Solution of an integer programming problem related to convergence of rows of Padé approximants (Q1181525)

An algorithm is developed by which the maximization of the expression \(\sum^ r_{i=1}(\omega_ i\sigma_ i-\sigma^ 2_ i)\), subject to the constraints \(\sum^ r_{i=1}\sigma_ i=\tau\), \(0\leq\sigma_ i\leq\omega_ i\), \(i=1,\ldots,r\), can be constructed in a noniterative fashion \((\omega_ i,\sigma_ i,\tau\) are integers, \(\omega_ i\) and \(\tau\) are given, \(\sigma_ i\) are integer unknowns). This article is related to that by the second author [J. Comput. Appl. Math. 29, No. 3, 257-291 (1990; Zbl 0693.41015)] in which the convergence of the so called intermediate rows of the Padé table for meromorphic functions was analyzed. The algorithm is based on a sequence of at most \(r-1\) reductions which decrease the dimension of the problem. It also enables to decide whether the solution is unique, and in case of nonuniqueness it provides all possible solutions.