ABOUT THE FUNCTIONING OF THE SPECIAL-PURPOSE CALCULATING UNIT BASED ON THE LINEAR SYSTEM SOLUTION USING THE FIRST ORDER DELTA- TRANSFORMATIONS

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1 ABOUT THE FUNCTIONING OF THE SPECIAL-PURPOSE CALCULATING UNIT BASED ON THE LINEAR SYSTEM SOLUTION USING THE FIRST ORDER DELTA- TRANSFORMATIONS 1 LIUBOV VLADIMIROVNA PIRSKAYA, 2 NAIL SHAVKYATOVISH KHUSAINOV 1, 2 Institute of Computer Technoogy and Information Security, Southern Federa University, Rostov-on-Don, Russia E-mai: 1 yubov.pirskaya@gmai.com, 2 khusainov@sfedu.ru ABSTRACT This paper discusses the theoretica representations of specia-purpose cacuating unit functioning for the iteration soution of inear systems using the first order deta-transformations and variabe quantum. It is considered the agorithm for iterative soution of inear systems based on the first order detatransformations and variabe quantum is adapted for impementation in a specia-purpose cacuating unit. A specia feature of specia-purpose cacuating unit functioning based on this agorithm is the impementation of the introduction at the beginning of each cyce of a new variabe quantum vaue that is refected in the current cyce when the residua vaues and the unknown variabe are formed by shifting them to the eft by 1 or 2 bits. Formation of unknown variabes in the unit is carried out by adding or subtracting the signs of quantum of the first and second variabes differences, taking at each iteration the vaues ± 1. This feature of variabe normaization represents the possibiity of organizing a computationa process on the basis of an integer data representation. At the fina step of the agorithm operation in the unit, it is possibe to bring the vaues of the variabes to the origina rea form, taking into account the weight of the minimum transformation quantum. With the orientation to FPGA, comparative estimates are obtained for the hardware and time resources of the deveoped agorithm and comprehensive comparative estimate of the effectiveness for specia-purpose cacuating unit functioning. In this paper for the deveoped agorithm of the unit functioning, it is shown that it is possibe to reduce the execution of one iteration and the iterative process as a whoe, the amount of hardware resources and generay improve the efficiency in comparison with the specia-purpose cacuating unit functioning based on the simpe iteration method. Keywords: Specia-Purpose Cacuating Unit, Linear System Soution, First Order Deta-Transformation. 1. INTRODUCTION The issues study of quaitative and quantitative improvement of computing devices perfomence for soving compex practica probems in rea time shows that using the most modern universa computer technoogy there are various difficuties: the need to organize effective parae computing processes with the decision to reduce the amount of transferred information, the number of simutaneousy operating mutipiers, muti-bit registers for storing information, simpification of compex systems of information exchange, ensuring high performance of cacuators in processing information. Such probems arise, in particuar, in the organization in rea time of a parae soution of practica probems, that can be reduced to probems of computationa mathematics, to design of on-board computer systems, speciaized contro devices, and so on. One of the ways to sove these probems is the creation of speciaized computing devices and systems, taking into account the design of their probem-oriented purpose and using specia effective methods for impementing the computationa process [1]. The necessity of probem-oriented computations arises, for exampe, when in rea time soving probems of oca navigation [2]-[4], in particuar, determining the aircraft coordinates, which reduces to the probem of soving inear agebraic equations systems (inear systems). The 7187

