Multilinearly

Multilinearly is an adverb derived from the adjective multilinear, which describes an operation, relationship, or function that preserves linearity in more than one independent variable or dimension simultaneously. The term is used primarily in mathematics, physics and computer science to describe phenomena that behave in a linear fashion when observed along multiple axes, but not necessarily on a single line.

In mathematics, multilinear functions are mappings from a product of vector spaces to a single vector space that are linear in each argument when the others are held fixed. When discussing such functions in prose, one may say that a tensor varies multilinearly with its indices, meaning that each component depends linearly on each argument. When properties hold for all arguments jointly, one may say the function operates multilinearly.

In physics, the concept is applied to interactions that depend linearly on several fields simultaneously, such as a stress tensor that is multilinear in strain components. In computer science, multilinearly is used in the analysis of algorithms that exhibit linear growth with respect to several input parameters independently. For example, an algorithm might take a time that is linear in the number of users and also linear in the number of operations, so it operates multilinearly with respect to both dimensions.

The prefix “multi‑” indicates involvement of more than one variable, while “linearly” refers to the linear relationship with each variable. Therefore, multilinearly is an adverbial form used to express the simultaneous linear dependence on multiple variables.