In this paper we try to show the relations between the Lanczos algorithm and Pad'e approximations as used e.g. in identification and model reduction of dynamical systems. We also explore the use of variants of the Lanczos method in order to obtain approximations with better properties than the ones resulting from the standard Lanczos algorithm. 1 Introduction For simplicity we assume here that all systems are SISO, although some results do extend to the MIMO case. Let a n-th order dynamical system be described by x = Ax + bu (1.1) y = cx + du (1.2) where A is a square, b is a column vector, c is a row vector, and d is a scalar. It is well-known that the transfer function of this system : h(s) = c(sI Gamma A) Gamma1 b + d has a Taylor expansion around s = 1 that looks like : h(s) = d + cbs Gamma1 + cAbs Gamma2 + cA 2 bs Gamma3 + cA 3 bs Gamma4 + : : : : The coefficients m Gammai of the powers of s Gammai satisfy thus m 0 = d ; m Gammai = cA iGamma1 b ...