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Convenient recursive prediction error algorithms for identification and adaptive state estimation are proposed, and the convergence of these algorithms to achieve off-line prediction error minimization solutions is studied. To set the recursive prediction error algorithms in another perspective, specializations are derived from significant simplifications to a class of extended Kalman filters. The latter are designed for linear state space models with the unknown parameters augmenting the state vector and in such a way as to yield good convergence properties. Also, specializations to approximate maximum likelihood recursions, Kalman filters with adaptive gains, and connections to the extended least squares algorithms are noted.