2 use of probem-oriented computations under the given conditions aows organizing the computing process in such a way that the processing of information at the eve of one iteration is performed with sufficient accuracy, high speed and with an extremey ong time step. Then, it is possibe to demand the owest requirements to the performance of computing toos, taking into account the possibiity of simutaneous impementation of other agorithms and programs. The functioning of a specia-purpose cacuating unit based on known iterative agorithms for soving inear systems [5]-[8] is characterized by a huge amount of hardware resources, which is associated with the need to impement a arge number of muti- bit mutipiers of coefficients and variabes, as we as to transfer of muti- bit codes between the equations of the system at parae impementation. The number of hardware and time resources increases, when it is necessary to simutaneousy sove a arge number of inear systems. The deveoped agorithms for soving inear systems with constant and variabe free terms based on the first order deta-transformations with constant [9]-[10] and variabe quantum [11]-[16] when impemented in a specia-purpose cacuating unit aow organizing a computing process with the exception of operations of a muti-bit mutipication and obtaining a resut in one iteration of the steadystate process. However, in papers [9]-[11] the use of a constant quantum is characterized by a much arger number of iterations. The use of an variabe quantum greaty reduces the number of iterations [12]-[18]. The essence of these agorithms with a variabe quantum ies in the representation of the iterative process in the form of iteration cyces, in each of which a parae for a equations of a inear system is performed the formation of variabes at a constant moduus of transformation quantum [14]- [16]. In the paper [11] the process of inear systems soution must be preceded by the number of iterations in the cyce, using, in particuar, the division operation that under the conditions of the perform these cacuations with the hep of the specia-purpose cacuating unit is not appropriate. Amost a the papers [11]-[13] concerned the agorithmic organization of the iterative process for inear systems soving with the use of the first order deta-transformations with the variabe quantum has not got any information about the theoretica justification for the seection of the best reation between the quanta of the adjacent cyces; on the one hand the ways to specify these reations are introduced heuristicay, on the other hand they are not fuy cover the theoretica and practica interest of these reations. In addition, the foowing observations shoud be noted. In the papers [11]-[13] the proposed agorithms are characterized by high computationa compexity when impementing the iteration process due to the necessity of performing the squaring of the discrepancies across a equations at each iteration, summing them and highight the smaest of the current according to the amount of vaues across the iterations. The squaring of the discrepancies is associated with the need to use the mutibit mutipiers, which is contrary to the origina goa impementation of the specia-purpose contro units of inear systems soution with the possibe of the exception of such devices. The iterations termination moment is fixed either with the use of a certain constant, the determining vaue of which is uncertain, or according to the number of iterations that is aso a probem of the preiminary estimate. In the papers [14]-[16] is deveoped a theoretica substantiation of the seection of the number of cyces and the vaues of the variabes quantum, aimed at minimizing the number of iterations in the inear systems soution on the basis of the first order deta-transformations. The agorithms which showed efficiency on ensuring timey competion of iterations, simpe in reaization and not demanding any numerica assessment with use of specia basic data are introduced for the formation of the competion moment of the rea iterative processes in the oops. The architecture of the specia-purpose cacuating unit can undoubtedy impact on fina agorithmic efficiency. And in the design of specia-purpose cacuating unit using the FPGA pecuiarities of agorithmic software aow to achieve the highest quaity performance indicators to minimize the costs of resources (equipment) compared with known [19]-[21]. Thus, on the basis of the agorithm presented in [14]-[16], the paper presents the agorithm adapted to the impementation in a 7188

