normalizeL

normalizeL is a function found in several numerical and machine learning libraries that scales a real-valued vector to have unit length under a specified Lp norm. The designation “L” refers to the Lp norm parameter, commonly with a default p = 2, corresponding to the Euclidean norm. In practice, normalizeL computes a normalized vector v_hat = v / ||v||_p, where ||v||_p = (sum_i |v_i|^p)^{1/p} for p >= 1, and ||v||_∞ = max_i |v_i| for p = ∞.

The operation requires handling of edge cases. If the input vector is the zero vector, the norm is zero and normalization is undefined; implementations typically return the original vector, a zero vector, or raise a warning. To improve numerical stability, a small epsilon can be added to the denominator. Different libraries may offer options for behavior when the norm is near zero or for nonstandard p values (e.g., p < 1), though p < 1 yields a quasi-norm that does not satisfy the triangle inequality.

Typical uses of normalizeL include preprocessing for machine learning, where features are scaled to a common length, or in optimization and computer graphics where direction vectors are required without altering magnitude. It is also used in projections and similarity measures where the direction rather than the magnitude is important. Variants such as normalizeL2, normalizeL1, and normalizeLinf correspond to p = 2, p = 1, and p = ∞, respectively, and some libraries expose normalizeL with a parameter to select among these norms.

Time complexity for normalizeL is linear in the vector size, O(n), with constant extra space, assuming a single pass to compute the norm and a second pass to scale.