Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 43', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 262 Limits and Derivatives c lim an lim bn and the estimate n x n œ 0 c - an -1 b - a is valid. The following two methods are more efficient. . Regula falsi method false position method . Suppose that the derivatives f x and f x exist on the interval a 6 and the inequalities f x 0 and f x 0 hold for all x e a 6 . If f a f a 0 then we take x0 a for the zero approximation the subsequent approximations are given by the formulas f xn - - b-xn n 0 1 . f 6 - f xn If f a f a 0 then we take x0 b for the zero approximation the subsequent approximations are given by the formulas f xn - - a - xn n 0 1 . f a - f xn The regula falsi method has the first order of local convergence as n to xn 1 xn Xn 1 Xn Xn 1 C k Xn C where k is a constant depending on f x and c is the root of equation . The regula falsi method has a simple geometric interpretation. The straight line secant passing through the points a f a and b f b of the curve y f x meets the abscissa axis at the point x1 the value xn 1 is the abscissa of the point where the line passing through the points x0 f x0 and xn f xn meets the x-axis see Fig. a . Figure . Graphical construction of successive approximations to the root of equation by the regula falsi method a and the Newton-Raphson method b . . Newton-Raphson method. Suppose that the derivatives f x and f x exist on the interval a b and the inequalities f x 0 and f x 0 hold for all x e a b . . Functions of Several Variables. Partial Derivatives 263 If f a f a 0 then we take x0 a for the zero approximation if f b f b 0 then x0 b. The subsequent approximations are computed by the formulas _ f xn _n . xn 1 xn n 0 1 f xn If the initial approximation x0 is sufficiently close to the desired root c then the Newton-Raphson method exhibits quadratic convergence xn 1 - C M xn - C 2 2m where M max f z x and m min fz x . The Newton-Raphson method has a simple geometric interpretation. The tangent to the curve y f x through the point xn f xn
