Jozef Mikloško, Vadim Evgenich Kotov (auth.), Jozef's Algorithms, Software and Hardware of Parallel Computers PDF

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

ISBN-10: 3662111063

ISBN-13: 9783662111062

ISBN-10: 366211108X

ISBN-13: 9783662111086

Both algorithms and the software program . and of computerized pcs have passed through a fast improvement some time past 35 years. The dominant consider this improvement used to be the improvement in desktop expertise. desktop parameters have been systematically more advantageous 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 location in desktops improvement was once that no extra enhancement in their functionality should be completed through expanding the rate in their logical parts, as a result of actual barrier of the utmost move velocity of electrical signs. one other enhancement of computing device functionality has been completed by means of 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. study into parallel computations has been performed for numerous years in lots of nations and plenty of result of basic value were got. Many parallel desktops were designed and their algorithmic and software­ ming structures equipped. Such pcs comprise ILLIAC IV, DAP, STARAN, OMEN, STAR-100, TEXAS tools ASC, CRAY-1, C mmp, CM*, CLIP-3, PEPE. This pattern is supported by means of the truth that: a) many algorithms and courses are hugely parallel of their constitution, b) the hot LSI and VLSI applied sciences have allowed processors to be mixed into huge parallel constructions, c) higher and larger calls for for velocity and reliability of pcs are made.

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The inversion of a matrix of order N or the solution of a system of N linear equations with N unknows requires a minimum of s of data transfer steps. ;. N 2 • If the processors are connected in a two-dimensional regular network - each with four neighbours - then g(k) = 2k 2 + 2k + l. 35N, approximately. ;. 7 N, approximately. Thus, the solution of the above problems on the given parallel architecture requires O(N) data transfer steps, even though the number of parallel steps of the arithmetic operations is significantly smaller, O(log N) or O(log 2 N), respectively.

Instead, we transform the iterative process (7), using the matrix T = (Q, Q, ... , Q)n 2 , where [2] 0;; = (2/(n + 1)) 112 sin (ij:rrl(n + 1)) for i, j = 1, ... , n. We obtain 0Ck+IJ = TU(k+IJ = T[2wl- 2(1- w ) 2A- 2 L]UCkl- w2TUCk-l) + (1- w ) 2Tq = =2(1 - w )wO(k)- (1 - w ) 2 0(k-1)- 2(1 - w yock+l) + (1 - w yq (k = 1, 2, ... )J (9) where ocoJ= 0, 0C 1l= (1- w)TA- 1 S. The transformation does not change the spectral radius of the original iterative process (7). Hence, the asymptotic number of iterations is also O(n 112 log n), if optimal w and (j = O(n- 2 ) are assumed for the process (9).

Univ. of California, Berkeley, 1974. : A determinant theorem with applications to parallel algorithms. SIAM J. Numer. , 11, 1974, 559-568. , STEVENSON, D. K. and TRAUB, J. : Accelerated iterative methods for the solution of tridiagonal systems on parallel computers. J. ACM, 23, 1976, 636-654. : A survey of parallel algorithms in numerical linear algebra. Tech. Report, Comp. Sci. , 1976. [11] HYAFIL, L. and KUNG, H. : Parallel algorithms for solving triangular systems with small parallelism. Tech.

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Algorithms, Software and Hardware of Parallel Computers by Jozef Mikloško, Vadim Evgenich Kotov (auth.), Jozef Mikloško, Vadim Evgenich Kotov (eds.)

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