This is a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are examined. A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented. Concrete applications to optimization problems, idempotent linear algebra and interval analysis are indicated. It is known that some nonlinear problems (and especially optimization problems) become linear over appropriate semirings with idempotent addition (the so-called idempotent superposition principle). This linearity over semirings is convenient for parallel computations. Key words and phrases: semirings, idempotent semirings, universal algorithms, generic programs, correspondence principle, superposition principle, optimization on graphs, linear algebra over semirings, interval analysis, parallel computations, harwdware and software design.

Abstract. This paper is a survey on universal algorithms for solving the matrix Bellman equations over semirings and especially tropical and idempotent semirings. However, original algorithms are also presented. Some applications and software implementations are discussed.

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A brief description and examples of using a program package for handling non-commutative variables in the REDUCE-2 system are presented. The package facilitates work with arbitrary commutation rules. The user may declare operators, variables, and procedures to be non-commutative, and introduce the rules for their rearrangement. Tools are included for reducing delta functions and Kronecker symbols in expressions involving non-commutative variables. During calculation, expressions are rearranged according to the rules introduced by the user, and similar terms are grouped together with explicit singling out of the non-commutative part.

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