Automatic differentiation of the vector that solves a parametric linear system

Generalizing Cramer's Rule: Solving Uniformly Linear Systems of Equations

“…By the same reasoning as in the previous section, we obtain Equation (8). Thus, if for every (α, β) in S m,n,k, , we multiply both sides of Equation (8) by µ α,β t |α|−|β| and add up all these equalities, we obtain the following expression:…”

“…Solving parametric linear systems of equations is an interesting computational issue encountered, for example, when computing the characteristic solutions for differential equations (see [5]), when dealing with coloured Petri nets (see [13]) or in linear prediction problems and in different applications of operation research and engineering (see, for example, [6], [8], [12], [15] or [19] where the elements of the matrix have a linear affine dependence on a set of parameters). More general situations where the elements are polynomials in a given set of parameters are encountered in computer algebra problems (see [1] and [18]).…”

“…such that (1, 1, 0) is a concrete zero of our system and 1 t+3 (0, 8,6) is solution of the homogenous linear system defined by A.…”