and Mitchell Luskin for organizing this workshop
and editing the proceed ings. The financial support of the National
Science Foundation and the Min nesota Supercomputer Institute made
the workshop possible. A vner Friedman Willard Miller, Jr. xi
PREFACE The solution of very large linear algebra problems is an
integral part of many scientific computations."
Numerical Methods provides a clear and concise exploration of
standard numerical analysis topics, as well as nontraditional ones,
including mathematical modeling, Monte Carlo methods, Markov
chains, and fractals. Filled with appealing examples that will
motivate students, the textbook considers modern application areas,
such as information retrieval and animation, and classical topics
from physics and engineering. Exercises use MATLAB and promote
understanding of computational results. The book gives instructors
the flexibility to emphasize different aspects--design, analysis,
or computer implementation--of numerical algorithms, depending on
the background and interests of students. Designed for
upper-division undergraduates in mathematics or computer science
classes, the textbook assumes that students have prior knowledge of
linear algebra and calculus, although these topics are reviewed in
the text. Short discussions of the history of numerical methods are
interspersed throughout the chapters. The book also includes
polynomial interpolation at Chebyshev points, use of the MATLAB
package Chebfun, and a section on the fast Fourier transform.
Supplementary materials are available online. * Clear and concise
exposition of standard numerical analysis topics * Explores
nontraditional topics, such as mathematical modeling and Monte
Carlo methods * Covers modern applications, including information
retrieval and animation, and classical applications from physics
and engineering * Promotes understanding of computational results
through MATLAB exercises * Provides flexibility so instructors can
emphasize mathematical or applied/computational aspects of
numerical methods or a combination * Includes recent results on
polynomial interpolation at Chebyshev points and use of the MATLAB
package Chebfun * Short discussions of the history of numerical
methods interspersed throughout * Supplementary materials available
online
Much recent research has concentrated on the efficient solution of
large sparse or structured linear systems using iterative methods.
A language loaded with acronyms for a thousand different algorithms
has developed, and it is often difficult even for specialists to
identify the basic principles involved. Here is a book that focuses
on the analysis of iterative methods. The author includes the most
useful algorithms from a practical point of view and discusses the
mathematical principles behind their derivation and analysis.
Several questions are emphasized throughout: Does the method
converge? If so, how fast? Is it optimal, among a certain class? If
not, can it be shown to be near-optimal? The answers are presented
clearly, when they are known, and remaining important open
questions are laid out for further study. Greenbaum includes
important material on the effect of rounding errors on iterative
methods that has not appeared in other books on this subject.
Additional important topics include a discussion of the open
problem of finding a provably near-optimal short recurrence for
non-Hermitian linear systems; the relation of matrix properties
such as the field of values and the pseudospectrum to the
convergence rate of iterative methods; comparison theorems for
preconditioners and discussion of optimal preconditioners of
specified forms; introductory material on the analysis of
incomplete Cholesky, multigrid, and domain decomposition
preconditioners, using the diffusion equation and the neutron
transport equation as example problems. A small set of recommended
algorithms and implementations is included.
LAPACK is a library of numerical linear algebra subroutines
designed for high performance on workstations, vector computers,
and shared memory multiprocessors. Release 3.0 of LAPACK introduces
new routines and extends the functionality of existing routines.
The most significant new routines and functions include: a faster
singular value decomposition computed by divide-and-conquer; faster
routines for solving rank-deficient least squares problems: using
QR with column pivoting; using the SVD based on divide-and-conquer;
new routines for the generalized symmetric eigenproblem: faster
routines based on divide-and-conquer; routines based on
bisection/inverse iteration, for computing part of the spectrum;
faster routine for the symmetric eigen problem using "relatively
robust eigenvector algorithm"; new simple and expert drivers for
the generalized nonsymmetric eigenproblem, including error bounds;
solver for generalized Sylvester equation, used in computational
routines Each Users' Guide comes with a "Quick Reference Guide"
card.
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