D esign and Imp lem en tation of Parallel Genetic A lgo rithm

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1 : (2004) , 2, 1 2, (1., ; 2., ) : PGA, : ; ; : T P : A D esign and Imp lem en tation of Parallel Genetic A lgo rithm fo r F inding Roo ts Comp lex Functional Equation L IU Feng 1, 2, CH EN Guo2liang 1, W U H ao 2 ( 1. D epartm ent of Computer Science and Engineering, U niversity of Science and T echno logy of Ch ina, H efei , Ch ina; 2. D epartm ent of Computer Science and Engineering, U niversity of A nh iu, H efei , Ch ina) Abstract: A parallel genetic algo rithm fo r finding all roo ts of comp lex functional equation is p resent, research are m ade in som e technical p roblem s fo r realizing. W e describe the design and imp lem ent of genetic algo rithm fo r finding roo ts com p lex functional equation. Key words: parallel genetic algo rithm; roo t; comp lex functional equation 1,, :, ; ;, ;,,,, [ 1, 2 ],,,, (PGA ), [ 3 Grefen stette GA ], PGA ;M uh lenbei PGA 64 [ R astrigin ] [ 5 ; Sp iessen s GA ] ; Po th s GA [ 6 ], PGA : : 973 (G ) : (1962- ),,,, Em ail: fengliu@m ail. ustc. edu. cn; (1938- ),,,, ; (1973- ),,

2 , PGA, ;,, 2 PGA PGA,,,, PGA, [ 8, 9 PGA : ],, PGA PGA, PGA,,, COW,,, COW, LAN, E thernet, Token- R ing COW, COW : 1) M PP, COW 2),, C, C + + 3), COW, COW 4), COW 5) COW,,,,, COW PGA PGA PGA PGA,,, GA PGA : 1),, 2) GA,, 3 1), Ζ f (z ) = 0, : m ingf (z ) gζ 2) [ 9 ] f (z ) = u + iv C C, C, f (z ) C 1g(2Π) z C f (z ) Ζ f (z ) = u+ iv C C, C,, z 0= x 0+ iy 0 C, (u, v) N p, N n, C f (z ) = 0 N = N p - N nζ C (Z 1, Z 2,, Z n), Z u (x, y ), v (x, y ), Z, 2, 2, 3, 3, 3, 4, 4, 4, 4, 1, 1, 2, 2, Ζ (2, 3, 4, 1, 2, ) Ζ 3, 4, 4,

3 6 63 1, 2, 2, 3, 3, 4, 4, 1, 1, 2, 2, 3, 3, (3, 4, 1, 2, 3, 4, 1, 2, 3) Ζ N p = 2 ( (1, 2, 3, 4) 2 ), N n= 0 ( (4, 3, 2, 1) 0 ) Ζ N = N p - N n= 2Ζ ), : in t selection (Cumm u lative fitnesses) { in t i = 0; doub le selection= RAND _ NUM (0. 0, fitnesses[po PULA T ION _ S IZE- 1 ]) ; w h ile (fitnesses[ i ] < i + + ; selection) retu rn i ; gg } fitnesses, A [a0, a1,, a i,, an ], a0 1, a1 3, a2 4, fitnesses[0 ]= 1, fitnesses[1 ]= 1+ 3= 4, fitnesses[2 ]= = 8 2),, : alpha= RANDOM ; ch ild1- > arg= alpha3 father- > arg+ (1- alpha) 3 m o ther- > arg; ch ild2- > arg= (1- alpha) 3 father- > arg+ alpha3 m o ther- > arg; 3), : individual- > arg= RAND _ NUM (from arg, endarg) ; 4) GA GA,,, GA,, f (z ) = 0, x y,, (x, y ) gf (x, y ) g, p ( ) gf (p ) g GA 4. 2 PGA 1) GA : P c P m, m ax g enζ P c ; P m ; m ax g en GA, e1, e2 (e1, e2 ) ; SR, SX, SY (SR, SX, SY ) ; T 0 d L (T 0, d, L M arkov ) 2) ( ) : typedef struct { doub le fitness; gg doub le values; gg

4 doub le arg; gg, } Geno type; 3) typedef Geno type Genes[PO PULA T ION _ S IZE+ 1 ]; typedef struct { in t cu rren t; gg genes[ ] Genes genes[2 ]; gg } Pheno type; Pheno type ( ), ( ) Pheno type cu rren t genes, cu rren t= 0, genes[0 ] GA ) ) 4) # define RANDOM - M A SK # define RANDOM ISE (srand (tim e (NULL ) ) ) # define RAND (rand () & RANDOM - M A SK) # define RANDOM ( ( (doub le) (rand () & RANDOM - M A SK) ) g( ( (doub le) RANDOM - M A SK) # define RAND - NUM (M,N ) ( (RANDOM 3 (N - M ) ) + M ) doub le arg; arg= RAND - NUM (from arg, endarg) ; individual- > arg= arg;, RAND - NUM (M, N ) RANDOM, RAND - NUM (M, N ) [ M, N ], RANDOM [0, 1 ] 5),,,, so rt- pheno type (individuals) ; gg individuals fo r ( i = 0; i < PO PULA T ION - S IZE; i+ + ) individuals- > genes[ individuals- > cu rren t ][ i]. fitness= 23 ( i + 1) ; gg, ), PGA, GA : ( N ) C ( p m, p c, T 0, d ), L ; : M etropo lis [4 ] ; N 3 (L + 1), N T 0 d M arkov L 7) PGA PGA (SPM D ), PGA (, GA, ) ;

5 ( GA ) ; ( ) GA, (ge) ( ge 5 ), ; (m o list), ; m o list ; ge= gm ax (gm ax ), (7), (3) ; 5 4 P III 450, L inux PVM, M = 4, N = 50, G = 100, p c = 0. 92, p m = 0. 02, T 0 = 0. 9; d = 0. 5,M arkov L = 10 1 f (z ) = co sz - sinz = 0 z = Θ(co sη+ isinη), Θ [0, 20 ], Η [0, 2Π) : e1 = , e2 = = g - g i i i i i i i i i i i i i gm ax= 200,,, 2 f (z ) = z 5 + ( i) z 4 + (7-40i) z 3 + (4 + 70i) z 2 + ( i) z - 80i z = Θ(co sη+ isinη), Θ [0, 20 ], Η [0, 2Π) : e1 = , e2 = = g - g gm ax= 100,,,

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