Editing Newton's method (section)

==Applications==
===Minimization and maximization problems===
{{main|Newton's method in optimization}}
Newton's method can be used to find a minimum or maximum of a function {{math|{{var|f}}({{var|x}})}}. The derivative is zero at a minimum or maximum, so local minima and maxima can be found by applying Newton's method to the derivative.<ref>{{cite book|mr=2061575|title=Convex optimization|last1=Boyd|first1=Stephen|last2=Vandenberghe|first2=Lieven|publisher=[[Cambridge University Press]]|location=Cambridge|year=2004|isbn=0-521-83378-7|author-link1=Stephen P. Boyd|doi=10.1017/CBO9780511804441|zbl=1058.90049}}</ref> The iteration becomes:

<math display="block">x_{n+1} = x_n - \frac{f'(x_n)}{f''(x_n)}. </math>

===Multiplicative inverses of numbers and power series===
An important application is [[Division algorithm#Newton–Raphson division|Newton–Raphson division]], which can be used to quickly find the [[Multiplicative inverse|reciprocal]] of a number {{mvar|a}}, using only multiplication and subtraction, that is to say the number {{mvar|x}} such that {{math|1={{sfrac|1|{{var|x}}}} = {{var|a}}}}. We can rephrase that as finding the zero of {{math|1={{var|f}}({{var|x}}) = {{sfrac|1|{{var|x}}}} − {{var|a}}}}. We have {{math|1={{var|{{prime|f}}}}({{var|x}}) = −{{sfrac|1|{{var|x}}{{sup|2}}}}}}.

Newton's iteration is

<math display="block">x_{n+1} = x_n-\frac{f(x_n)}{f'(x_n)} = x_n+\frac{\frac{1}{x_n}-a}{\frac{1}{x_n^2}} = x_n(2-ax_n).
</math>

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

This method is also very efficient to compute the multiplicative inverse of a [[power series]].

===Solving transcendental equations===
Many [[transcendental equation]]s can be solved up to an arbitrary precision by using Newton's method. For example, finding the cumulative [[probability density function]], such as a [[Normal distribution]] to fit a known probability generally involves integral functions with no known means to solve in closed form.  However, computing the derivatives needed to solve them numerically with Newton's method is generally known, making numerical solutions possible.  For an example, see the numerical solution to the [[Normal distribution#Using the Taylor series and Newton's method for the inverse function|inverse Normal cumulative distribution]].

===Numerical verification for solutions of nonlinear equations===
A numerical verification for solutions of nonlinear equations has been established by using Newton's method multiple times and forming a set of solution candidates.{{cn|date=June 2024}}