MATRIX INVERSE ON CONNEX PARALLEL ARCHITECTURE

Transcription

1 U.P.B. Si. Bull., Series C, Vol. 75, Iss. 2, ISSN MATRIX INVERSE ON CONNEX PARALLEL ARCHITECTURE An-Mri CALFA, Gheorghe ŞTEFAN 2 Designed for emedded omputtion in system on hip design, the Connex Prllel Arhiteture n ompete with generl purpose devises in the domin of mtrix omputtion, improving exeution time nd GIPS/Wtt euse of its speifi rhiteturl fetures. The tul performnes of this rhiteture in the omputtion of mtrix inversion re investigted using oth nlytil lultion nd numeril simultion. Keywords: prllel rhiteture, emdded systems, inverse of mtrix. Introdution In this pper we will ontinue to investigte the Connex Prllel Arhiteture [], strted on [2] for the first omputtionl motif from Berkeley s view [3] - liner lger. Two prllel lgorithms to lulte the inverse of mtrix [4] re desried in the seond setion, one optimized for energy use nd the other one for re effiieny. In the third setion the pek performnes of Connex Prllel Arhiteture re ompred with generl purpose CPU nd generl purpose miroontroller, in order to emphsize the dvntges of Connex Arhiteture. We onlude in the fourth setion tht, even if the Connex Arhiteture is very simple rhiteture ompred with its ompetitors, re nd power gin re signifint. 2. Mtrix Inverse In this pper, only dense mtries will e investigted nd will e lter mentioned s mtries. The inverse of mtrix A is the mtrix A - tht stisfies the property: A A - = A - A = I () PhD student, Fulty de Eletronis, Teleommunitions nd Informtion Tehhnology, University POLITEHNICA of Buhrest, Romni, e-mil: nnie_lf@yhoo.om 2 Prof., Fulty de Eletronis, Teleommunitions nd Informtion Tehhnology, University POLITEHNICA of Buhrest, Romni, e-mil: gstefn@rh.pu.ro

2 92 An-Mri Clf, Gheorghe Ştefn where I is the unity mtrix. Inverse of mtrix n e defined only for squre mtries. Not ll squre mtries hve inverses, ut if they do, they re unique! A mtrix with inverse is lled invertile or nonsingulr nd one without inverse is lled noninvertile or singulr. The inverse of mtrix n e lulted using two methods:. Guss-Jordn elimintion trnsforms [A I] into [I A - ]. To exemplify, we onsider mtries of 4x4: (2) The method of Guss-Jordn elimintion involves reting the index mtrix insted of the initil mtrix nd its inverse insted of the unit mtrix. This implies to form zeros, under nd upper the min digonl of the initil mtrix. The mtrix to e inverted nd the index mtrix tthed re onsidered unit nd ll the opertions re going to e pplied to it. To use Connex speifi rhiteturl fetures, opertion must e performed on olumn. On the first olumn, first step is to divide the first row y the element of the min digonl of the mtrix tht needs to e inverted (3) If this element is zero then we need to interhnge this row with nother one of grter index tht doesn t hve this element zero. If ll the rows hve the first element zero it mens tht the mtrix is singulr. To rete zeros on the first olumn under the min digonl we need to multiply, one y one, the first row with the first element of the row where we intend to rete zero nd then derese it from the orresponding row. Applying the proedure listed ove for the seond row:

3 Mtrix inverse on onnex prllel rhiteture = (4) nd then, deresing the result from the seond row, results: (5) Continuing this opertion for ll remining rows, it will e otined zeros on the first olumn: (6) The zeros on the seond olumn will e first reted under the min digonl, this mens tht we should strt from the seond row where we hve to generte on the min digonl. This is done dividing the seond row y 2 2 = α. To simplify the equtions we mke the following nottions x y xy = xyα (7) Applying (7) to (6):

