sqrtss

sqrtss stands for the square root of the sum of squares. For a vector v = (x1, ..., xn), sqrtss(v) = sqrt(x1^2 + x2^2 + ... + xn^2). This quantity is the length or magnitude of the vector and is the Euclidean norm, commonly denoted ||v||2.

Properties of sqrtss include nonnegativity, with sqrtss(v) = 0 if and only if all components are zero. It is homogeneous of degree 1, meaning sqrtss(k·v) = |k|·sqrtss(v) for any scalar k. It satisfies the triangle inequality: sqrtss(u + w) ≤ sqrtss(u) + sqrtss(w). The value depends only on the magnitudes of the components and is invariant under sign changes of individual elements.

Interpretation and computation: in two dimensions, sqrtss(a, b) = sqrt(a^2 + b^2), giving the ordinary length of the vector in the plane. It also represents the Euclidean distance from the origin to the point (a, b). The distance between two vectors u and v can be expressed as sqrtss(u − v). In programming, this operation is often implemented with stable routines such as hypot in several languages, or as the norm function in numerical libraries.

Examples: sqrtss(3, 4) = 5; sqrtss(1, 2, 2) = 3. Applications include vector normalization, distance calculations in geometry and data analysis, and as the L2 norm in optimization and machine learning (where its square is used in regularization).