3 specia-purpose cacuating unit for the parae iterative soution of inear systems based on the first order deta-transformations and variabe quantum. Next, the paper contains the deveoped architecture, features and a study of the effectiveness of the specia-purpose cacuating unit functioning based on the inear system soution using the first order deta-transformations and variabe quantum. 2. THE ALGORITHM OF A LINEAR SYSTEM SOLUTION ADAPTED UNDER REALIZATION IN A SPECIAL-PURPOSE CALCULATING UNIT There is an inear system that has a matrix of constant coefficients and, in genera case, variabe free terms, fufiing convergence conditions described in papers [5]-[8]: BY () t G() t. (1.1) Let's transform the system Y () t AY () t D() t, and transfer to writing with residua zt () for appying the iteration method: zt () Yt () AYt () Dt (). (1.2) In the given systems B [ b rj ], A [ brj / brr ] are matrices of dimensiona coefficients n n; Gt (), D() t are coumn-vectors of absoute system terms (in particuar for the system with fixed absoute terms Gt () G [ g r ], Dt () D [ gr / brr] ); Y () t are coumn-vectors of system unknowns; zt (), Yt () are coumn-vectors of residuas and approximate unknown vaues; t is an independent variabe; det A 0. Based on the resuts obtained in [14]-[16], an agorithm for the parae soution of inear system (1.1) with constant free terms using the first order deta-transformations and variabe quantum oriented for specia-purpose cacuating unit is presented beow in the foowing difference form for i-step under the initia conditions Yr 01 0, zr01 [17]-[18]: D, r 01 max z, r 1, n, c P 2 s, s N - cacuating the vaues of residuas and unknowns before each iteration cyce: z z R ; (1.3.1) r0 rr ( 1) Y Y R ; (1.3.2) r0 rr ( 1) r 1, n, 1,2,, P ; - creation of first difference quantum signs at each iteration in cyces: sign( z ); (1.3.3) ri r( i1) { 1, 1} ; r 1, n, i 1,2,, R ; ri 1,2,, P ; - demoduation: ri r( i1) ri Y Y ; (1.3.4) - creation of residua vaues at each iteration in cyces: z Ф (, j 1, n) ; (1.3.5) ri r ji z z z ; (1.3.6) ri r( i1) ri - conditions for competion of iterative processes in the iteration cyce: 1. sign z sign z rr r ( Rint,(1,2 ) 1) 2. ( ) ( ) or (1.3.7) z 0 ; rrint,(1,2 ) 1,2,, P, r 1, n. sign( z sign( z )) rrint,(1,2 ) rr sign( z ) rr or (1.3.8) sign( z sign( z )) 0 ; rrint,(1,2 ) rrint,(1,2 ) 1,2,, P, r 1, n. In the agorithm (1.3) P is the number of iteration cyces performed at a constant moduus of quantum; R a constant vaue, that refects the change in the quantum of transformation and the redefinition of a variabes of the inear system during the transition from cyce to cyce. The vaues of the constant vaues of P и R 7189

4 are set in accordance with those obtained in papers [14]-[16] reations: P z n n R 01 max cp, (1.4) where c P is the weight of the minimum transformation quantum on the ast cyce ( c 0 ), Rint,1 2 и Rint,2 4. The number of iterations of a rea computationa process in a cyce R, in accordance with the concusions obtained in papers [14]-[16], may be arger or smaer reative vaue R. In the reations (1.3.1) and (1.3.2), vaues z r0 and Yr0 are cacuated in the current cyce by shifting them by 1 bits at Rint,1 2 or by 2 bits at Rint,2 4. This procedure refects the introduction at the beginning of each cyce before the fufiment of the reations (1.3.3) - (1.3.8) of the new vaue of the variabe quantum. At each iteration, the quantum signs are formed as ± 1 vaues and are used ater in next steps of the agorithm to cacuate the unknown variabe Y ri (1.3.4) by adding or subtracting a unit from this current vaue. This feature of variabe normaization represents the possibiity of organizing a computationa process on the basis of an integer representation of data. Upon competion of the agorithm, it is possibe to form the variabes vaues to the origina rea representation using the weight of the minimum transformation quantum c : Y Y c, (1.5) rr P P rr P where the combination R P in the index denotes the formation of the most recent vaue of the unknown, that is, the fina resut. Competion of iterative processes in the cyces in the agorithm (1.3) is carried out on the basis of the reations (1.3.7) or (1.3.8), when in the cyce for a equations of a inear system P P simutaneousy or distributed in time, at east one of them is fufied. 3. ARCHITECTURE OF THE SPECIAL- PURPOSE CALCULATING UNIT FUNCTIONING USING THE FIRST ORDER DELTA-TRANSFORMATIONS AND VARIABLE QUANTUM Figure 1 shows the bock diagram of the specia-purpose cacuating unit functioning for the parae iterative soution of inear systems based on the agorithm (1.3) using the first order detatransformations and variabe quantum. Bocks 1, 4 are r registers ( r 1, n), which contain residua z ri and unknown Y ri vaues, r 1, n with initia vaues zr01 Dr and Yr 01 0, respectivey. Bocks 2, 3 are registers that contain the vaues of the iterative cyces number performed at a constant moduus of quantum, determined before the agorithm (1.3) starts and assumes one of the two vaues R, respectivey. Bocks 5, 7 are r shift registers ( r 1, n), reaizing the shift of the vaues z r0 (1.3.1) and Y r0 (1.3.2) into 1 bit at Rint,1 2 or 2 bits at Rint,2 4 at the beginning of