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

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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.

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  1. Institut für Angewandte Mathematik und Statistik der Universität Würzburg, Am Hubland, 8700, Würzburg, FRG

    M. v. Golitschek

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Supported by Deutsche Forschungsgemeinschaft Grant No. GO 270/3

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v. Golitschek, M. Optimal cycles in doubly weighted graphs and approximation of bivariate functions by univariate ones. Numer. Math. 39, 65–84 (1982). https://doi.org/10.1007/BF01399312

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