Newton's method

a method of approximating a root x₀ of the equation f(x) = 0; also called the method of tangents. In Newton’s method, the initial (“first”) approximation x = a₁ is used to find a second, more accurate, approximation by drawing the tangent to the graph of y = f(x) at the point A[a₁, f(a₁)] up to the intersection of the tangent with the Ox-axis (see Figure 1). The point of intersection is x = a₁ – f(a₁)/f’(a₁) and is adopted as the new value a₂ of the root. By repeating this process as necessary, we can obtain increasingly accurate approximations a₂, a₃, … of the root x₀ provided that the derivative f’(x) is monotonic and preserves its sign on the segment containing x₀.

The error ε₂ = x₀ – a₂ of the new value a₂ is related to the old error ε₁ = x₀ – a₁ by the formula

where f”(ξ) is the value of the second derivative of the function f(x) at some point ξ that lies between x₀ and a 1. It is sometimes recommended that Newton’s method be used in conjunction with some other method, such as linear interpolation. Newton’s method allows generalizations, which makes it possible to use the method for solving equations f(x) = 0 in normed spaces, where F is an operator in such a space, in particular, for solving systems of equations and functional equations. This method was developed by I. Newton in 1669.