ceiln

Ceiln is a mathematical function used to round a real number up to the nearest multiple of a given positive integer n. For real x and integer n > 0, ceiln(x, n) is the smallest multiple of n that is greater than or equal to x. Equivalently, ceiln(x, n) can be expressed as n multiplied by the ceiling of x divided by n: ceiln(x, n) = n · ceil(x / n). If x is already a multiple of n, ceiln returns x. The function works for all real x, including negatives; for example, ceiln(-5, 3) = -3 and ceiln(-1, 3) = 0.

In integer arithmetic, a common implementation for nonnegative x is ceiln(x, n) = ((x + n - 1) // n) * n, where // denotes integer division. This form is convenient in programming when x and n are integers and the result must be aligned to the next multiple of n.

Common uses of ceiln include grid alignment, resource budgeting, and discretization tasks where values must conform to a fixed multiple. It is often contrasted with floor-to-multiple functions, which round down to the nearest multiple (floorn).

Notes and variants: ceiln is not a standard universally named mathematical operator; in some libraries it is implemented as a utility function named differently (e.g., ceiling-to-multiple, align-to, or using a combination of division and ceiling). When implementing, care should be taken to define the domain (n > 0) and to handle integer versus real inputs consistently.