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6. Conclusions
One of the purposes of this text was to demonstrate the fundamental role the digital feedback-loop principle plays in computer arithmetic. We chose fundamental problems in algebra namely: solving linear system of algebraic equations and computing real roots of polynomials. The former is a typical example of an explicit function computation, the latter represents a method of an implicit function computation. As a result it is possible to build a recurrent sequence of values for the residuals in the algorithm of solving the linear systems of algebraic equations. On the contrary, in the algorithm of extracting polynomials' roots the consecutive values of the residuals do not form a recurrent sequence, i.e. - the old value of the residual is not used to compute the new value at each iteration step. Nevertheless the algorithm of computing roots of polynomials utilizes the digital feedback-loop principle the way it was described at the end of the pr evious section.
A new text treating a formal theoretical model of the digital feedback-loop principle in computer arithmetic for radix-2 number system is under preparation. We called this model a division scheme.
On the other hand the limits of application of the digit-recurrence ( or shift-and-add, or digit-by-digit ) approach for developing new algorithms of computer arithmetic have never been discussed in literature. In his dissertation [14] B. G. De Lugish noticed that "the generalization and extension of the "normalization" technique presents, probably, the most interesting topic for further investigation ... It is not known what general class of functions may be evaluated through such a formulation" ( i. e. formulation based on multiplicative or additive representation of arguments or functions). More then thirty years later we must admit that this is still an open problem for further research.
Lee and Morf [15] proposed an ambitious plan to develop generalized CORDIC algorithms. As far as we know S.-F. Hsiao and J. -M. Delosme followed their guideline to offer the Householder CORDIC method [2]. But true generalized CORDIC methods for computation of orthogonal systems of special functions have not been developed yet.
We suspect that perhaps there exists no systematic way to describe all operations and functions that can be computed using additive or multiplicative representation of numbers. Nevertheless a theoretical model formalizing the digital feedback-loop principle might be useful in this area of future research.
