Adjoint methods and sensitivity analysis for recurrence relations

where subscripts denote partial derivatives, and should be thought of as row vectors, vs. column vectors x and p. So, for example, g x is a 1×K matrix, and f p is aK×P matrix. Equation (1) is derived simply by applying the chain rule to g = g(x,p) = g(f(xn−1,p),p) = g(f(f(xn−2,p),p),p) = · · ·. 1Note that if xn depends on xn−` for ` = 1, . . . , L, the recurrence can still be cast in terms of xn−1 alone by expanding x into a vector of length KL, in much the same way that an Lth-order ODE can be converted into L 1st-order ODEs.