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 well-known result of Bernstein states that a Chebyshev quadrature formula of the form $$\int\limits_{ - 1}^1 {f(x) dx = \frac{2}{n}\sum\limits_{k - 1}^n {f(t_k ) + R_n (f), t_k real,} }$$ can have algebraic degree of exactnessp=n only if 1?n?7 orn=9. The nodest k are necessarily symmetric with respect to the origin, so that in factp=2[n/2]+1. If symmetry of the nodes is imposed, it is known from work of Gautschi and Yanagiwara and others that next-to-highest algebraic degreep=2[n/2]?1, beyond the classical cases above, can be attained only whenn=8, 10, 11, and 13. For these values ofn, "optimal" formulas have been obtained which minimize |R n (x p+1)| among all symmetric Chebyschev quadratures of degreep=2[n/2]?1. We show here that these same formulas in fact minimize |R n (x i )| for eachi?p+1.

On optimal Chebyshev-type quadratures

Procedures and corresponding Matlab software are presented for generating Gauss---Turan quadrature rules for the Laguerre and Hermite weight functions to arbitrarily high accuracy. The focus is on the solution of certain systems of nonlinear equations for implicitly defined recurrence coefficients. This is accomplished by the Newton---Kantorovich method, using initial approximations that are sufficiently accurate to be capable of producing n-point quadrature formulae for n as large as 42 in the case of the Laguerre weight function, and 90 in the case of the Hermite weight function.

High-precision Gauss---Turán quadrature rules for Laguerre and Hermite weight functions

This c,'~se cau have a bint~ry coded table consisting of two rows. The first row, s, is used to keep a record of which sequence value has been generated by using the second row. Let t~ = p~ if Pi ¢ O; otherwise let tg = 1. Let J3.~: = 1 for j = t~, t!+t2,... ,t~+ t2+ \"'\" + &. Let B2y = 1 for j = 1+ ~2}t~,... ,pv+ ~-~ & where k = 1, 2, ... , m. The s row contains m bits while the second row contains ~'~:, p~ bits. This method requires 2 ~*~*_J t, bit positions of memory. For an example of this method, Sequence B can have the binary coded table given in Table I by using Rows 8 and 9, where 8 is the s row. In this case the coding to find x,, , given n, is done so that the x0 accumulator is incremented by C according to the second row, to the nth bit in the s row. If p~ ~ 0 for i = 1, 2, ... , m tile second row would not have a blank position and therefore by suitable programming would not be necessary. sequence could have the same binary coded table as d~ = p~C and p~ >_ 0 for i = 1, 2, ..-, m if in the program D was added to the x0 accumulator for each B,y = 1 where j =< n. is possible to use one row when binary coding q sequences using Method II. Let L~ = max(hi,-, tq0 where ti~ is t~ of the jth sequence. comment This procedure will generate successive \"prime twins,\" i.e. pairs of primes Twin1, Twin2 which differ by 2. Storage is the maximum number of primes that can be stored. Act is any procedure for recording, examining, or utilizing each pair of twins as it is generated, t is a serial number for the twins. P[Storage] T 2 is the last number examined; begin integer array P[I: Storage]; integer j, m, previous, current; comment P[j] is the jth prime; P[1] := 2; P[2] := 3; j := 2; previous := 3; t := 0; for current := 5 step 2 until P[j] X P[j] do begin m := 1; for m := m q-1 while P[m] X P[m] _< current do if current = (current + P[m]) X P[m] then go …

Certification of algorithm 222: Incomplete beta function ratios

We now prove separately the inequality for every term of the summation. Let us assume the thesis is wrong > ([(XLk+:-Xi,k) 2 + (yLk+:-yl.~)~] t \"Jr [(X2.k+x-X2,k) 2 + (y2,k+l-Y2,k)2]t). which is contradictory. We can have equality to zero iff i.e. if all the corresponding sides of the two polygons are parallel. (e, d) We must prove that the minimal polygon has at least one vertex on the boundary of the corresponding domain. In fact, consider a minimal polygon: there will be at least a couple of adjaeent noncollinear sides. The corresponding vertex must lie on the boundary, otherwise a shorter perimeter polygon could be found (see Figure 7 (a), (b)). Furthermore, from simple geometrical considerations it can be deduced that the normal to the boundary must bisect the angle between the two adjacent sides (Figure 7 (b)). Conversely, if a constraint is not active, i.e. a vertex is not on the boundary, the adjacent sides in the minimal polygon must be collinear. (e) From the convexity properties of the domain D and of the function f it can be deduced that: (i) all local minima are global; (ii) any convex linear combination of minima is a minimum as well; (iii) Jensen's relation (3) applied to two minima obviously holds with equality. We have proved in (b) that in our ease Jensen's relation holds with equality if and only if the corresponding sides of the two polygons are parallel. Now assume to have found a (global) minimum: any vertex between two noncollineax sides will satisfy the bisection property proved in (d). If no straight line segments axe present on the boundary of domain C (e.g. if domain C is a circle), it is not possible to translate two adjacent noncollinear sides still satisfying the bisection property. Therefore, all minimal polygons have the vertices between two noncollinear sides in the same position. Thus they differ only in the position of vertices corresponding to nonactive constraints: the minimal reduced polygon is unique. * The work forms part of a research program supported by the Bundesministerium fiir wissenschaftliche Forsehung and the Fritz ter Meer-Stiftung. c o m m e n t If a Laplace transform P(s) is given in the form of a real procedure, L/nv produces an approximate value Fa of the inverse F(t) at T. Fa is evaluated according to Fa =-~-~=: N must be even. Since the V~ depend on …

Walter Gautschi

Papers

On optimal Chebyshev-type quadratures

High-precision Gauss---Turán quadrature rules for Laguerre and Hermite weight functions

Certification of algorithm 222: Incomplete beta function ratios

Remark on algorithm 282 [S22] derivatives of ex/x, cos(x)/x, and sin (x)/x

Reminiscences of my involvement in de Branges’s proof of the Bieberbach conjecture