4 94 An-Mri Clf, Gheorghe Ştefn 2 α α α α α α (8) α α 3 α α α α 4 To rete zeros on the seond olumn under the min digonl, we pply the sme method s for the first olumn. Multiplying it, one y one, with the elements of grter index on the sme olumn: α α 2 α = α α α α α α α 2 α α α α = (9) α α α α nd then deresing it from the orresponding row, results: 2 α α 3 α α α α α α α α α α 4 α α α α α α α α 2 α 3 α ( + α 4 α ( + α 2 2 ) ) α α α α () Continuing these opertions for ll olumns, insted of initil mtrix will e otined mtrix tht hs ones on the min digonl nd zeros under it. Applying the sme method for the upper prt of the mtrix, strting with the lst row, the unit nd the inverse mtrix will e otined: ' 2' 3' 4' 2' ' ' ' 3' ' ' ' 4' ' ' ' ' 2' 3' 4' 2' ' ' ' 3' ' ' ' 4' ' ' ' ()

5 Mtrix inverse on onnex prllel rhiteture 95. Crmer s rule is defined y : A - = djoint( A) oftor _ mtrix( A) = ( ) det( A) det( A) T (2) First step is to lulte the determinnt for ll defined mtries. It will e lulted using the Gussin elimintion method presented ove, tht will e pplied until the originl mtrix eomes tringulr mtrix, without reting s on the min digonl. Under this ondition the determinnt is defined s the produt of ll the elements on the min digonl. If the determinnt is zero then the mtrix A is noninvertile. Beuse, with this rhiteture n e lulted up to 52 inverses in prllel (for mtries of 2x2), those mtries tht hve the determinnt zero will e mrked nd the inverse will not e lulted for them. For the rest of the mtries the lgorithm will ontinue until the inverses re found. The oftors of mtrix re those determinnts tht results fter deleting the row nd the olumn of n element from mtrix, with the sign given y the sum of the indexes of the given element. If the sum is n even numer then the sign is +, otherwise the sign is -. If we onsider three mtries for whih we wnt to ompute the inverse in prllel: (3) first these mtries will e trnsposed with the method presented in [], resulting: (4) Then, in uxiliry mtries will e stored those mtries resulted fter deleting the row nd the olumn of speifi element, for whih will e lulted the determinnts (oftors fter dding the sign) using the method of Guss-Jordn elimintion.

6 96 An-Mri Clf, Gheorghe Ştefn for(i = ; i < mtr_size; i++){ Index is in position i from mtr_size; WHERE(Index == ) Tmp = Mtr; Tmp is shifted one position to the left; for(j = ; j < mtr_size; j++){ if(( j!= ) (j!= mtr_size)){ for(k = j; k< mtr_size; k++){ Tmp[k+] = Tmp[k]; } Clulte determinnts for the Tmp mtries using Guss-Jordn elimintion; CoftorMtr[i][j] = (-) i+j det(tmp); } } } If i =, the Index vetor is: pplied to (4), it eomes: ( )( )( ) (5) (6) After shifting it one position to the left, the uxiliry mtries re: (7)

7 Mtrix inverse on onnex prllel rhiteture 97 (8) For these mtries of size mtr_size- will e lulted the determinnts. It represents the oftors vlues for (,). Repeting these tions for ll olumns nd rows the djoint mtries will e otined. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (9) The inverse of mtries will e otined dividing djoint mtries with the determinnt of initil mtries. 3. Performne of Connex Arhiteture The figures in Tle I represents the lok yles per opertion fter running the presented lgorithms on the emultor of Connex Arhiteture for slr vetors of floting points, without onsidering the input-output trnsfer of dt. To emphsize the performnes of Connex, the results will e ompred with relesed figures of generl purpose CPU nd miroontroller used in DSP pplition:

