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SHOWING PROPERTIES OF NUMERIC METHOD ON THE BASIS OF HOMOGRAPHIC FUNCTIONS Belyankov A.Ya. Institution of Russian Academy of Sciences Dorodnicyn Computing Centre of RAS, Russia, 119333 Moscow, Vavilov st. 40, Tel.: (499) 135-42-50, E-mail: belyankov@ccas.ru Let function f : R R be sufficiently smooth, f ( x* ) = 0 , f ( x* ) 0 , and let successive approximations x0 , x1 , x2 , be generated by a numeric method for solution of equation f ( x ) = 0 . Recall well known asymptotic convergence behavior of certain three methods described in [1], namely Newton method, secant method and secant method with the recurrency x k + 1 = F ( xk , x k - 1 ) : f ( xk ) = x* + ( xk - x* ) 2 ( + o(1)) , ( xk + 1 = ) xk - (1) f ( xk ) af ( xk ) - xk f ( a) = x* + ( xk - x* )( q + o(1)) , ( xk + 1 = ) (2) f ( xk ) - f ( a ) x f ( x k ) - xk f ( xk - 1 ) = x* + ( xk - 1 - x* )( xk - x* )( + o(1)) . ( xk + 1 = ) k - 1 (3) f ( xk ) - f ( xk - 1 ) For homographic functions A ( x) = ( a11 x + a12 ) /( a21 x + a22 ) , det( A) 0 , the following is established. If f = A then formulae (13) hold exactly i.e. o(1) can be thrown off; corresponding values for , q are = f ( x* ) / 2 f ( x* ) , q = ( a - x* ) . These properties are specific for homographic functions: if we throw off o(1) in any formula (13) and consider this formula as functional equation over unknown function f with independent variable xk (formulae (12)) or with independent variables xk -1 , xk (formula (3)), then we deduce that f is homographic. It should be noted also that solving equation x = ( x) using simple iteration method i.e. creating sequence x0 , x1 = ( x0 ) , x2 = ( x1 ) = ( ( x0 )), is quite transparent in the case of homographic function = A because A ( A ( x)) = A ( x) , A ( A ( A ( x))) = A ( x), and powers of A are easy to calculate: Ak = V kV - 1 where A = V V - 1 is canonical decomposition of 2 by 2 matrix A .