Maxabsx

Maxabsx is a mathematical notation used to denote the maximum absolute value among the components of a vector or sequence. It is commonly identified with the L-infinity norm and is widely used in numerical analysis, optimization, and related fields to quantify the size or bound of a set of numbers.

Formal definition: For a vector x = (x1, x2, ..., xn) with real or complex entries, maxabsx = max_i |x_i|, where |x_i| is the absolute value (for complex numbers, the modulus). If x is an empty vector, the value is undefined in strict terms, though some contexts adopt a convention such as zero.

Properties: Maxabsx is nonnegative and equals zero only when all components are zero. It defines a norm, ||x||∞ = max_i |x_i|, which satisfies homogeneity, the triangle inequality, and subadditivity. The unit ball for this norm is a hypercube aligned with the coordinate axes in n-dimensional space.

Applications and computation: In practice, maxabsx measures the largest magnitude among components, helping to bound vector magnitudes, assess numerical stability, or normalize data. In programming and data analysis, functions or operations compute the maximum of the absolute values across elements, sometimes labeled maxabs or implemented as a routine for efficiency. Examples: x = (3, −4, 2) yields maxabsx = 4; x = [1.2, −5.6, 3.0] yields 5.6; for a complex vector (3+4i, 2−5i), the magnitudes are 5 and sqrt(29), so the maximum is sqrt(29).