linearequadratische

Linearequadratische is a German compound adjective used in mathematics and related fields to describe concepts that combine linear and quadratic components. As a descriptive label, it is not a single formal theory, but a way to identify equations, functions, models, or cost functions that mix linear terms with quadratic terms. The term is commonly encountered in algebra, analysis, optimization, and control theory, where mixed linear-quadratic structures arise naturally.

Typical forms described by this term include simple univariate functions such as f(x) = a x + b x^2 + c, and multivariate or matrix representations like f(x) = x^T A x + b^T x + c, with A symmetric. In dynamics and differential equations, linearequadratische terms may appear in equations of motion or growth models, for example dy/dt = a y + b y^2 + c. In optimization and control, the phrase often points to linear-quadratic problems, such as the Linear-Quadratic Regulator (LQR), where the objective is a quadratic function of the state and control variables while the system dynamics are linear. The concept also appears in statistics and physics where models combine linear predictors with quadratic response terms.

Relation to other terms: linearequadratische emphasizes a mixed structure rather than a purely linear or purely quadratic model. When precision is required, more specific terminology is used, for example lineare-quadratische Kostenfunktion in optimal control, lineare Quadratische Programmierung (quadratic programming) in optimization, or simply linear-quadratic models in various applied contexts.