Numeric code optimization in computer algebra systems and recurrent relations technique

An important problem of symbolic-numeric interface is the optimization of computations generated by formulae that are obtained in computer algebra system [1]. This problem concerns not only the case of numeric code generation, because necessity in numeric computations can appear immediately in computer algebra system. It often means that large-scale scalar computations in cycles must be evaluated. Cycles that require numeric computation can appear in the program immediately as loop statements. For example in the time of symbolic-numeric integration [2]. Another case of the numeric cycle appearance is the case of 2D or 3D plotting by expressions that are obtained in the computer algebra system. The time for 3D plotting by large expressions is often unacceptable [3]. Optimization of computations in the cycles is the main problem that is considered in this paper. Computer algebra systems (such as Reduce, Maple, etc. ) have flexible tools for numeric programs generation [4, 5, 1]. Some optimizing transformations can be performed when the code has been generated. In SCOPE package for Reduce and in Maple these transformations consist in finding of common subexpressions of arithmetic expressions given and in the reduction of computational complexity on this basis. Specialized systems for code generation (such as AL PAL [6]) have more wide collection of optimizing transformations, including transformations of cycles. However, collection of cycles’ transformations in ALPAL consists of loop fusion and constant folding (or code motion in the terms of [7]) only. When numeric computations are performed in the