Smooth maximum

In mathematics, a smooth maximum of an indexed family x₁, ..., x_n of numbers is a smooth approximation to the maximum function ${\displaystyle \max(x_{1},\ldots ,x_{n}),}$ meaning a parametric family of functions ${\displaystyle m_{\alpha }(x_{1},\ldots ,x_{n})}$ such that for every α, the function ⁠${\displaystyle m_{\alpha }}$⁠ is smooth, and the family converges to the maximum function ⁠${\displaystyle m_{\alpha }\to \max }$⁠ as ⁠${\displaystyle \alpha \to \infty }$⁠. The concept of smooth minimum is similarly defined. In many cases, a single family approximates both: maximum as the parameter goes to positive infinity, minimum as the parameter goes to negative infinity; in symbols, ⁠${\displaystyle m_{\alpha }\to \max }$⁠ as ⁠${\displaystyle \alpha \to \infty }$⁠ and ⁠${\displaystyle m_{\alpha }\to \min }$⁠ as ⁠${\displaystyle \alpha \to -\infty }$⁠. The term can also be used loosely for a specific smooth function that behaves similarly to a maximum, without necessarily being part of a parametrized family.