In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function ${\displaystyle f}$, which are solutions to the equation ${\displaystyle f(x)=0}$. However, to optimize a twice-differentiable ${\displaystyle f}$, our goal is to find the roots of ${\displaystyle f'}$. We can therefore use Newton's method on its derivative ${\displaystyle f'}$ to find solutions to ${\displaystyle f'(x)=0}$, also known as the critical points of ${\displaystyle f}$. These solutions may be minima, maxima, or saddle points; see section "Several variables" in Critical point (mathematics) and also section "Geometric interpretation" in this article. This is relevant in optimization, which aims to find (global) minima of the function ${\displaystyle f}$.

The central problem of optimization is minimization of functions. Let us first consider the case of univariate functions, i.e., functions of a single real variable. We will later consider the more general and more practically useful multivariate case.

Given a twice differentiable function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$, we seek to solve the optimization problem

Newton's method attempts to solve this problem by constructing a sequence ${\displaystyle \{x_{k}\}}$ from an initial guess (starting point) ${\displaystyle x_{0}\in \mathbb {R} }$ that converges towards a minimizer ${\displaystyle x_{*}}$ of ${\displaystyle f}$ by using a sequence of second-order Taylor approximations of ${\displaystyle f}$ around the iterates. The second-order Taylor expansion of f around ${\displaystyle x_{k}}$ is

The next iterate ${\displaystyle x_{k+1}}$ is defined so as to minimize this quadratic approximation in ${\displaystyle t}$, and setting ${\displaystyle x_{k+1}=x_{k}+t}$. If the second derivative is positive, the quadratic approximation is a convex function of ${\displaystyle t}$, and its minimum can be found by setting the derivative to zero. Since

the minimum is achieved for

Putting everything together, Newton's method performs the iteration

The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of ${\displaystyle f(x)}$ at the trial value ${\displaystyle x_{k}}$, having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below. Note that if ${\displaystyle f}$ happens to be a quadratic function, then the exact extremum is found in one step.