ceilmax

Ceilmax is a mathematical function defined for finite collections of real numbers. Given numbers x1, x2, ..., xn, ceilmax(x1, ..., xn) returns the smallest integer that is not less than the largest value among them. In formal terms, ceilmax(x1, ..., xn) = ⌈max{x1, ..., xn}⌉. If the input collection is empty, ceilmax is generally considered undefined.

For finite inputs, ceilmax has an equivalent formulation: ceilmax(x1, ..., xn) = max{⌈x1⌉, ..., ⌈xn⌉}. This follows because the maximum value max{x1, ..., xn} is attained by some xk, and ⌈max{x1, ..., xn}⌉ = ⌈xk⌉, which is the largest among the ceilings.

Examples illustrate the concept: ceilmax(0.3, 0.7, −1.2) = ⌈0.7⌉ = 1. ceilmax(−2.0, −3.5) = ⌈−2.0⌉ = −2. ceilmax(2, 3) = ⌈3⌉ = 3.

Computation is straightforward: one can either compute the maximum first and then apply the ceiling, or take the ceilings of each input and select their maximum. Both approaches yield the same result for finite inputs and run in linear time relative to the number of inputs.

Applications of ceilmax arise in discretization and rounding up worst-case values in resource planning, scheduling, and numerical algorithms where an integer bound tied to the largest input value is required.

See also: ceiling function, maximum, floor function. For infinite sets, ceilmax can be extended as ⌈sup S⌉, with notes on how it relates to the supremum and the ceilings of individual elements.