CH4非线性方程求根3埃特金算法正版.pdf

MATLAB 1 Aitken Ste?ensen

? 3.1. (Aitken Algorithm) 2 1 (Aitken Algorithm) x k +1 = ? (x k ) , Taylor ? ? x k +1 - x = ? ( ξ)( x k - x ) (3 .1)

yk = x k +1 = ? (x k ) , L ≈? ( ξ), yk - x ? ≈ L (x k - x ? ) (3 .2) zk = yk +1 = ? (yk ) = ? (? (x k )), zk - x ? ≈ L (yk - x ? ) (3 .3) L 3 ?

yk - x ? yk - x

? (yk - x ? )2 ≈(x k - x ? )( zk - x ? )

? y2 - 2y x ? ≈x z - x x ? - z x ? k k k k k k

? (x k + zk - 2yk )x ? ≈x k zk - y2 k 2 ? x k zk - yk

? x ≈ x k + zk - 2yk 2 ? (x k - y k )

? x ≈x k - ? x k ? (? (x k )) - (? (x k )) 2

x ≈ (3 .5) x k + ? (? (x k )) - 2? (x k ) ? x 5 2

x k = ? (x k ) - x k , x k = x k + ? (? (x k )) - 2? (x k ) (3 .6) ? ( x k )2 x ≈x k - 1 ? ? ? yk - Lx k yk - x ≈ L (x k - x ) ? x ≈ - L + 1 - L