Accuracy and Stability of Numerical Algorithms gives a
thorough, up-to-date treatment of the behavior of numerical
algorithms in finite precision arithmetic. It combines
algorithmic derivations, perturbation theory, and rounding error
analysis, all enlivened by historical perspective and informative
quotations.
This second edition expands and updates the coverage of the first
edition (1996) and includes numerous improvements to the original
material. Two new chapters treat symmetric indefinite systems and
skew-symmetric systems, and nonlinear systems and Newton's method.
Twelve new sections include coverage of additional error bounds
for Gaussian elimination, rank revealing LU factorizations,
weighted and constrained least squares problems, and the fused
multiply-add operation found on some modern computer
architectures.
This text may become the new 'Bible' about accuracy and stability
for the solution of systems of linear equations. It covers 688
pages carefully collected, investigated, and written ...One will
find that this book is a very suitable and comprehensive
reference for research in numerical linear algebra, software
usage and development, and for numerical linear algebra courses.
N. Kockler, Zentrallblatt fur Mathematik, Band 847/96.
Contents
List of Figures; List of Tables; Preface to Second Edition;
Preface to First Edition; About the Dedication; Chapter 1:
Principles of Finite Precision Computation; Chapter 2: Floating
Point Arithmetic; Chapter 3: Basics; Chapter 4: Summation;
Chapter 5: Polynomials; Chapter 6: Norms; Chapter 7: Perturbation
Theory for Linear Systems; Chapter 8: Triangular Systems; Chapter
9: LU Factorization and Linear Equations; Chapter 10: Cholesky
Factorization; Chapter 11: Symmetric Indefinite and Skew-Symmetric
Systems; Chapter 12: Iterative Refinement; Chapter 13: Block LU
Factorization; Chapter 14: Matrix Inversion; Chapter 15:
Condition Number Estimation; Chapter 16: The Sylvester Equation;
Chapter 17: Stationary Iterative Methods; Chapter 18: Matrix
Powers; Chapter 19: QR Factorization; Chapter 20: The Least
Squares Problem; Chapter 21: Underdetermined Systems; Chapter 22:
Vandermonde Systems; Chapter 23: Fast Matrix Multiplication;
Chapter 24: The Fast Fourier Transform and Applications; Chapter
25: Nonlinear Systems and Newtonfs Method; Chapter 26:
Automatic Error Analysis; Chapter 27: Software Issues in Floating
Point Arithmetic; Chapter 28: A Gallery of Test Matrices;
Appendix A: Solutions to Problems; Appendix B: Acquiring
Software; Appendix C: Program Libraries; Appendix D: The Matrix
Computation Toolbox; Bibliography; Name Index; Subject Index.
2002 / xxx + 680 pages / Hardcover / ISBN 0-89871-521-0
1st ed. 1997. Corr. online-edition 2002 Online Book, XI, 247 p.
27 illus.
3-540-14949-X
These lecture notes are an introduction to several ideas and
applications of noncommutative geometry. It starts with a not
necessarily commutative but associative algebra which is thought
of as the algebra of functions on some 'virtual noncommutative
space'. Attention is switched from spaces, which in general do
not even exist, to algebras of functions. In these notes,
particular emphasis is put on seeing noncommutative spaces as
concrete spaces, namely as a collection of points with a topology.
The necessary mathematical tools are presented in a systematic
and accessible way and include among other things, C'*-algebras,
module theory and K-theory, spectral calculus, forms and
connection theory.
Application to Yang--Mills, fermionic, and gravity models are
described. Also the spectral action and the related invariance
under automorphism of the algebra is illustrated.
Some recent work on noncommutative lattices is presented. These
lattices arose as topologically nontrivial approximations to
'contuinuum' topological spaces. They have been used to construct
quantum-mechanical and field-theory models, alternative models to
lattice gauge theory, with nontrivial topological content.
This book will be essential to physicists and mathematicians with
an interest in noncommutative geometry and its uses in physics.
Keywords: Algebras, Field Theories, Gravity Models,
Noncommutative Lattices, Noncommutative Spaces, Quantum
Mechanical Models, Topology
Contents: Introduction.- Noncommutative Spaces and Algebras of
Functions.- Projective Systems of Noncommutative Lattices.-
Modules as Bundles.- A Few Elements of K-theory.- The Spectral
Calculus.- Noncommutative Differential Forms.- Connections on
Modules.- Field Theories on Modules.- Gravity Models.- Quantum
Mechanical Models on Noncommutative Lattices.- Basic Notions of
Topology.- The Gel'fand-Naimark- Segal Construction.- Hilbert
Modules.- Strong Morita Equivalence.- Partially Ordered Sets.-
Pseudodifferential Operators.
2003 VIII,499 p. Softcover
3-540-00072-0
The 36th Seminaire de Probabilites contains an advanced course on
Logarithmic Sobolev Inequalities by A. Guionnet and B.
Zegarlinski, as well as two shorter surveys by L. Pastur and N.
O'Connell on the theory of random matrices and their links with
stochastic processes. The main themes of the other contributions
are Logarithmic Sobolev Inequalities, Stochastic Calculus,
Martingale Theory and Filtrations. Besides the traditional
readership of the Seminaires, this volume will be useful to
researchers in statistical mechanics and mathematical finance.
