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Newton's method uses curvature information (i.e. the second derivative) to take a more direct route. In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function, which are solutions to the equation =.
From this, it can be seen that the rate of convergence is superlinear but subquadratic. This can be seen in the following tables, the left of which shows Newton's method applied to the above f(x) = x + x 4/3 and the right of which shows Newton's method applied to f(x) = x + x 2. The quadratic convergence in iteration shown on the right is ...
Newton's method assumes the function f to have a continuous derivative. Newton's method may not converge if started too far away from a root. However, when it does converge, it is faster than the bisection method; its order of convergence is usually quadratic whereas the bisection method's is linear. Newton's method is also important because it ...
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
Then the intervals containing one root may be further reduced for getting a quadratic convergence of Newton's method to the isolated roots. The main computer algebra systems ( Maple , Mathematica , SageMath , PARI/GP ) have each a variant of this method as the default algorithm for the real roots of a polynomial.
The four 4th roots of −1, none of which are real The three 3rd roots of −1, one of which is a negative real. An n th root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose nth power is x:
(The single changes variable x = (2By – B) may also be used.) The method requires an algorithm for testing whether an interval has zero, one, or possibly several roots, and for warranting termination, this testing algorithm must exclude the possibility of getting infinitely many times the output "possibility of several roots".
The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial p(z) ∈ [z] or transcendental function. It is the Julia set of the meromorphic function z ↦ z − p ( z ) / p′ ( z ) which is given by Newton's method.