sqrtl2

sqrtl2 is not a standard mathematical object with a universally fixed meaning. In mathematical writing or code, it may appear as an informal notation for the square root of an L2-related quantity, and its precise definition depends on context. Two common informal interpretations are described below.

One common interpretation is that sqrtl2(x) denotes the square root of the L2 norm of a vector x in R^n. If x = (x1, ..., xn), then the L2 norm is ||x||2 = sqrt(sum xi^2). In this usage, sqrtl2(x) = sqrt(||x||2) = (sum xi^2)^{1/4}. This function is nonnegative and satisfies sqrtl2(t x) = sqrt(|t|) sqrtl2(x). It can arise in contexts where a secondary scaling of the Euclidean length is desired.

A related interpretation defines sqrtl2(x, y) as the square root of the L2 distance between two points x and y: sqrtl2(x, y) = sqrt(||x - y||2) = (sum (xi - yi)^2)^{1/4}. This emphasizes a softened or more rapidly scaled measure of separation between points.

In computing, sqrtl2 may also be a user-defined function name or a misnomer for existing functions. The standard square root function sqrt is widely available, and sqrtl refers to the long double variant in some programming languages; sqrtl2 would typically require a specific, explicit definition in the given codebase.

Examples: for x = (3, 4), sum xi^2 = 25, sqrtl2(x) = sqrt(5) ≈ 2.236. For x = (1, 0, 0), sqrtl2(x) = sqrt(1) = 1.

See also: L2 norm, Euclidean distance, Lp norms, root mean square.