8 98 An-Mri Clf, Gheorghe Ştefn Proessor : 45 MHz, W Miroontroller : 3 MHz The Guss-Jordn elimintion hs the dvntge tht it is quiker thn the Crmer s rule, for ig mtries, ut the numer of inverted mtries is hlved. Depending on the pplitions requests one of the two methods n e used. Results of mtrix invert on Connex Prllel rhiteture Guss-Jordn elimintion Crmer s rule No. of mtries Clok yles No. of mtries Clok yles 2x x x x Tle I In Fig. it n e oserved tht the lok yles re funtion of O(n 3 ) for smll n, nd funtion of O(n 4 ) for n igger thn 8. An improvement of t lest n is otined ompred with sequentil method. Compring oth inversion methods on Connex Arhiteture with sequentil proessor [5] in Tle II, for mtries of 4x4, it n e oserved tht n elertion of t lest 3 times is otined if is used Crmer s rule method, nd effiieny of power (rry_inverse/wtt) is inresed round 5 times. Tle II Performne of Connex Arhiteture ompred with Intel proessor Method Connex CPU Performnes (lk y/mtr) (lk y/mtr) Aelertion Power eff. Guss-Jordn Elimintion, ,74x 8,4x Crmer s rule 64, ,x 48,5x If the results re ompred with generl purpose miroontroller [6] when the dimension of the mtries is squred, for Crmer s rule, the elertion is still tens [7] (see Tle III). Performne of Connex Arhiteture ompred with PIC miroontroller for Crmer s rule Method Connex uc Performne Aelertion Crmer s rule ,x Tle III Connex Arhiteture, esides lowering the exeution time lso power is more effiient used ompred with urrent solutions. In the domin of ig mtries, performnes re even higher due to rhiteturl fetures.

9 Mtrix inverse on onnex prllel rhiteture 99 lg(yles Guss-Jordn elimintion Crmer's rule lg(n^3) lg(n^4) Mtrix dim Fig. Evolution of lok yles funtion of mtrix rnge Nowdys, there re mny seurity nd ommeril pplition tht uses mtrix inversion: mthemtil reserh systems, ryptogrphy, serh engines, utomti pilot s.. These pplition proesses lrge mounts of dt, nd due to its performne, Connex Arhiteture n e integrted. The methods nd results presented ove pply only for dense mtries. Due to high proportion of non useful dt, spre mtries hve dedited methods to store the dt in the memory nd to ompute it. Spre mtries re prt of future investigtion. 4. Conlusion Due to lrge mount of low power pplition tht uses mtrix inversion, improvements of time exeution nd power effiieny (osts lowering) in the domin of liner lger re mndtory. Even if prllel proessors re usully used s elertors where they re integrted, nd most of the times re dedited to pplition tht proesses lrge mount of dt, due to its performnes trnslted in lower osts, Connex Prllel Arhiteture n e used in vriety of other pplitions outside of its design domin.

10 An-Mri Clf, Gheorghe Ştefn R E F E R E N C E S [] Gheorghe Ştefn, The CA: SoC with Integrl Prllel Arhiteture for HDTV Proessing", 4th Interntionl System-on-Chip (SoC) Conferene & Exhiit, Novemer & 2, 26, Rdisson Hotel Newport Beh, CA [2] An-Mri Clf, Gheorghe Stefn, Mtrix Computtion on Connex Prllel Arhiteture, in ICSES 2 Proeedings, Gliwie Polnd, Septemer 2 [3] K. Asnovi, et. l., The Lndspe of Prllel Computing Reserh: A View from Berkeley, Tehnil Report No. UCB/EECS-26-83, Deemer 8, 26 [4] H. Anton, Elementry Liner Alger, 7 th Edition, John Wiley & Sons In., 994 [5] Intel Corportion, Streming SIMD Extensions - Inverse of 4x4 Mtrix, AP-928, Order Numer 5-, Mrh 999 [6] Mirohip, dspic Lnguge Tools Lirries, DS5456B, [7] Artit C. Jrptnkul, A Multi-Sensor Emedded Miroontroller System for Condition Monitoring of RC Heliopters, The Pennsylvni Stte University, 25