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Convergence with = and any is called cubic convergence. However, it is not necessary that q {\displaystyle q} be an integer. For example, the secant method , when converging to a regular, simple root , has an order of the golden ratio φ ≈ 1.618.
Halley's method is a numerical algorithm for solving the nonlinear equation f(x) = 0.In this case, the function f has to be a function of one real variable. The method consists of a sequence of iterations:
Both use the polynomial and its two first derivations for an iterative process that has a cubic convergence. Combining two consecutive steps of these methods into a single test, one gets a rate of convergence of 9, at the cost of 6 polynomial evaluations (with Horner's rule). On the other hand, combining three steps of Newtons method gives a ...
Bicubic interpolation can be accomplished using either Lagrange polynomials, cubic splines, or cubic convolution algorithm. In image processing , bicubic interpolation is often chosen over bilinear or nearest-neighbor interpolation in image resampling , when speed is not an issue.
Ridders' method — fits a linear function times an exponential to last two iterates and their midpoint; Halley's method — uses f, f' and f''; achieves the cubic convergence; Householder's method — uses first d derivatives to achieve order d + 1; generalizes Newton's and Halley's method; Methods for polynomials: Aberth method; Bairstow's method
If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x 1 in the basin of attraction of x, and let x n+1 = f(x n) for n ≥ 1, and the sequence {x n} n ≥ 1 will converge to the solution x.
This guarantees quadratic convergence; that is, the number of correct digits is roughly doubled with each iteration. Using Halley's method, Ω can be approximated with cubic convergence (the number of correct digits is roughly tripled with each iteration): (see also Lambert W function § Numerical evaluation).
The Weierstrass approximation theorem states that for every continuous function f(x) defined on an interval [a,b], there exists a set of polynomial functions P n (x) for n=0, 1, 2, ..., each of degree at most n, that approximates f(x) with uniform convergence over [a,b] as n tends to infinity, that is,