Acta Numerica 1997 (Volume 6) by Arieh Iserles

By Arieh Iserles

Acta Numerica surveys every year an important advancements in numerical research. the topics and authors, selected by way of a individual foreign panel, supply a survey of articles striking of their caliber and breadth. This quantity contains articles on multivariate integration; numerical research of semiconductor units; quickly transforms in utilized arithmetic; complexity matters in numerical research.

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COOLS J. N. Lyness and E. de Doncker (1993), 'Quadrature error expansions, II. The full corner singularity', Numer. Math. 64, 355-370. J. N. Lyness and E. de Doncker-Kapenga (1987), 'On quadrature error expansions, part F, J. Comput. Appl. Math. 17, 131-149. J. N. Lyness and D. Jespersen (1975), 'Moderate degree symmetric quadrature rules for the triangle', J. Inst. Math. Appl. 15, 19-32. J. N. Lyness and B. J. J. McHugh (1970), 'On the remainder term in the Ndimensional Euler-Maclaurin expansion', Numer.

1 we obtained a lower bound for the number of points N of a cubature formula that is exact on a vector space of functions V^. 1, depends only on V^, restricted to Q. In this section we will see that this bound is in general too low for odd degrees d. Higher lower bounds have to take into account more information on the region $7 and weight function u;(x). 1, and let 21 be the corresponding ideal. Then H(fc;a) = dimPJJ = N = W(d;»). Hence the ideal contains dim / P^ + 1 — dim'PJJ linearly independent polynomials of degree k + 1.

Let / € 21 U V2- Then, with #, as given in the definition of H-basis, Schmid managed to give a characterization of cubature formulae with real points and positive weights using real ideals. 8 Let ,Rt} C Vj+i be a set of linearly independent d-orthogonal polynomials that is fundamental of degree d + 1. , Rt}. Let N + t = d i m P ^ i and U an arbitrary but fixed vector space such that V^+i = V®U- Then the following statements are equivalent. ,y( JV )}c NG(B). 21 and U are characterized by: (i) 2int/ = (ii) I[f2 - R+] > 0 for all / e U, where R+ G 21 is chosen such that f-R+e Vnd.

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