each cyce. The obtained vaues z r0 and Y r0 after the shift come to the registers of bocks 1 and 4, respectivey. In bock 6, the cyces of the agorithm (1.3) are executed, the conditions for the termination of the agorithm (1.3) as a whoe are verified in bock 9 and the fufiment of this condition shows the resut of soving the inear system from bock 4. Bocks 8, 15 determine, according to the reations (1.3.3), the quantum signs of the first differences of the previous iteration zri ( 1) and the current iteration z ri for a equations of the system. In bock 10, by the reations (1.3.4), the vaues of the unknowns in the group of adders ( r 1, n) are cacuated, with the input quantum signs of the first differences of variabes ri and the unknown vaues Y ri. Further, the resuts obtained go to bock

5 In the diagram of Figure 1, the current vaue of the residua zri is cacuated on the basis of the tabuar method in the form (1.3.5), where Ф (, j 1, n) a r ji ri rj ji j1 ( jr ) and storage devices ( r 1, n) of the bock 11 for storing the tabes. The tabe is organized as a sums coection of mutipication of inear system coefficients by transformation quantum for each equation. Choosing vaues of the pre-formed mutipication sums is performed based on the set of current vaues ji, j 1, n coming from bock 8. Thus, it is possibe to excude the mutipication operation due to the given organization of cacuations and to obtain the resut for 1 unit of time. For a arge dimension of the matrix A, it is expedient to partition the sum Фr( ji, j 1, n) into m bocks: m r ( ji, 1, ) r ( ji, 1, ) g 1 Ф j n Ф j n and store the tabe vaues for each bock. Bock 12 is r adders ( r 1, n), where the input of each receives the residuas z ri and vaues obtained in bock 11 corresponding to the system equations. This bock ensures the fufiment of reation (1.3.6). Bocks 13 and 16 are designed to verify the conditions (1.3.7), (1.3.8) that fix the moments of iterative processes competion in the -th iteration cyce for each equation of inear system. In the bock 17, the generation of these moments for a equations of the system is verified either by (1.3.7) or (1.3.8). If one of the conditions is successfuy met, the counter is incremented in the bock 6, and the agorithm is organized on a new iterative cyce. In bock 14, the number of iterations is counted within one iteration cyce. This counter i is reset to zero for each next cyce. n Bocks 12, 18 are a group of adders ( r 1, n), in which vaues of the current residuas z ri (1.3.6) and additiona ones zadd ri are cacuated, where z z sign( z ). add ri ri ri Performing agorithmic sequence as part of a singe iteration of the agorithm (1.3) and according to that shown in Figure 1 a bock diagram is carried out for about 3 units of time: at the first cock it is performed the actions of bocks 5,7; at the second cock - bocks 10,12; at the third cock - bock EXPERIMENTAL RESEARCH OF THE ALGORITHM OF A LINEAR SYSTEM SOLUTION ADAPTED UNDER REALIZATION IN A SPECIAL-PURPOSE CALCULATING UNIT The agorithm for the parae soution of inear system with constant free terms using the first order deta-transformations and variabe quantum oriented for specia-purpose cacuating unit has been tested on the soution of various inear systems, characterized by different convergence in the performance of simpe iteration method. Foowing are the resuts of individua experiments based on inear systems given beow on (norm of the matrix coefficient A of exampes (26) and (27) greater than one): y y2 0.13y3 0.97; 0.12 y1 y2 0.11y3 1.13; (24) 0.16 y y2 y y1 0.6y2 0.08y ; 0.12y1 y2 0.7 y ; (25) 0.11y1 0.4y2 y y1 y2 0.2y3 1.37; 0.8y1 y2 0.2y3 0.98; (26) 0.4y1 0.7 y2 y y1 0.9y2 0.4y3 0.85; y1 y2 0.6y3 0.69; 1.3y1 0.3y2 y (27) The obtained resuts are presented in Tabe 7191

6 The comparative anaysis between methods of inear systems soutions on the 14 condition of ensuring identica accuracy (~ 2 ) based on the first order deta-transformations and constant quantum c 2 14, by the simpe iteration method and by the method which is aso considered in this paper based on the first order detatransformations and variabe quantum at Rint,1 2, 14 R int = 4, c P 2 was carried out during the research. Tabe 1: The resuts of the experiments The method of organizing the iterative process of the inear systems soution on the basis of the first order detatransformations and constant quantum on the basis of the first order detatransformati ons and variabe quantum simpe iteration method R int,1 = 2 Rint,2 4 The number of iterations (24) (25) (26) (27) The anaysis of the data in the Tabe 1 shows that the deveoped agorithm of iterative process of the inear systems soution with the variabe quantum, based on the optimized assessments Rint,1 2, R int = 4, and providing the rea time optimization of the iterative processes differs significanty (by the hundreds thousands times) with the reduction of the number of iterations in reation to the method of the inear systems soution based on the first order detatransformations and constant quantum, as we as substantia proximity on the number of iterations to the simpe iteration method and in some cases represents the advantage over the simpe iteration method. According to the Tabe 1, the ratio of the number of iteration method for the simpe iteration method to the method with the variabe quantum is as foows ~ The data presented in Tabe 1 shows that iterative processes whie ensuring the convergence of inear systems may be successfuy impemented even in the norm of the matrix of coefficients greater than one. 5. RESEARCH OF EFFICIENCY OF THE SPECIAL-PURPOSE CALCULATING UNIT FUNCTIONING BASED ON THE DEVELOPED ALGORITHM Compex efficiency of specia-purpose cacuating unit functioning based on the agorithm (1.3) using deta the first order detatransformations and variabe quantum in comparison with using the simpe iteration method can be considered as an interreated set of comparative estimates of time and hardware resources. Taking into account, as shown in [14]- [16], that the number of iterations using the first order deta-transformations and variabe quantum may be greater or ess than using the simpe iteration method, in the study this amount is assumed to be the same for the methods studied. The impementation of the bock diagram of the unit, shown in Figure 1, can be considered with using FPGA. Resource characteristics of the impementation of the basic arithmetic operations with their hardware performance are significanty unequa. Especiay a huge hardware resources, expressed in ogica ces, are required for mutipiers. The mutipication operation wi be considered as the execution of mutipication using singe-cock hardware circuits of parae impementation, as we as with the expansion of the factors bit grid - efficient parae-seria impementation by performing mutipication at severa units of time using agorithm "shift with the accumuation". The simpest to reaize the structure of simpe matrix summation, which forms a parae mutipier as an array of singe-bit adder connected by oca interconnections [22]. This scheme is the most effective at sma bit operands (4 or ess), where a parae mutipier 4 4 requires 12 adders for its impementation. With a further increase the bit capacity, the matrix of singe-bit adder grows significanty and the critica propagation path of the transport signa increases, accordingy the performance wi imited, and the reaization of the 7192

7 mutipier becomes ess rationa. Thus, in this work, mutipiers of higher bit capacity were considered as a combination of mutipiers 4 4 [22]. The formation of reative estimates for hardware resources was based on the number of ogica gates invoved [23], [24]. In accordance with the agorithm (1.3) and the bock diagram in Figure 1, the soution of the equations is carried out in parae. Aso, the equations soution is reaized in parae using the simpe iteration method. Let Q be an estimate characterizing the agorithm hardware resources, Q d. p.1is the estimate for the agorithm (1.3) based on the first order deta-transformations and variabe quantum, Q p. r. is the estimate for the simpe iteration method. For the comparative hardware estimate, it is entered the Qp. r. ratio. Q d. p.1 Based on the obtained estimates, it was constructed the dependence graph of comparative Qp. r. estimates on the order of inear systems n Qd. p.1 ( n 3 ), when operating with 32-bit data is shown in Figure 2. method, when operating with 32-bit data. At increasing order of the system n, the efficiency increases. The estimate of time resources amount was performed by units of time. Time resources were cacuated in the framework of a singe pass through the cyce of the agorithm, reaizing agorithmic sequence of actions for a equations of the system. Let T be an estimate characterizing the agorithm time resources, T d. p.1 is the estimate for the agorithm (1.3) based on the first order detatransformations and variabe quantum, T p. r. is the estimate for the simpe iteration method. For the comparative time estimate, it is entered the ratio Tp. r.. T d. p.1 Based