Generalized interpolation polynomials

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“…Note that if f (x) ∈ C [2] [a, b], then the error of interpolation ( 1)-( 2) is distributed more uniformly over the entire interval [a, b] than that of interpolation ( 10)- (11). Hence, one can expect that the rate of convergence of method ( 1)-( 6), in the general case, is not slower than the rate of convergence in Newton's method.…”

“…Definition 1. The endpoints of the interval [a, b] are said to be chosen with equal probability, when solving the equation f (x) = 0 for f (x) ∈ H [2] [a, b] by Newton's method, if the approximation errors e…”

“…[a,b] is the subclass of functions f (x) ∈ C [2] [a, b] that are strictly monotone and strictly convex for all x ∈ (a, b); ε > 0 is some small quantity; x (C) is an approximate value of a root of the equation f (x) = 0 in the chord method and x (N ) that in Newton's method; e (C) and e (N ) are the corresponding approximation errors. (2) [a, b], found by using method (10)-( 16) has the approximation error smaller than the error in the chord-Newton method. In the general case, for f (x) ∈ H (2) [a, b] we have e (inv.it.)…”

“…(2) [a, b], found by using method (10)-( 16) has the approximation error smaller than the error in the chord-Newton method. In the general case, for f (x) ∈ H (2) [a, b] we have e (inv.it.) < max e (C) , e (N ) .…”

“…For the sake of definiteness, let a point c be the midpoint of the interval (u, v). Let f ∈ H (2) [a, b]. Expand f (v) and f (c) in the Taylor series up to order two in a neighborhood of the point u ∈ (a, b).…”