MATHEMATICS OF COMPUTATION, VOLUME 33, NUMBER 148 OCTOBER 1979, PAGES 1251-1256 A Polynomial Representation of Hybrid Methods for Solving Ordinary Differential Equations By G. K. Gupta Abstract. A polynomial representation of the hybrid methods for solving ordinary differential equations is presented. The advantages of the representation are briefly discussed. Also it is shown that one step taken using a hybrid method is equivalent to two steps of the usual multistep methods; one step taken using an explicit method and the other taken using an implicit method. Therefore, the hybrid methods are really a special case of cyclic methods. 1. Introduction. A linear fc-step multistep formula for the solution of the differential equation (i-i) y'=f(x,y), y(x0)=y0 is usually written as k k (1.2) ^ + 1 = EVn + l-r-4^ Zfy',I + 1_,- r=l r=0 The above formula has 2fc + 1 unknown a's and |3's and, therefore, can be of order up to 2fc. However, it was shown by Dahlquist (1956) that the order of the above formula cannot exceed fc + 1 (if fc is odd) or fc + 2 (if fc is even) for the for- mula to be stable. It was suggested by several authors, e.g., Butcher (1965), Gear (1965), and Gragg and Stetter (1964), that a slight modification of the above formula can overcome the stability condition imposed. The modified formulas were given the name 'hybrid methods' by Gear (1965), because they seem to combine the features of Runge- Kutta and the linear multistep methods by evaluating the function / at 'off-step' points. Thus, the hybrid method with one 'off-step' point looks like (there is no reason why the number of off-step points should be restricted to one) O-3) yn + l =E «ryn + l-r+hi:ßSn + l-r + «ß8fn+l-e- r=l r=0 The order of the above formula can be as high as 2fc + 2 if we include 0 as a variable. In practice, it has been shown by Kohfeld and Thompson (1967) that stable formulas of order 2fc + 2 can be obtained only for fc < 6. Stable formulas of order 2fc + 1, Received November 21, 1978. AMS (MOS) subject classifications (1970). Primary 65L05; Secondary 65D30. Key words and phrases. Linear multistep methods, hybrid methods, numerical solution of ordinary differential equations. © 1979 American Mathematical Society 0025-571 8/79/0000-01 57/$02.50 1251 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use