Keywords: 60GXX, 60HXX, 60JXX, Stochastic Processes, Logarithmic
Sobolev Inequalities, Random Matrices, Martingales
Contents:
Cours specialises et exposes thematiques: A. Guionnet, B.
Zegarlinski: Lectures on Logarithmic Sobolev inequalities.- A.
Lejay, L. Pastur: Matrices aleatoires : statistique asymptotique
des valeurs propres.- N. O'Connell: Random matrices, non-colliding
particle systems and queues.- Exposes: A. Dermoune, O. Moutsinga:
Generalized variational principles.- D. Chafai: Gaussian maximum
of entropy and reversed log-Sobolev inequality.- L. Miclo: About
projections of logarithmic Sobolev inequalities.- L. Miclo: Sur
l'inegalite de Sobolev logarithmique des operateurs de Laguerre a
petit parametre.- A. Bentaleb: Sur les fonctions extremales des
inegalites de Sobolev des operateurs de diffusion.- C. Donati-Martin,
Y. Hu: Penalization of the Wiener measure and principal values.-
C. Leuridan: Theoreme de Ray-Knight dans un arbre : Une approche
algebrique.- R. Bass: Stochastic differential equations driven by
symmetric stable processes.- T. Simon: Support d'une equation d'
Ito avec sauts en dimension 1.- N. Eisenbaum: A Gaussian sheet
connected to symmetric Markov chains.- C. Leuridan: Filtration
d'une marche aleatoire stationnaire sur le cercle.- S. Beghdadi-Sakrani:
Une martingale non pure, dont la filtration est brownienne.- J.
Hannig: On filtrations related to purely discontinuous
martingales.- S. Beghdadi-Sakrani: Calcul stochastique pour des
mesures signees.- J. Jacod: On processes with conditional
independent increments and stable convergence in law.- V. Grecea:
Duality and quasi-continuity for supermartingales.- Y. Kabanov, C.
Stricker: On the true submartingale property, d'apres
Schachermayer.- C. Stricker: Simple strategies in exponential
utility maximization.- M. Arnaudon, A. Thalmaier: Horizontal
martingales in vector bundles.- D. Kurtz: Representation
nucleaire des martingales d'Azema.- S. Attal: Approximating the
Fock space with the toy Fock space.- Corrections aux volumes
anterieurs.
Series: Lecture Notes in Mathematics. Volume. 1801
2002 XII, 491 p. Hardcover
0-7354-0071-7
Neuroscience was the topic of several talks and papers. Emerging
themes appear to be the control of neuronal activity and the
construction of hybrid neuron-circuit systems, and that of using
nonlinear data analysis techniques for the characterization of
physiological pathologies. It was shown that these approaches can
be applied to cardiac arrhythmias and detection of problems in
EEG signals. Synchronization of chaos in optical systems was also
presented. New topics that were presented were Ecology and
Traffic dynamics. Also, ecological dynamics showed that state
space concepts can also be applied to populations and insect
networks. Topics include: Chemistry, Electronics, Geophysics and
Astrophysics, Neuroscience and Physiology, Data Analysis,
Applications, Traffic, Combustion, Optics, Hydrodynamics,
Condensed Matter, Synchrony, Dynamo, and Ecology.
Series: AIP Conference Proceedings. Volume. 622
2002 XII,346p. Hardcover
3-540-44071-2
The outcome of a close collaboration between mathematicians and
mathematical physicists, these lecture notes present the
foundations of A. Connes noncommutative geometry as well as its
applications in particular to the field of theoretical particle
physics. The coherent and systematic approach makes this book
useful for experienced researchers and postgraduate students
alike.
Contents: Foundations of Noncommutative Geometry and Basic Model
Building.- The Lagrangian of the Standard Model Derived From
Noncommutative Geometry.- New Directions in Noncommutative
Geometry and Mathematical Physics 233.
Series: Lecture Notes in Physics. Volume. 596
2002 VIII, 119 p. Softcover
3-540-00001-1
Based on a graduate course given by the author at Yale University
this book deals with complex analysis (analytic capacity),
geometric measure theory (rectifiable and uniformly rectifiable
sets) and harmonic analysis (boundedness of singular integral
operators on Ahlfors-regular sets). In particular, these notes
contain a description of Peter Jones' geometric traveling
salesman theorem, the proof of the equivalence between uniform
rectifiability and boundedness of the Cauchy operator on Ahlfors-regular
sets, the complete proofs of the Denjoy conjecture and the
Vitushkin conjecture (for the latter, only the Ahlfors-regular
case) and a discussion of X. Tolsa's solution of the Painleve
problem.
Keywords: analytic capacity, rectifiability, menger curvature,
singular integral operators, Cauchy integral, 28A75, 30C85, 42B20
Contents: Preface.- Notations and conventions.- Some geometric
measures theory.- Jones' traveling salesman theorem.- Menger
curvature.- The Cauchy singular integral operator on Ahlfors-regular
sets.- Analytic capacity and the Painleve Problem.- The Denjoy
and Vitushkin conjectures.- The capacity gamma (+) and the
Painleve Problem.- Bibliography.- Index.
Series: Lecture Notes in Mathematics. Volume. 1799