on the obtained estimates, it was constructed the dependence graph of comparative Tp. r. estimates on the order of inear systems n Td. p.1 ( n 3 ), when operating with 32-bit data is shown in Figure 3. Figure 2: Dependence graph of Qp. r. comparative estimates on the order of inear Qd. p.1 systems n. The anaysis of the obtained estimates showed that using the agorithm based on the first order deta-transformations and variabe quantum in a soution of inear systems of the order n = 3 has an advantage in terms of hardware resources of ~2,7 times in comparison with the simpe iteration Figure 3: Dependence graph of Tp. r. comparative estimates on the order of inear Td. p.1 systems n. The anaysis of the obtained estimates showed that using the agorithm based on the first order deta-transformations and variabe quantum in a soution of inear systems of the order n = 3 has an advantage in terms of time resources of ~2,5 times in comparison with the simpe iteration method, when operating with 32-bit data. At increasing order of the system n, the efficiency increases. 7193

8 Compex efficiency of specia-purpose cacuating unit functioning based on the agorithm (1.3) using deta the first order detatransformations and variabe quantum The comparative compex estimate of the agorithm impementation efficiency based on using deta the first order deta-transformations and variabe quantum was formed as the mutipication of the reative estimates, obtained above, for the time and hardware resources of the system: Qp. r. Tp. r. E Q T. (1.6) d. p.1 d. p.1 In accordance with (1.6), the comparative compex efficiency estimate were obtained. Based on these resuts, it was constructed the dependence graph of the comparative compex efficiency estimate E on the order of inear systems n and is shown in Figure 4. Figure 4: Dependence graph of the comparative compex efficiency estimate E of specia-purpose cacuating unit functioning on the order of inear systems n. In accordance with this estimate E, the agorithm based on the first order detatransformations and variabe quantum, when soving inear systems of the n = 3 order, has an advantage ~6.7 times compared to using the simpe iteration method, when operating with 32-bit data. The estimate with an increase in the order of the system n sharpy increases. 5. CONCLUSION In paper it is considered principes of specia-purpose cacuating unit functioning for the iteration soution of inear systems using the first order deta-transformations and variabe quantum. A specia feature of specia-purpose cacuating unit functioning based on this agorithm is the impementation of the introduction at the beginning of each cyce of a new variabe quantum vaue that is refected in the current cyce when the residua vaues and the unknown variabe are formed by shifting them to the eft by 1 or 2 bits. Formation of unknown variabes in the unit is carried out by adding or subtracting the signs of quantum of the first and second variabes differences, taking at each iteration the vaues ±1. This feature of variabe normaization represents the possibiity of organizing a computationa process on the basis of an integer data representation. At the fina step of the agorithm operation in the unit, it is possibe to bring the vaues of the variabes to the origina rea form, taking into account the weight of the minimum transformation quantum. In addition, the feature of specia-purpose cacuating unit functioning is using in the agorithm for forming the moment of iterative processes competion in cyces the condition that requires when in the cyce for a equations of a inear system simutaneousy or distributed in time the residuas sign wi be changed. The architecture of the specia-purpose cacuating unit impact on fina agorithmic efficiency: as it is shown in this work the possibiities of the organization of simutaneous shifts of coefficients with preservation of shift of the previous cyce or without preservation that can be connected with need of essentia expenses of the hardware or time resources. And there is no doubt that the eve of the paraeization processes has an impact on the performance of the specia-purpose cacuating unit. The utimate effectiveness is estimated by the aggregate of the assessments of the hardware resources and the speed. 7194

