MATHEMATICS OF COMPUTATION, VOLUME 32, NUMBER 141 JANUARY 1978, PAGES 13-18 Implementing Second-Derivative Multistep Methods Using the Nordsieck Polynomial Representation By G. K. Gupta Abstract. A polynomial representation for the second-derivative linear multistep methods for solving ordinary differential equations is presented. This representation leads to an implementation of the second-derivative methods using the Nordsieck poly- nomial representation. Possible advantages of such an implementation are then discussed. 1. Introduction. In this paper we are concerned with the second-derivative linear multistep methods (formulae). The differential equation being solved is (1.1) y'=f(x,y), y(0)=y0. The second-derivative fc-step multistep formula may be written as (1-2) yn+l = ¿ vn+i-, + * i ßry'n+1-r + "2 £ y/n+l-r- r=l r=0 r=0 Several authors have recently studied such formulae and other formulae which include higher derivatives, for example, Makinson (1968), Genin (1974), Makela et al. (1974), Enright (1974a, b) and Liniger and Willoughby (1970). Also Lambert (1973, Sections 7.2 and 8.11) discusses such methods and calls them Obrechkoff methods. The moti- vation for studying second-derivative and higher derivative formulae is that the usual multistep methods cannot be A -stable for orders higher than 2 (Dahlquist (1963)), while .4-stable multi-derivative formulae of higher orders exist as shown by Genin (1974) and Jeltsch (1975). Therefore, higher derivative multistep formulae may be suitable for solving stiff equations. Also while solving stiff equations, the Jacobian df/dy is required in the corrector iterations anyway; and therefore,./' = (df/by)f + df/dx can be computed quite easily. Enright (1974a, b) has presented a subroutine SDBASIC for solving stiff equa- tions and it was shown by Enright et al. (1975) that this subroutine is efficient and reliable for solving a wide range of stiff test problems. In this paper, we present a polynomial representation of the second-derivative multistep methods and discuss how this representation may be used in implementing the second-derivative methods using the Nordsieck representation. Advantages of such implementations are then briefly discussed. The discussion in this paper is easily ex- tended to higher derivative multistep methods. Received March 31, 1977. AMS (MOS) subject classifications (1970). Primary 65L05; Secondary 65D30. Key words and phrases. Linear multistep methods, stiff differential equations, multideriv- ative methods, numerical solutions of ordinary differential equations. Copyright e 1978. American Mithenutlcil Society 13 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use