In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
The algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions and to systems of equations.
The Newton-Raphson method in one variable is implemented as follows:
Given a function ƒ defined over the reals x, and its derivative ƒ ', we begin with a first guess x0 for a root of the function f. Provided the function satisfies all the assumptions made in the derivation of the formula, a better approximation x1 is
Geometrically, (x1, 0) is the intersection with the x-axis of a line tangent to f at (x0, f (x0)).
The process is repeated as
until a sufficiently accurate value is reached.
Read more about Newton's Method: Description, History, Practical Considerations, Analysis, Failure Analysis
Famous quotes containing the words newton and/or method:
“Glorious things of thee are spoken, Zion city of our God!
He, whose word cannot be broken, Form’d for thee his own abode:
On the rock of ages founded, What can shake thy sure repose?
With salvation’s walls surrounded Thou may’st smile at all thy foes.”
—John Newton (1725–1807)
““English! they are barbarians; they don’t believe in the great God.” I told him, “Excuse me, Sir. We do believe in God, and in Jesus Christ too.” “Um,” says he, “and in the Pope?” “No.” “And why?” This was a puzzling question in these circumstances.... I thought I would try a method of my own, and very gravely replied, “Because we are too far off.” A very new argument against the universal infallibility of the Pope.”
—James Boswell (1740–1795)