A Genetic Algorithm for Solving General System of Equations

Transcription

1 A Geetic Algorithm for Solvig Geeral System of Equatios Győző Molárka, Edit Miletics Departmet of Mathematics, Szécheyi Istvá Uiversity, Győr, Hugary Abstract: For solvig liear system of equatios is kow several algorithms. Iteratio algorithms are recommeded for the large liear systems with sparse matrix. But i the case of geeral x m matrices the classic iterative algorithms are ot applicable with a few exceptios. For example i some cases the Laczos type algorithms are adequate. The algorithm preseted here based o the miimizatio of residuum of solutio ad it has some geetic character. Therefore this algorithm seems to be applicable for costructio of parallel algorithm. Here we describe a sequetial versio of proposed algorithm ad give its theoretical aalysis. Moreover we show some umerical test results of the sequetial algorithm. Keywords: liear system of equatios, iterative algorithms, geetic algorithms, parallel algorithm. 1 Itroductio Let A be a geeral xm matrix. The basic problem is to solve the followig liear system of equatios: Au = b, (1) u R m b R where ad are the solutio ad the give right had side vector. The existece ad uiqueess of the solutio of (1) ca be determied from matrix A ad the vector b. Theoretically the Gaussia or Gauss-Jorda elimiatio algorithm is appropriate tool to solve the system (1) ad decide the questio of solvability. But practically for large systems whe we use floatig poit arithmetic, these direct algorithms are iapplicable. For these cases the iteratio algorithms are suitable. Effective iteratio algorithms are kow for symmetric positive defiite liear systems called Hermitia. Most of kow iterative algorithms i geeral form ca be writte i the form of: ( ) ( 1) u Gu k = +, = 1,,..., ()

2 where G ad k are such matrix ad vector that () for its statioary solutio must be equivalet with (1), see ([1]). The most effective algorithms for such system are the precoditioed cojugate gradiet (CG) oes see ([]). These iterative algorithms ca be applied for geeral osymmetric liear systems too if we solve the followig ormal system T T AAx Ab v = = (3) istead of the origial oe. A disadvatage of this approach is that the resultig liear system (3) for matrices with full rak will be Hermitia oes but its coditio umber will be the square of the origial coditio umber, therefore the covergece will be very slow. For geeral o-hermitia liear systems istead of geeralizatio of some variat of the CG algorithms oe of the most successful scheme is the geeralized miimal residual algorithm (GMRES) see ([5]) ad the bicojugate gradiet algorithm (BCG) see ([6]). A more effective approach was suggested by Freud ad Nachtigal ([7]) for the case of geeral osigular o-hermitia systems which calls the quasi-miimal residual algorithm (QMR). I the followig we describe a iterative miimal residual algorithm which is slightly differet from the above oes, but this differece ca be very importat for further developmet of these algorithms for parallel implemetatio. A Iterative Miimal Residual Algorithm It is kow, that the most of the iterative algorithms for the solutio of liear systems based o some miimizatio algorithm see ([]). The ormal system (3) ca be obtaied by the least square miimizatio i the followig way. We have to solve the followig problem: x R x R x R mi Ax b = mi( Ax b, Ax b) = mi( r, r), (4) where r = Ax b is the residual belogig to the vector x. The ormal system (3) ca be derived easily from (4) by a simple calculatio. More precisely we obtai, that the ecessary coditio of the existece ad uiqueess of the solutio of (4) is the fulfilmet of (3). A sufficiet coditio for T the uiqueess is the Hermitia property of the ormal matrix AA. For geeral o-hermitia matrices this coditio will ot fulfil i geeral. To solve the problem (4) we chose the followig way. Oe possible algorithm ca be obtaied from the observatio formulated by the followig theorem.

3 m Theorem 1 Let A R R ad b R be a arbitrary matrix ad vector. m Moreover let x α m R ad x β R be arbitrary, but such differet vectors for α β which Ax ( x ) 0. Let us itroduce the followig otatios: s s r = Ax b, s= α, β, ad r αβ = cr α + (1 c) r β αβ α β, x = cx + (1 c) x, (5) αβ αβ where c R. It is easy to see, that Ax b = r. The the solutio of the costraied miimizatio problem of (4) alog the lie defied by the vectors x α ad x β is the vector x αβ with c, where β α β ( r r, r ) c = α β r r (6) Moreover αβ, { } r αβ < mi r α, r β, where we use the Euclidea orm. Proof of Theorem 1 The costraied miimizatio problem of (4) with (5) is a oe-dimesioal problem. So we have to solve the followig oe-dimesioal problem: mi f ( c) = mi( cr α + (1 c) r β, cr α + (1 c) r β ) = mi r αβ ( c). (7) c c c From here the result (6) as the ecessary coditio ca be obtaied after a simple calculatio. The sufficiet coditio from (7) is: d f c () α β = r r > 0, (8) dc which coditio is fulfilled if r α ad r β are differet vectors. The meaig of the theorem 1 is the followig: If we have some approximate solutio x α for the problem (1), the by arbitrary x β vector satisfyig the coditios of theorem (1) we ca get a better approximate solutio i the form of x αβ vector, where the appropriate costat c is defied by (4). Remark: As agaist the classical iterative algorithms, such as gradiet ad the cojugate gradiet oes, here the directios of miimizatio are chose by chace. The algorithm ca be cotiued i every step by choosig a ew idepedetly

