Approximate message passing (AMP) is an effective iterative sparse recovery algorithm for linear system models. Its performance is characterized by the state evolution (SE) which is a simple scalar recursion. However, depending on a measurement matrix ensemble, AMP may face a convergence problem. To avoid this problem, orthogonal AMP (OAMP), which uses de-correlation linear estimation and divergence-free non-linear estimation, was proposed by Ma and Ping. They also provide the SE analysis for OAMP. In their SE analysis, the following two assumptions were made: (i) The estimated vector of the de-correlation linear estimator consists of i.i.d. zero-mean Gaussian entries independent of the vector to be estimated and (ii) the estimated vector of the divergence-free non-linear estimator consists of i.i.d. entries independent of the measurement matrix and the noise vector. In this paper, we derive a simple scalar recursion to characterize iterative sparse recovery algorithms with divergence-free estimators without such assumptions of independence of messages by using the generating functional analysis (GFA), which allows us to study the dynamics by an exact way in the large system limit.