AOL ARITHMETICA ONLINE

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Introduction

Linear systems

Division

Polynomials

Conclusions

References

The on-line computer arithmetic was invented in 1975 by M.D.Ercegovac and K.S.Trivedi, who proposed on-line algorithms for the basic arithmetic operations: addition, multiplication and division of numbers. There was a considerable interest in this area of a theoretical research in the eighties. New algorithms for on-line computations were developed and some efforts to start their practical implementation were made. In the course of time the on-line computer arithmetic seemed to have lost its attractiveness for the researchers. The rapid development of the new technologies keeps the cost-effectiveness ratio of the on-line algorithms unacceptably high, perhaps with the only exception of the on-line addition/subtraction of sign-digit numbers.

We believe in the following two points: first of all the on-line computer arithmetic contributed significantly to better understanding the nature of redundant numbers computations both from a theoretical and a practical point of view. Introducing a redundant representation of numbers is beneficial to computer arithmetic algorithms because either the speed of computations or their fault-tolerance can be improved.

On the other hand the synthesis of on-line algorithms still is an open area. Despite the fact that such algorithms can be completely inconvenient for practical implementation they offer the opportunity to obtain some new interesting and even rather exotic theoretical results.

We took courage to prepare the following text not only for the sake of having fun but also to demonstrate the fact that using a redundant representation of numbers in numerical computations has always been a taboo for the professional mathematicians.