19 Facts About Newton's method

In numerical analysis, Newton's method, known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots of a real-valued function.

The Newton's method can be extended to complex functions and to systems of equations.

Name "Newton's method" is derived from Isaac Newton's description of a special case of the method in De analysi per aequationes numero terminorum infinitas and in De metodis fluxionum et serierum infinitarum.

Newton's method used each correction to rewrite the polynomial in terms of the remaining error, and then solved for a new correction by neglecting higher-degree terms.

Newton's method did not explicitly connect the method with derivatives or present a general formula.

Related searches

Newton's method was used by 17th-century Japanese mathematician Seki Kowa to solve single-variable equations, though the connection with calculus was missing.

Newton's method was first published in 1685 in A Treatise of Algebra both Historical and Practical by John Wallis.

Raphson applied the method only to polynomials, but he avoided Newton's tedious rewriting process by extracting each successive correction from the original polynomial.

Newton's method is a powerful technique—in general the convergence is quadratic: as the method converges on the root, the difference between the root and the approximation is squared at each step.

Newton's method requires that the derivative can be calculated directly.

For situations where the Newton's method fails to converge, it is because the assumptions made in this proof are not met.

In some cases, Newton's method can be stabilized by using successive over-relaxation, or the speed of convergence can be increased by using the same method.

Newton's method is only guaranteed to converge if certain conditions are satisfied.

Newton–Fourier method is Joseph Fourier's extension of Newton's method to provide bounds on the absolute error of the root approximation, while still providing quadratic convergence.

Newton's method can be generalized with the -analog of the usual derivative.

Hirano's modified Newton Newton's method is a modification conserving the convergence of Newton Newton's method and avoiding unstableness.

Newton's method can be used to find a minimum or maximum of a function.

Therefore, Newton's method iteration needs only two multiplications and one subtraction.

Newton's method is applied to the ratio of Bessel functions in order to obtain its root.