Muller's method is a root-finding algorithm, a numerical method for solving equations of the form f(x) = 0. It was first presented by David E. Muller in 1956.

Muller's method proceeds according to a third-order recurrence relation similar to the second-order recurrence relation of the secant method. Whereas the secant method proceeds by constructing a line through two points on the graph of f corresponding to the last two iterative approximations and then uses the line's root as the next approximation at every iteration, by contrast, Muller's method uses three points corresponding to the last three iterative approximations, constructs a parabola through these three points, and then uses a root of the parabola as the next approximation at every iteration.

Muller's method uses three initial approximations of the root, ${\displaystyle x_{0},x_{1}}$ and ${\displaystyle x_{2}}$, and determines the next approximation ${\displaystyle x_{3}}$ by considering the intersection of the x-axis with the parabola through ${\displaystyle (x_{0},f(x_{0}))}$, ${\displaystyle (x_{1},f(x_{1}))}$ and ${\displaystyle (x_{2},f(x_{2}))}$.

Consider the quadratic polynomial

that passes through ${\displaystyle (x_{0},f(x_{0}))}$, ${\displaystyle (x_{1},f(x_{1}))}$ and ${\displaystyle (x_{2},f(x_{2}))}$. Define the differences

$${\displaystyle h_{0}=x_{1}-x_{0},\quad h_{1}=x_{2}-x_{1}}$$

and $${\displaystyle \delta _{0}={\frac {f(x_{1})-f(x_{0})}{h_{0}}},\quad \delta _{1}={\frac {f(x_{2})-f(x_{1})}{h_{1}}}.}$$

Substituting each of the three points ${\displaystyle (x_{0},f(x_{0}))}$, ${\displaystyle (x_{1},f(x_{1}))}$ and ${\displaystyle (x_{2},f(x_{2}))}$ into equation (1) and solving simultaneously for ${\displaystyle a,b}$ and ${\displaystyle c}$ gives