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- Krylov subspace methods for solving linear systems
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Michael Eiermann |
| Institut für Numerische Mathematik und Optimierung |
| Technische Universität Bergakademie Freiberg |
| D-09596 Freiberg |
| Germany |
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| “The simplest model in applied mathematics is a system of linear equations. It
is also by far the most important . . . ” [Gilbert Strang, Introduction to Applied
Mathematics, Wellesley-Cambridge Press 1986, p. 1]. Krylov subspace methods
are projection methods for solving a variety of large problems in numerical linear
algebra. In these lectures, we focus on recent developments of Krylov subspace
methods for the solution of linear systems of algebraic equations but other tasks
such as eigenproblems will also play a role. It is our objective to develop the theory
and algorithms on which all Krylov subspace methods are based in a unified way,
to emphasize their connections to other areas of applied mathematics, but to treat
also problems one encounters in practical computations.
The main emphasis is on the convergence properties of Krylov subspace methods
(linear versus superlinear convergence, error bounds based on angles between subspaces,
convergence results based on potential theory, bounds based on the field
of values or on pseudospectra) and on issues relevant for the implementation of
these algorithms (effects of finite precision arithmetic, error estimates and stopping
criteria, choice of the initial approximation, preconditioning). |
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- Matrix methods in data mining
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Lars Eldén |
| Department of Mathematics |
| Linköping University |
| SE-58183 Linköping |
| Sweden |
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| Powerful linear algebra techniques are available for solving problems in data mining
and pattern recognition. We give a brief introduction to matrix theory and decompositions.
The matrix methods are then applied to classification of handwritten digits,
text mining, text summarization, and pagerank computation. |
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- Mechanics and linear algebra
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| Richard B. Lehoucq |
| Computational Mathematics and Algorithms Department |
| Sandia National Laboratories |
| Albuquerque, NM 87185-1320 |
| New Mexico (USA) |
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| The well known conjugate gradient algorithm represents a linear algebra abstraction of
the principle of minimum potential energy. The minimum is determined by a linear set of equations
that represents the principle of virtual work. The purpose of these lectures is to
introduce the symbiotic relationship between mechanics and linear algebra. Topics include:
constrained and unconstrained energy principles that lead to nonlinear and linear sets
of equations, saddle point systems, and eigenvalue problems. The lectures provide
an introduction to the linear algebra culture prevalent in the computational mechanics community and literature. |
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- Structured eigenvalue problems: modern theory and computational practice
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| David S. Watkins |
| Department of Mathematics |
| Washington State University |
| Pullman, WA 99164-3113 |
| USA |
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| This course will provide a coherent, unified overview of modern methods for solving
eigenvalue problems, with emphasis on methods that preserve various structures.
Both theory and computational practice will be addressed.
Brief tentative contents: review of basic concepts; overview of important matrix stuctures;
numerical methods for small, medium, and large eigenvalue problems,
including the QR and related algorithms and Krylov subspace methods;
product eigenvalue problems; Hamiltonian, skew-Hamiltonian, symplectic,
and palindromic eigenvalue problems. |
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