# Newton's method

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## Newton's method

[′nüt·ənz ‚meth·əd]## Newton’s Method

a method of approximating a root *x*_{0} of the equation *f(x*) = 0; also called the method of tangents. In Newton’s method, the initial (“first”) approximation *x* = *a*_{1} is used to find a second, more accurate, approximation by drawing the tangent to the graph of *y* = *f(x*) at the point *A[a*_{1}, *f(a*_{1})] up to the intersection of the tangent with the *Ox*-axis (see Figure 1). The point of intersection is *x* = *a*_{1} – *f(a*_{1})/*f*’(*a*_{1}) and is adopted as the new value *a*_{2} of the root. By repeating this process as necessary, we can obtain increasingly accurate approximations *a*_{2}, *a*_{3}, … of the root *x*_{0} provided that the derivative *f*’(*x*) is monotonic and preserves its sign on the segment containing *x*_{0}.

The error ε_{2} = *x*_{0} – *a*_{2} of the new value *a*_{2} is related to the old error ε_{1} = *x*_{0} – *a*_{1} 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 *x*_{0} 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.