Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

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Iterative Methods for Sparse Linear Systems

In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed Orthogonal polynomials on the unit circle are not discussed

Orthogonal Polynomials

Introduction to support vector learning roadmap. Part 1 Theory: three remarks on the support vector method of function estimation, Vladimir Vapnik generalization performance of support vector machines and other pattern classifiers, Peter Bartlett and John Shawe-Taylor Bayesian voting schemes and large margin classifiers, Nello Cristianini and John Shawe-Taylor support vector machines, reproducing kernel Hilbert spaces, and randomized GACV, Grace Wahba geometry and invariance in kernel based methods, Christopher J.C. Burges on the annealed VC entropy for margin classifiers - a statistical mechanics study, Manfred Opper entropy numbers, operators and support vector kernels, Robert C. Williamson et al. Part 2 Implementations: solving the quadratic programming problem arising in support vector classification, Linda Kaufman making large-scale support vector machine learning practical, Thorsten Joachims fast training of support vector machines using sequential minimal optimization, John C. Platt. Part 3 Applications: support vector machines for dynamic reconstruction of a chaotic system, Davide Mattera and Simon Haykin using support vector machines for time series prediction, Klaus-Robert Muller et al pairwise classification and support vector machines, Ulrich Kressel. Part 4 Extensions of the algorithm: reducing the run-time complexity in support vector machines, Edgar E. Osuna and Federico Girosi support vector regression with ANOVA decomposition kernels, Mark O. Stitson et al support vector density estimation, Jason Weston et al combining support vector and mathematical programming methods for classification, Bernhard Scholkopf et al.

Advances in kernel methods: support vector learning

(2005). Applied Multivariate Statistical Analysis. Technometrics: Vol. 47, No. 4, pp. 517-517.

Applied Multivariate Statistical Analysis

The electronic ground state of a periodic system is usually described in terms of extended Bloch orbitals, but an alternative representation in terms of localized "Wannier functions" was introduced by Gregory Wannier in 1937. The connection between the Bloch and Wannier representations is realized by families of transformations in a continuous space of unitary matrices, carrying a large degree of arbitrariness. Since 1997, methods have been developed that allow one to iteratively transform the extended Bloch orbitals of a first-principles calculation into a unique set of maximally localized Wannier functions, accomplishing the solid-state equivalent of constructing localized molecular orbitals, or "Boys orbitals" as previously known from the chemistry literature. These developments are reviewed here, and a survey of the applications of these methods is presented. This latter includes a description of their use in analyzing the nature of chemical bonding, or as a local probe of phenomena related to electric polarization and orbital magnetization. Wannier interpolation schemes are also reviewed, by which quantities computed on a coarse reciprocal-space mesh can be used to interpolate onto much finer meshes at low cost, and applications in which Wannier functions are used as efficient basis functions are discussed. Finally the construction and use of Wannier functions outside the context of electronic-structure theory is presented, for cases that include phonon excitations, photonic crystals, and cold-atom optical lattices.

Maximally-localized Wannier Functions: Theory and Applications

where f is a function whose derivatives can be formed readily. Analytic differentiation in (1.1), while elementary, is obviously tedious, and the resulting expressions are of doubtful practical value. In the following we present a simple and effective recursive algorithm to generate these derivatives. As an example, we consider the cases where f(z) = eZ, f(z) = cos z, and f(z) = sin z. Our main observation may be paraphrased in the following surprising way. The calculation of a large number of derivatives (1.1) at a fixed point z is a stable process if the function g(?) = f(r)/? has a pole at r = 0, and an unstable process if g(?) is regular at r = 0.

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Computation of Successive Derivatives of f(z)/z*

The generation of generalized Gauss–Radau and Gauss–Lobatto quadrature formulae by methods developed by us earlier breaks down in the case of Jacobi and Laguerre measures when the order of the quadrature rules becomes very large. The reason for this is underflow resp. overflow of the respective monic orthogonal polynomials. By rescaling of the polynomials, and other corrective measures, the problem can be circumvented, and formulae can be generated of orders as high as 1,000.

High-order generalized Gauss–Radau and Gauss–Lobatto formulae for Jacobi and Laguerre weight functions

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Leonhard Eulers Umgang mit langsam konvergenten Reihen

Algorithms are developed for computing the coefficients in the three-term recurrence relation of repeatedly modified orthogonal polynomials, the modifications involving division of the orthogonality measure by a linear function with real or complex coefficient. The respective Gaussian quadrature rules can be used to account for simple or multiple poles that may be present in the integrand. Several examples are given to illustrate this.

Repeated modifications of orthogonal polynomials by linear divisors

Inequalities recently conjectured for all zeros of Jacobi polynomials \(P_n^{(\alpha,\beta)}\) of all degrees n are modified and conjectured to hold (in reverse direction) in considerably larger domains of the (α,β)-plane.

Walter Gautschi

Papers

Computation of Successive Derivatives of f(z)/z*

High-order generalized Gauss–Radau and Gauss–Lobatto formulae for Jacobi and Laguerre weight functions

Leonhard Eulers Umgang mit langsam konvergenten Reihen

Repeated modifications of orthogonal polynomials by linear divisors

New conjectured inequalities for zeros of Jacobi polynomials