4 chose arbitrary x β vector. This is the mai ew property of the proposed algorithm, because this property allows, that i parallel arbitrary umber of approximatio vector ca be geerated idepedetly, ad all these vectors ca be used to improve the earlier best approximatio of the solutio. The realizatio of the theoretical results as a algorithm ca be made i several ways. Here we describe a simple (maybe the most simple) versio of a sequetial algorithm..1 The Algorithm 1 Usig the results of the Theorem 1 we ca formulate a algorithm, which geerate k a approximate solutio sequece x, k = 1,,3,... ad i parallel its residual k vectors r, k = 1,,3, Let Calculate 1 x be a arbitrary vector ad ε the error tolerace. 1 1 r = Ax b. 3 Geerate a arbitrary vector, 4 c 1 : = 1 ( r r, r ) 1 r r 5 Calculate the ew r : = c r + (1 c ) r vectors., x such that 1 r r x : = c x + (1 c ) x ad 6 x : = x és 1 1 r : = r If 1 r < ε the go to 8. else go to 3. 8 The approximate solutio is 1 x, ed of algorithm. The algorithm 1 is the simplest oe which ca be formulated by the result of Theorem 1. Therefore the covergece of this algorithm is ot better tha the covergece of the classical oes. The ovelty of this algorithm is ot i this property.

5 Figure 1 The covergece of the Algorithm 1. o the dimesio problems.. Test Results for the Covergece Several arbitrary dimesio A R R matrix ad b R was geerated as test problems. The Algorithm 1 was realized usig the Maple8 Liear Algebra library. The results of the algorithm are show o the Fig. 1. O the Fig. 1 oe ca see that the covergece of the algorithm is strictly mooto but very slow. The attaied residual orms are: r = , , , , These results are such oes what we could expect because the test problems are such liear systems where the matrix is ot a square matrix, ad the test vectors were chose by hasard. By aalysig of the algorithm it is possible to elaborate better strategy for the geeratio of test vectors, but this ca be a topic of a ew work. O the Fig. we show the speed of covergece depedig o the coditio umber of the test matrices. Oe ca see, that the covergece is very slow for the test problems with large coditio umbers, which is ot a surprisig result too.

6 Figure The covergece results of the Algorithm 1. for dimesio test problems with differet coditio umbers. O the Table 1 we show the results for the differet test problems. From this table we ca coclude that the Algorithm 1 because of its hazard character ca produce quite differet speed of covergece, but this speed depeds o the coditio umber of the test matrices. Table 1 Numeratio of the curve o Fig. 1. Coditio umber of the matrix. Residual orm attaied Summary We have formulated a ew geetic like algorithm for the solutio of geeral liear systems of equatios, which based o the residual miimizatio techique. The test results cofirm the theoretical results. The covergece speed of the sequetial algorithm proposed is very slow, but the idea suggested is appropriate for costructig more effective parallel algorithm too combiig this idea with the results published i ([8-11]). This problem ca be the subject of some forthcomig paper.

7 Refereces [1] Louis A. Hagema, Davis M. Joug,: Applied Iterative Methods, Computer Sciece ad Applied Mathematics, Academic Press, (1981) [] P. G. Ciarlet, Itroductio à l aalyse umérique matricielle aet à l optimisatio, MASSON, Paris, 198 [3] G. Golub, A. Greebaum, M. Luski, eds., Recet Advaces is Iterative Methods, The IMA Volumes i Math. ad its Applicatios Vol. 60, Spriger Verlag, 1994 [4] J. Gilbert, D. Kershaw, Large-Scale Matrix Problems ad the Numerical Solutio of Partial Differetial Equatios, Advaces i Numerical Aalysis, Vol. III, Claredo Press, Oxford, 1994 [5] J. Saad ad M. H. Schultz, GMRES: A geeralized miimal residual algorithm for solvig osymmetric liear systems, SIAM J. Sci. Stat. Comput., 7 (1986) pp [6] C. Laczos, Solutio of systems of liear equatios by miimized iteratios, J Res. Nat. Bur. Stadards, 49 (195) pp [7] R. W. Freud ad N. M. Nachtigal, QMR: a quasi-miimal residual method for ohermitia liear systems, Numer. Math., 60 (1991) pp [8] G. Molárka ad B. Török, Residual Elimiatio Algorithm for Solvig Liear Equatios ad Applicatio to Sparse Systems. Zeitschrift für Agewadte Mathematik ud Mechaik (ZAMM), Issue 1, Numerical Aalysis, Scietific Computig, Computer Sciece. pp , 1996 [9] J. K Tar, I. J. Rudas, J. F. Bitó, L. Madarász: A Emergig Brach of Computatioal Cyberetics Dedicated to the Solutio of Reasoably Limited Problem Classes, AT & P Joural Plus 001, pp [10] J. K. Tar, I. J. Rudas, L. Madarász, J. F. Bitó: "Simultaeous Optimizatio of the Exteral Loop Parameters i a Adaptive Cotrol Based o the Cooperatio of Uiform Procedures", Joural of Advaced Computatioal Itelligece, Vol. 4, No. 4, 000, pp [11] A. Lotfi, A H-matrix Type Precoditioer for Frictioal Cotact Problems, PAMM Proc. Appl. Math. Mech. 4, (004) This work was supported by OTKA N o T04358 ad CBC PHARE 00/