Acta Numerica 1996 (Volume 5) by Arieh Iserles

By Arieh Iserles

The 5th quantity of Acta Numerica offers "state of the paintings" research and methods in numerical arithmetic and medical computing. This assortment encompasses a number of very important points of numerical research, together with eigenvalue optimization; thought, algorithms and alertness of point set tools for propagating interfaces; hierarchical bases and the finite point process. it is going to be a precious source for researchers during this vital box.

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Extra resources for Acta Numerica 1996 (Volume 5)

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11) can be arbitrary. It is convenient, however, to define it by Po = (TTOJ^O) = / dA(i). 13) Note that by construction, TTJV is orthogonal to all polynomials of degree < N. If dA = dAjv is a discrete measure with exactly N points of increase, there can be at most N orthogonal polynomials, TTQ, TTI, . . , TTJV­I, which implies that (ITN^N) = 0, that is, TTJV vanishes at all the support points of d\N. 11) holds for all k G No, with f}k > 0, then by a well­known 52 W. GAUTSCHI theorem of Favard (see, for instance, Natanson 1964/65, Volume II, Chapter VIII, Section 6) the system of polynomials {TT^} is orthogonal relative to some positive measure dA having infinitely many support points.

For surveys, see Monegato (1982), Gautschi (1988) and Notaris (1994). (4) Consider s > 1 different measures dXa, a = 1,2,... 1) with a common set of nodes {TU} but individual weights { A ^ a } , a = 1,2,... ,s. Assume n = ms to be an integer multiple of s. Find s such quadrature rules, each having degree of exactness n—l+m. ,s. 8) un(t)p{t) d\a(t) = 0, a l l p G P m ­ i , JR One obtains the shared-nodes quadrature rules recently introduced by Borges (1994) in connection with computer graphics illumination models, where the models dACT are colour matching functions.

3) holds for any weight function w since the underlying quadrature rule turns out to be simply the (n + l)­point Gaussian rule for w (all nodes Tj receive the weight zero). Another interesting choice for the nodes tj, first proposed by Bellen (1981, 1988), is the set of zeros of 7r n+ i( ) = 7rn_|_i( ; TT^IU), = 0, j = 1,2, . . , n + l ( 7 r n ( ­ ) = nn(-;w)). 5) Here the polynomial 7rn+i is the (n + l)st­degree polynomial of an infinite sequence of polynomials nm( ; TT^IO), m = 0,1, 2 , . .

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