Numerical Mathematics - A Collection of Solved Problems

132 NELINEARNE JEDNAČINE I SISTEMIRešenje. Newton–Kantorovičev iterativni postupak za rešavanje sistema nelinearnihjednačinaf (x) = 0 ,gde jedat je formulomx =2643x 1. .x n75 , f (x) =23f 1 (x 1 , . . . , x n )6745 ,.f n (x 1 , . . . , x n )(1) x(k + 1) = x(k) − W −1 (x(k)) f (x(k)) (k = 0,1, . . .),gde je W (x) Jacobieva matrica za f, tj.2∂f 1· · ·∂x 1 W(x) =6.4∂f n∂x 1pa je∂f31∂x n7∂f5 n∂x nDake, za sistem nelinearnih jednačina dat zadatkom, imamo2x = 4 x 32y 5 , f (x) = 4 x2 + y 2 + z 2 323− 12x 2y 2z2x 2 + y 2 − 4z 5 , W (x) = 44x2y −4 5 ,z3x 2 − 4y + z 2 6x −4 2z2f 0 = f(x(0)) = 4 −0.2532−1.25 5 , W 0 = W(x(0)) = 4 1 1 132 1 −4 5 .−1.003 −4 1Kako je det W 0 = −40, nalazimo inverznu matricu23W −1 (x(0)) = W0 −1 = − 1 −15 −5 −54 −14 −2 6 5 ,40−11 7 −1pa, na osnovu (1), imamo2x(1) = x(0) − W0 −1 f 0 = 4 0.53 23 20.5 5 + 1 −15 −5 −54 −14 −2 6 5 4 −0.253−1.25 5400.5 −11 7 −1 −1.002= 4 0.53 20.5 5 + 4 0.3753 20.000 5 = 4 0.87530.500 5 .0.5 −0.125 0.375.