In mathematics, a smooth maximum of an indexed familyx1, ..., xn of numbers is a smooth approximation to the maximum function meaning a parametric family of functions such that for every α, the function is smooth, and the family converges to the maximum function as . 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, as and as . 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.
Smoothmax applied on '-x' and x function with various coefficients. Very smooth for =0.5, and more sharp for =8.
For large positive values of the parameter , the following formulation is a smooth, differentiable approximation of the maximum function. For negative values of the parameter that are large in absolute value, it approximates the minimum.