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Digitale Medien

The ɛ-algorithm and multivariate Padé-approximants (1982)

Cuyt, Annie A. M.

Springer

Numerische Mathematik 40 (1982), S. 39-46

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ISSN: 0945-3245

Schlagwort(e): AMS(MOS): 65D15 ; CR: 5.13

Quelle: Springer Online Journal Archives 1860-2000

Thema: Mathematik

Notizen: Summary In the univariate case the ɛ-algorithm of Wynn is closely related to the Padé-table in the following sense: if we apply the ɛ-algorithm to the partial sums of the power series $$f(x) = \sum\limits_{i = 0}^\infty {c_i x^i } $$ then ε 2m l−m is the (l, m) Padé-approximant tof(x) wherel is the degree of the numerator andm is the degree of the denominator [1 pp. 66–68]. Several generalizations of the ɛ-algorithm exist but without any connection with a theory of Padé-approximants. Also several definitions of the Padé-approximant to a multivariate function exist, but up till now without any connection with the ɛ-algorithm. In this paper, we see that the multivariate Padé-approximants introduced in [3], satisfy the same property as the univariate Padé-approximants: if we apply the ɛ-algorithm to the partial sums of the power series $$f\left( {x_1 ,...,x_n } \right) = \sum\limits_{i_1 + ... + i_n = 0}^\infty {c_{i_1 ...i_n } x_1^{i_1 } ...x_n^{i_n } } $$ then ε 2m (l−m) is the (l, m) multivariate Padé-approximant tof(x 1, ...,x n ).

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1007/BF01459073

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Digitale Medien

Solving complex approximation problems by semiinfinite - finite optimization techniques: A study on convergence (1982)

Opfer, Gerhard

Springer

Numerische Mathematik 39 (1982), S. 411-420

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ISSN: 0945-3245

Schlagwort(e): AMS(MOS): 65D15 ; CR: 5.13

Quelle: Springer Online Journal Archives 1860-2000

Thema: Mathematik

Notizen: Summary We transform a complex approximation problem into an equivalent semiinfinite optimization problem whose constraints are described in terms of a quantity ϕ∈[0,2π[=I. We study the effect of disturbing the problem by replacingI by a compact subsetM⊂I which includes as special case the discrete case whereM consists only of finitely many points. We introduce a measure ɛ for the deviation ofM fromI and show that in any complex approximation problem the minimal distance of the disturbed problem converges quadratically with ɛ→0 to the minimal distance of the undisturbed problem which is a generalization of a result by Streit and Nuttall. We also show that in a linear finite dimensional approximation problem the convergence of the coefficients of the disturbed problem is in general at most linear. There are some graphical representations of best complex approximations computed with the described method.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1007/BF01407872

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Digitale Medien

Quotienten-Differenzen-Algorithmus: Beweis der Regeln von Rutishauser (1982)

Seewald, Wolfgang

Springer

Numerische Mathematik 40 (1982), S. 93-98

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ISSN: 0945-3245

Schlagwort(e): AMS(MOS): 65D15 ; CR: 5.15

Quelle: Springer Online Journal Archives 1860-2000

Thema: Mathematik

Beschreibung / Inhaltsverzeichnis: Zusammenfassung Der Quotienten-Differenzen-Algorithmus nach Rutishauser ist geeignet zur Bestimmung von Polen meromorpher Funktionen, gegeben durch eine Taylorreihe. Sind mehrere Pole betragsgleich, so kann eine Polynomfolge bestimmt werden, deren Grenzpolynom diese Pole als Nullstellen hat. Die Konvergenz wurde von Rutishauser jedoch nicht bewiesen. Ein Beweis wird in der vorliegenden Arbeit präsentiert.

Notizen: Summary The Quotient-Difference Algorithm of Rutishauser can be used for the determination of poles of a meromorphic function given by its power series. If some of the poles have same modulus, a sequence of polynomials can be determined such that the limiting polynomial has exactly these poles as zeros. The convergence has not been proved by Rutishauser, however. A proof is presented in this paper.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1007/BF01459078

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Digitale Medien

Unicity of best one-sidedL 1-approximations (1982)

Strauß, Hans

Springer

Numerische Mathematik 40 (1982), S. 229-243

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ISSN: 0945-3245

Schlagwort(e): AMS(MOS): 65D15 ; CR: 5.13

Quelle: Springer Online Journal Archives 1860-2000

Thema: Mathematik

Notizen: Summary This paper deals with the problem of uniqueness in one-sidedL 1-approximation. The chief purpose is to characterize finite dimensional subspacesG of the space of continuous or differentiable functions which have a unique best one-sidedL 1-approximation. In addition, we study a related problem in moment theory. These considerations have an important application to the uniqueness of quadrature formulae of highest possible degree of precision.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1007/BF01400541

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Digitale Medien

Optimal cycles in doubly weighted graphs and approximation of bivariate functions by univariate ones (1982)

Golitschek, M.

Springer

Numerische Mathematik 39 (1982), S. 65-84

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ISSN: 0945-3245

Schlagwort(e): AMS(MOS): 65D15 ; CR: 5.13

Quelle: Springer Online Journal Archives 1860-2000

Thema: Mathematik

Notizen: Summary The problem of finding optimal cycles in a doubly weighted directed graph (Problem A) is closely related to the problem of approximating bivariate functions by the sum of two univariate functions with respect to the supremum norm (Problem B). The close relationship between Problem A and Problem B is detected by the characterization (7.4) of the distance dist (f, t) of Problem B. In Part 1 we construct an algorithm for Problem A where the essential role is played by the minimal lengthsy j(k) defined by (2.3). If weight functiont≡1 then the minimum of Problem A is computed by equality (2.4). Ift≡1 then the minimum is obtained by a binary search procedure, Algorithm 3. In Part 2 we construct our algorithms for solving Problem B by following exactly the ideas of Part 1. By Algorithm 4 we compute the minimal pseudolengthsh k(y, M) defined by (7.5). If weight functiont≡1 then the infimum dist(f,t) of Problem B is obtained by equality (7.12) which is closely related to (2.4). Ift≢1 we compute the infimum dist(f,t) by the binary search procedure Algorithm 5. Additionally, Algorithm 4 leads to a constructive proof of the existence of continuous optimal solutions of Problem B (see Theorem 7.1e) which is already known in caset≡1 but unknown in caset≢1. Interesting applications to the steady-state behaviour of industrial processes with interference (Sect. 3) and the solution of integral equations (Problem C) are included.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1007/BF01399312

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