Unimodal function

In mathematics, a function f(x) between two ordered sets is unimodal if for some value m (the mode), it is monotonically increasing for x ≤ m and monotonically decreasing for x ≥ m. In that case, the maximum value of f(x) is f(m) and there are no other local maxima (i.e. there is one mode as the name indicates).

Examples of unimodal functions:

• Quadratic polynomial with a negative quadratic coefficient
• Logistic map
• Tent map.

A function f(x) is "S-unimodal" if its Schwartzian derivative is negative for all $\ x\neq 0$.

In probability and statistics, a "unimodal probability distribution" is a probability distribution whose probability density function is a unimodal function, or more generally, whose cumulative distribution function is convex up to m and concave thereafter (this allows for the possibility of a non-zero probability for x=m). For a unimodal probability distribution of a continuous random variable, the Vysochanskii-Petunin inequality provides a refinement of the Chebyshev inequality. Compare multimodal distribution.

In computational geometry if a function is unimodal it permits the design of efficient algorithms for finding the extrema of the function.

A more general definition, applicable to a function f(X) of a vector variable X is that f is unimodal if there is a one to one differentiable mapping X = G(Z) such that f(G(Z)) is convex. Usually one would want G(Z) to be continuously differentiable with nonsingular Jacobian matrix.