By Jozef Mikloško, Vadim Evgenich Kotov (auth.), Jozef Mikloško, Vadim Evgenich Kotov (eds.)

Both algorithms and the software program . and of automated desktops have undergone a quick improvement some time past 35 years. The dominant consider this improvement was once the improvement in computing device expertise. machine parameters have been systematically superior via electron tubes, transistors and built-in circuits of ever-increasing integration density, which additionally prompted the improvement of latest algorithms and programming tools. a few years in the past the placement in desktops improvement was once that no extra enhancement in their functionality might be completed by means of expanding the rate in their logical components, as a result of actual barrier of the utmost move velocity of electrical indications. one other enhancement of laptop functionality has been completed via parallelism, which makes it attainable by means of an appropriate association of n processors to acquire a practice­ ance elevate of as much as n instances. examine into parallel computations has been conducted for a number of years in lots of nations and plenty of result of basic value were received. Many parallel pcs were designed and their algorithmic and application­ ming structures outfitted. Such desktops comprise ILLIAC IV, DAP, STARAN, OMEN, STAR-100, TEXAS tools ASC, CRAY-1, C mmp, CM*, CLIP-3, PEPE. This development is supported via the truth that: a) many algorithms and courses are hugely parallel of their constitution, b) the recent LSI and VLSI applied sciences have allowed processors to be mixed into huge parallel buildings, c) better and larger calls for for velocity and reliability of desktops are made.

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And KUNG, H. : The complexity of parallel evaluation of linear recurrences. Tech. Report, IRIA 1975, Rocquencourt, 96. 58 [11] KoGGE, P. : Parallel solution of recurrence problems. IBM J. Res. , 18, 1974, 138-148. [12] KucK, D. : Multioperation machine computational complexity. Proc. Complexity of Sequential and Parallel Numerical Algorithms. J. E. Traub (Editor). Academic Press, New York, 1973, pp. 17--47. [13] KucK, D. : Parallel processing or ordinary programs. Advances in Computers, Vol. 15.

This problem can only be speeded up by the constant. Let us consider, for example, the known relation for the computation of (i = 1, 2, ... , n), Va (8) which is of second degree. Let us assume that we calculate with real numbers and every arithmetic operation requires a time U. e. fn l U. Thus, the speedup of relation (8) is at most three-fold for every k and n. No parallel algorithm using an arbitrary number of processors can be essentially faster than the serial procedure in relation (8), which is completely different from linear relations which have a speedup O(nllog n).

Its disadvantage is also in its narrow applicability in numerical mathematics and in its restriction to iterative methods. Therefore, from the aspect of the effectiveness of the implementation and the range of application, the so-called asynchronous implementation of algorithms is preferable. We give below only some results from [2], making use of some facts and ideas from Chapter 12 of this book. Suppose we have to solve a task that requires sequences of n subtasks w~, w2 , ••. , Wn under the following conditions: c1: subtask W;, i = 2, 3, ...

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