Walter Gautschi

Bio: Walter Gautschi is an academic researcher from Purdue University. The author has contributed to research in topics: Orthogonal polynomials & Gauss–Kronrod quadrature formula. The author has an hindex of 38, co-authored 208 publications receiving 8766 citations. Previous affiliations of Walter Gautschi include ETH Zurich & Oak Ridge National Laboratory.

Orthogonal polynomials : computation and approximation

Abstract: BASIC THEORY 11 Orthogonal polynomials 12 Properties of orthogonal polynomials 13 Three-term recurrence relation 14 Quadrature rules 15 Classical orthogonal polynomials 16 Kernal polynomials 17 Sobolev orthogonal polynomials 18 Orthogonal polynomials on the semicircle 19 Notes to chapter 1 COMPUTATIONAL METHODS 21 Moment-based methods 22 Discretization methods 23 Computing Cauchy integrals of orthogonal polynomials 24 Modification algorithms 25 Computing Sobolev orthogonal polynomials 26 Notes to chapter 2 APPLICATIONS 31 Quadrature 32 Least squares approximation 33 Moment-preserving spline approximation 34 Slowly convergent series 35 Notes to chapter 3

Numerical analysis: an introduction

Abstract: This is a text in numerical analysis which is taken to mean the branch of mathematics that develops and analyzes computational methods dealing with problems arising in classical analysis, approximations theory, the theory of equations, and ordinary differential equations. The topics in this book are presented with a view towards stressing basic principles and maintaining simplicity and teachability as far as possible. Topics that require a level of technicality that goes beyond the standard of simplicity imposed are referenced in bibliographic notes at the end of each chapter. This book does not cover numerical linear algebra, nor the numerical solution of partial differential equations, as the author takes the view that these are now separate disciplines. It is intended that the student has a good background in calculus and advanced calculus and some knowledge of linear algebra, complex analysis, and differential equations.

Numerical integration of ordinary differential equations based on trigonometric polynomials

Abstract: There are many numerical methods available for the step-by-step integration of ordinary differential equations. Only few of them, however, take advantage of special properties of the solution that may be known in advance. Examples of such methods are those developed by BROCK and MURRAY [9], and by DENNIS Eg], for exponential type solutions, and a method by URABE and MISE [b~ designed for solutions in whose Taylor expansion the most significant terms are of relatively high order. The present paper is concerned with the case of periodic or oscillatory solutions where the frequency, or some suitable substitute, can be estimated in advance. Our methods will integrate exactly appropriate trigonometric polynomials of given order, just as classical methods integrate exactly algebraic polynomials of given degree. The resulting methods depend on a parameter, v=h~o, where h is the step length and ~o the frequency in question, and they reduce to classical methods if v-~0. Our results have also obvious applications to numerical quadrature. They will, however, not be considered in this paper. 1. Linear functionals of algebraic and trigonometric order In this section [a, b~ is a finite closed interval and C ~ [a, b~ (s > 0) denotes the linear space of functions x(t) having s continuous derivatives in Fa, b~. We assume C s [a, b~ normed by s (t.tt IIxll = )2 m~x Ix~ (ttt. a=0 a~t~b

Adaptive Quadrature—Revisited

Abstract: First, the basic principles of adaptive quadrature are reviewed. Adaptive quadrature programs being recursive by nature, the choice of a good termination criterion is given particular attention. Two Matlab quadrature programs are presented. The first is an implementation of the well-known adaptive recursive Simpson rule; the second is new and is based on a four-point Gauss-Lobatto formula and two successive Kronrod extensions. Comparative test results are described and attention is drawn to serious deficiencies in the adaptive routines quad and quad8 provided by Matlab.

Iterative Methods for Sparse Linear Systems

Abstract: 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.

Orthogonal Polynomials

Abstract: 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

Advances in kernel methods: support vector learning

Abstract: 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.

Applied Multivariate Statistical Analysis

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

Maximally-localized Wannier Functions: Theory and Applications

Abstract: 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.