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In this paper we propose a simple recursive algorithm to assign eigenvalues real or mixed (real and complex) in a Hessenberg matrix. This problem of assigning eigenvalues in a Hessenberg matrix is considered by many authors. Among these methods is the conceptually easy and simple algorithm introduced by Datta for real eigenvalue assignment case. Recently, the problem was reconsidered by Ramadan and El-Shazly where mixed eigenvalues are assigned by modifying the Datta algorithm. In the presented paper, we consider the general case: that is the controllable pair ðH, e 1 Þ where e 1 is the first column of the n  n identity matrix I considered in both Datta and El-Shazly is in the more general form: ðH, bÞ where b is an arbitrary vector. This will be carried out by modifying both algorithms. Numerical examples are given to demonstrate the performance and accuracy of the proposed algorithm.