Numerical Analysis

On the convergence rates of Legendre approximation

“…Interpolation is one of the most important tools of numerical analysis and almost every numerical analysis textbook has a chapter to introduce it (see, for instance, [2,10,16,33]). The Lagrange interpolation polynomial q n (x) which interpolates the function f (x) at the (n + 1) points x j may be written as…”

“…Moreover, Higham has shown in [22] that the barycentric Lagrange formula (3.2) is stable for any set of interpolating points with a small Lebesgue constant, such as Chebyshev and Gauss-Legendre points, and ought to be the standard method of polynomial interpolation [4]. For an exhaustive discussion of the theoretical advantage of the barycentric Lagrange formula, we refer the reader to [4,10,16,22]. If f (x) is continuous and of bounded variation on [−1, 1], then the Lagrange interpolation polynomial q n (x) which interpolates f (x) at the Gauss-Legendre points converges uniformly to f (x) on [−1, 1] [38].…”

Multiplicative Newton s Methods with Cubic Convergence

“…It is clear that the number of function evaluation in per iteration for methods in (3), (4), (5) and (7) is three. According to the definition of efficiency index [21] which is p √ NFE (where p is order of convergence of method and NFE is number of function evaluation), the efficiency index of methods defined in this work is 3 √ 3 ∼ = 1.442. Hence, the efficiency index of these new methods is better than the efficiency index of Newton method and multiplicative Newton method which are √ 2 ∼ = 1.414 and are same as the ones of methods defined in [6], [7] and [9].…”

Solving a Nonlinear Equation Using a New Two-Step Derivative Free Iterative Methods

“…Methods proposed by King in equation (2) and equation (3), combination of Newton and Helley methods in equation (4) and equation (5) and methods proposed by Weerakon and Fernando in equation (6) and equation (7) is methods for solving a nonlinear equation, they can fast to approximating of a roots, but they not success finding a root if '( ) n f x =0 or '( ) n f y =0, therefore must use derivative free iterative method. Some other paper have discuss about a two step iterative methods free derivative for solving a nonlinear equation is a method proposed by Dehghan-Hajarian can be call as Dehghan Method (DM), with order of convergence three and four evaluations of the function per iteration, so it possesses 1.316 as the efficiency index, can express their formula as [5]:…”