Linearity Analysis for Automatic Differentiation

Linearity analysis determines which variables depend on which other variables and whether the dependence is linear or nonlinear. One of the many applications of this analysis is determining whether a loop involves only linear loop-carried dependences and therefore the adjoint of the loop may be reversed and fused with the computation of the original function. This paper specifies the data-flow equations that compute linearity analysis. In addition, the paper describes using linearity analysis with array dependence analysis to determine whether a loop-carried dependence is linear or nonlinear.