9 In the paper it is shown the advantages of using the deveoped features of the specia-purpose cacuating unit functioning based on the inear system soution using the first order detatransformations and variabe quantum. So with the orientation to FPGA, comparative estimates are obtained for the hardware resources and the speed of the deveoped agorithm for specia-purpose cacuating unit functioning. For the deveoped agorithm of the unit functioning, it is shown that it is possibe to reduce the execution of one iteration and the iterative process as a whoe in ~2,5 times and the amount of hardware resources in ~2,7 times in comparison with the specia-purpose cacuating unit functioning based on the simpe iteration method, and these estimates increase with increasing order of the system n. Thus, using the adapted agorithm for buiding specia-purpose cacuating unit, the hardware resources are significanty reduced due to the abiity to excude the mutipication operator of the mutibit code and increasing the performance of the simpe iteration method at the eve of the iterative processes. These circumstances create prerequisites associated, in particuar, with the expansion of the resource capabiities of FPGAs for the simutaneous reaization in rea time of compex tasks, as separate components of which are used inear systems. The resuts obtained in this work are panned to be used in on-board contro and navigation systems by modern aircraft. As a separate component of the navigation task soved within the framework of the creation of on-board speciaized computing devices, the task of determining the aircraft coordinates is presented, represented as systems of inear agebraic equations. Based on the resuts of preiminary studies, using the principes of specia-purpose cacuating unit functioning, presented in this paper, in on-board speciaized devices for the oca navigation task makes it possibe to determine the aircraft coordinates at each time step of the steadystate process in one iteration and operate with practicay significant time steps of the system at a sufficienty arge distance of the beginning of the steady process from the origin. REFRENCES: [1] P.P. Kravchenko, L.V. Pirskaya and N.Sh. Khusainov, Deta-transformations and probem-oriented computations: monograph. Taganrog: SFedU Press, [2] Sh. Khusainov, P.P. Kravchenko, V.N. Lutai, S.A. Tarasov and V.V. Scherbinin, Radionavigation systems of currenttechnoogy and future-technoogy vehices. P. 1. Fixing methods and autonomous integrity monitoring: Monograph, Taganrog: SFedU Press, [3] O.O. Barabanov and L.P. Barabanova, Mathematica probems of r-r navigation, Мoscow: Phismatit, [4] O.N. Skrypnik, Aircraft radionavigation systems: Training manua, Мoscow: INFRA- M, [5] C. Vuik, Iterative soution methods, The Netherands: Deft Institute of Appied Mathematics, [6] A. Greenbaum, Iterative Methods for Soving Linear Systems, Phiadephia, PA: SIAM, [7] D.K. Faddeev and V.N. Faddeeva, Computationa methods of inear agebra, 4th ed., reprint, Saint-Petersburg: Lan', [8] N. Bakhvaov, et a., Numerica Methods, Laboratory of knowedge. Moscow: BINOM, [9] S. Tretyakov, The agorithms of the speciaized processors for soving systems of equations, Cybernetics, Vo. 5, 1978, pp [10] P.P. Kravchenko, The optimised second order deta-transformations. Theory and appication. Monograph, Moscow: Radio engineering, [11] P. Kravchenko, Incrementa methods for soving the systems of the inear agebraic equations, Mutiprocessor computing structures, Vo. 5(XIV), 1983, pp [12] B. Mainowskij, Agorithms for soving the systems of the inear agebraic equations, structura-oriented impementation, Contro systems and machines, Vo. 5, 1977, pp [13] О. Gomozov and Y. Ladyshchenskij, Incrementa agorithms for soving the systems of the inear agebraic equations, and architecture of mutiprocessors on programmabe ogic, Scientific works of DonNTU, Informatics, cybernetics and computer science, Vo. 12(165), 2010, pp [14] P.P. Kravchenko and L.V. Pirskaya, The iterative method of the system of the inear agebraic equations soution excuding the 7195

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11 -D r i P int(1,2) [2;4] 0 z ri 1 P int(1,2) 2 R int(1,2) 3 Y ri 4 Y r z ri z ri(-1) z ri(-1) <<[1;2] 5 z r(i-1) R int(1,2) P int(1,2) Inc() 6 end R int(1,2) Y ri(-1) Y ri(-1) <<[1;2] 7 Y ri Y r(i-1) z ri ri -sign(z r(i-1) ) 8 <=P int(1,2) 9 ri Sum Y ri 10 Y ri i Ф r Ф r ( ji ) 11 z r(i-1) res z ri z ri Sum z ri 12 z ri -sign(z ri ) 15 z ri z ri Ischange (z ri,z r(i-1) ) 13 Ischange (z add ri,z ri ) 16 or ( ) 17 res Inc(i) 14 Sum z add ri 18 z add ri Figure 1: The bock diagram of the specia-purpose cacuating unit functioning for the parae iterative soution of inear systems using the first order deta-transformations and variabe quantum 7197