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

The modified moments of the distribution do(x) = xa In (1 /x) dx on [ 0, 1 ], with respect to the shifted Legendre polynomials, are explicitly evaluated. The result in the theorem of Section 2 of [1] can be generalized as follows: Let vn(a)=S10xaln(l/x)P*(x)dx, ot>-l, « = 0,1,2,..., where P*(x) = P„(2x 1) is the shifted Legendre polynomial of degree n. Then (1) \"„(«) = / ,N«-m mr(n m l)\\ . . (-1) -*■-\", a = m<n,m>0an integer, (n + m + l)\\ 1 . A / 1 1 \\/ A OL+l -k otherwise. _LJ_L+ /_J_i_\\?A a+lja+1 ^yot + l+k a + l-k/L^ ! OL+l+k The result in [1] is the case a = 0 of (1). For the proof, we note that «'„(a) = -2-<°+1>£i(l +t)aln(K(l + t))Pn(t)dt (2) = -2-<a + 1> Urn {£(1 + t)a ln(l + t)Pv(t)dt-In 2 •£ (1 + t)aPv(t)dj, where /'„(i) is the Legendre function of degree v. It is well known [2, p. 316, Eq. (15)] that J '>a + lp2/ i ,\\ (I + t)aP (t)dt =^ ^a>-l. -i U ; pW r(a + v + 2)r(a + 1 -1») ' Differentiating (3) with respect to a gives £(1 +ffln(l +QPv(t)dt (4) 2a + 1r2(a+l) = r^ + . + ^a+i-,)0\"^2^^1)-^^^^-^^1^ with i//(x) = r'(x)/r(x) the logarithmic derivative of the gamma function. The assertion (1) now follows by inserting (3) and (4) in (2) and by using the recurrence relations r(x + 1) = xT(x), ip(x + l)= 0(x) + 1/x, together with the fact that for any integer Received May 31, 1978. AMS (MOS) subject classifications (1970). Primary 33A65. © 1979 American Mathematical Society 0025-571 8/79/0000-0069/S01.50 742 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE PRECEDING PAPER BY JAMES L. BLUE 743 r>0, ^L±A -(-irV! ase-0. r(-r + e) v ' The method of proof also allows the evaluation of integrals of the form \"„,*(«) = So X\"[ln Wxy\\*Pn*(x) dx> by repeatedly differentiating (4) with respect to a. Department of Computer Sciences Purdue University West Lafayette, Indiana 47907 1. J. L. BLUE, \"A Legendre polynomial integral,\" Math. Comp., v. 33, 1979, pp. 739-741. 2. A. ERDÉLYI (Ed.), Tables of Integral Transforms, Vol. II, McGraw-Hill, New York, 1954. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

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On the preceding paper “A Legendre polynomial integral” by James L. Blue

Series are considered whose general term is a rational function multiplied by a fractional power. The summation of such series is reduced, via Laplace transformation techniques, to a problem of quadrature, which is then solved by Gaussian quadrature relative to Einstein and Fermi weight functions. A number of examples are worked out in detail.

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A class of slowly convergent series and their summation by Gaussian quadrature

An estimate is given for the condition number of the coordinate map associating to each polynomial its coefficients with respect to a system of orthogonal polynomials. Let w(x) > 0 be a weight function on the finite interval [a, b], and {pk(x)} k=O the associated orthogonal polynomials. We consider the linear parametrization map Mn: Rn -> P1which associates to each (real) vector UT = [u0, ui, , u l-] E R' the (real) polynomial p(x) = Jk1 Ukpk(X) C Pn-1. The object of this note is to estimate the condition condc Mn = JiMnjlo. jiMn11o. of the map Mn, the infinity norms in Rn being defined by ||ulK., = maxo5k<n-1 lukkl and in Pn-l by I IplI 1 = maxa,,,.,b Jp(x) . Letting ,b rb /o = f w(x) dx, hk = f p'(x)w(x) dx, k = 0, 1, 2, we show in fact that / 1/2 n-1 ()cond < Mn _ max AO max E IPk(X)I. O'k_n-1 hk a<x<b k=O For Chebyshev polynomials Pk(X) = Tk(x) on [-1, 1], e.g., this gives condc, Mn < 21/2n (Pk = Tk), while for Legendre polynomials Pk(X) = Pk(x) on [-1, 1] one gets condOc Mn ? n(2n 1)1/2 (Pk = Pk). In order to prove (1), we first observe that, for any u C Rn, n-1 n-1 I|MnUi|| = Ukpk(X) < I Iu I o. max Z Ipk(X)I , k=O a< xb k=O so that n-1 (2) $ Mnll < max E IPk(X)I. a<x<b k=O On the other hand, if Mn lp = u, then, by orthogonality, Received December 27, 1971. AMS 1970 subject classifications. Primary 33A65; Secondary 65G05.

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The condition of orthogonal polynomials

Approximation and Computation

A good portion of Gatteschi’s research publications—about 65%—is devoted to asymptotics of special functions and their zeros. Most prominently among the special functions studied figure classical orthogonal polynomials, notably Jacobi polynomials and their special cases, Laguerre polynomials, and Hermite polynomials by implication. Other important classes of special functions dealt with are Bessel functions of the first and second kind, Airy functions, and confluent hypergeometric functions, both in Tricomi’s and Whittaker’s form. This work is reviewed here, and organized along methodological lines.

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Walter Gautschi

Papers

On the preceding paper “A Legendre polynomial integral” by James L. Blue

A class of slowly convergent series and their summation by Gaussian quadrature

The condition of orthogonal polynomials

Approximation and Computation

Luigi Gatteschi's work on asymptotics of special functions and their zeros