Varmat

Varmat is a term used in numerical computing to denote a matrix whose entries are differentiable variables, usually in the context of automatic differentiation. It is not a formal data type defined by a single library; rather, it describes a pattern where a matrix is built from variables that carry derivative information alongside function values.

In reverse-mode autodiff frameworks, a var-matrix enables derivative propagation through matrix expressions. A matrix of differentiable variables can be used to compute Jacobians and Hessians with respect to a parameter vector by composing matrix operations with derivative variables. In practice, developers may refer to a "varmat" when discussing arrays or matrices of autodiff variables, and libraries that implement automatic differentiation with scalar "var" types often support matrix-valued variables or wrappers.

Implementation notes: Each entry in a var-matrix stores a primal value and its derivatives; arithmetic operators are overloaded to propagate derivatives via the chain rule. Performance depends on memory layout, representation, and the efficiency of the underlying autodiff engine. Some frameworks provide specialized matrix types or views to support efficient differentiation of linear algebra operations.

Applications: varmat concepts appear in optimization, statistical modeling, and numerical analysis, where one needs derivatives of matrix-valued functions, such as Jacobians, sensitivities, or gradient-based solvers.

See also: automatic differentiation, Jacobian, Hessian, matrix calculus, Stan Math, matrix-valued differentiation.