Bulletin of the Japan Society for Industrial and Applied Mathematics

Network resource allocation is a central issue in modern networks. The main objective of the allocation is to share the available resource among users in the network so as to maximize the sum of their utilities subject to the feasible region for allocating the resource. In this paper, we discuss a network allocation problem in which the constraint set composed of the absolute set and the subsidiary sets is not feasible. We formulate a compromise solution of the problem by using a maximizer of the objective function over a subset of the absolute set with the elements closest to the subsidiary sets in terms of the norm. We propose fixed point optimization algorithms, based on iterative techniques for optimization over the fixed point set of a certain nonexpansive mapping, for solving the problem and perform convergence analyses on them. We apply the algorithms to concrete network resource allocation problems such as power and bandwidth allocation and provide numerical examples for these problems.

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This paper is concerned with interval arithmetic, especially, an enclosure of a matrix product is focused on. Using level 3 operations of matrix computations, an algorithm outputting a tight enclosure for matrix multiplication is proposed. Most of the algorithms for this purpose require switches of rounding modes defined in the IEEE standard 754. However some programing enviroments have not supported them. Our proposed method demands only rounding-to-nearest mode, so that it is very portable.

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BNCpack is a multiple precision numerical computation library built on the functions provided by MPFR/GMP which is currently fastest and the most reliable multiple precision floating-point arithmetic library. It can handle complex arithmetic, basic linear computation, polynomial computation, root finding, numerical integration, and initial value problem of ordinary differential equation in double precision and in multiple and mixed precision. Parts of these functions such as basic linear computation, the Durand-Kerner method and numerical integration are parallelized by using a message-passing interface (MPI) to obtain higher performance using parallelized programs provided by BNCpack on MPI PC clusters. In this paper, we first introduce MPFR/GMP and its functions. Second, we explain the numerical algorithms implemented in BNCpack and the method for parallelizing numerical computation using MPFR/GMP. Third, we describe benchmark tests performed using BNCpack to demonstrate its effectiveness. Finally, we discuss future works on BNCpack.

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