Differentiation algorithm based on the additional variables method

Symbolic, automatic and numerical differentiation are widely used in various fields of applied mathematics. Symbolic differentiation transforms the formula representation of a function into a formula representation for its derivative. By contrast, automatic differentiation is calculation of the derivative of a function (a computer program), and the result is again a function (a computer program). Numerical differentiation utilizes approximation formulas to derivatives derived by differentiating polynomial approximations of functions given at a few points. All three types of differentiation techniques are complementary and each has its own advantages and drawbacks. A major advantage of symbolic differentiation is that, in principle, evaluation of formulas gives exact values of the derivatives of the function. Its major drawback is that it may generate very complicated expressions containing many unnecessary instances of the same sub-expressions. It is for this reason that symbolic differentiation works well for simple expressions but necessary computation time and memory grow rapidly as a function of an expression size. In the present paper, we propose the new symbolic differentiation algorithm which is free from the drawback mentioned. It is applicable to the class of finite compositions of functions satisfying total system of differential equations with polynomial right-hand sides (note that an ODE system is a special case of a total system). In particular, this class includes four basic arithmetic operations, elementary functions and a wide variety of special functions of mathematical physics. Libraries of functions are used. To be more specific, libraries include names of functions and total polynomial systems of differential equations satisfied by these functions. Using these differential equations from the library to introduce a number of additional variables, the algorithm transforms a system of functions of several variables to the system of polynomials in original and additional variables, and then finds derivatives of additional variables as polynomials in the same variables. Derivatives of functions of the original system could be represented then as recurrent or explicit formulas (polynomials in the same variables too) using polynomial representations of the original functions and derivatives of additional variables. Bibliogr. 15. Il. 1. Table 3.