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

A Conjectured Inequality for Hermite Interpolation at the Zeros of Jacobi Polynomials

INERFC: Evaluation of the Repeated Integrals of the Coerror Function by Half-Range Gauss-Hermite Quadrature

All positive zeros of subrange Jacobi polynomials, orthogonal on [−c, c], 0 < c < 1, with respect to the weight function w(x) = (1 − x)
 α
 (1 + x)
 β
 , α > − 1, β > − 1, are shown in the ultraspherical case α = β, and partly conjectured in the general case α < β, to be monotonically increasing as functions of c.

On the zeros of subrange Jacobi polynomials

Series expansions are given which are rapidly convergent when x is small. These are derived by noting the relation to theta functions and applying Poisson- s transformation formulas which convert small arguments into large ones.

Diffusion Functions for Small Argument

There are two easy ways in Matlab to construct contour plots of analytic functions, i.e., lines of constant modulus and constant phase. One is to use the Matlab contour command for functions of two variables, another to solve the differential equations satisfied by the contour lines. This is illustrated here for the function f(z) = e n(z), 

$$ e_n \left( z \right) = 1 + z + \frac{{z^2 }} {{2!}} + \ldots + \frac{{z^n }} {{n!}} $$

(25.1)

where is the nth partial sum of the exponential series. The lines of constant modulus 1 of e n are of interest in the numerical solution of ordinary differential equations , where they delineate regions of absolute stability for the Taylor expansion method of order n and also for any n-stage explicit Runge-Kutta method of order n, 1 ≤ n ≤ 4 (cf. [4, §9.3.2]).

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Walter Gautschi

Papers

A Conjectured Inequality for Hermite Interpolation at the Zeros of Jacobi Polynomials

INERFC: Evaluation of the Repeated Integrals of the Coerror Function by Half-Range Gauss-Hermite Quadrature

On the zeros of subrange Jacobi polynomials

Diffusion Functions for Small Argument

Contour plots of analytic functions