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(y,dydx,nvar,&x,h,eps,yscal,&hdid,&hnext,derivs); if (hdid == h) ++(*nok); else ++(*nbad); if ((x-x2)*(x2-x1) = 0.0) { Are we done? for (i=1;i | 722 Chapter 16. Integration of Ordinary Differential Equations rkqs y dydx nvar x h eps yscal hdid hnext derivs if hdid h nok else nbad if x-x2 x2-x1 0.0 Are we done for i 1 i nvar i ystart i y i if kmax xp kount x Save final step. for i 1 i nvar i yp i kount y i free_vector dydx 1 nvar free_vector y 1 nvar free_vector yscal 1 nvar return Normal exit. if fabs hnext hmin nrerror Step size too small in odeint h hnext nrerror Too many steps in routine odeint CITED REFERENCES AND FURTHER READING Gear C.W. 1971 Numerical Initial Value Problems in Ordinary Differential Equations Englewood Cliffs NJ Prentice-Hall . 1 Cash J.R. and Karp A.H. 1990 ACM Transactions onMathematical Software vol. 16 pp. 201222. 2 Shampine L.F. and Watts H.A. 1977 in Mathematical Software III J.R. Rice ed. New York Academic Press pp. 257-275 1979 Applied Mathematics and Computation vol. 5 pp. 93-121. Forsythe G.E. Malcolm M.A. and Moler C.B. 1977 Computer Methods for Mathematical Computations Englewood Cliffs NJ Prentice-Hall . 16.3 Modified Midpoint Method This section discusses the modified midpoint method which advances a vector of dependent variables y x from a point x to a point x H by a sequence of n substeps each of size h h H n 16.3.1 In principle one could use the modified midpoint method in its own right as an ODE integrator. In practice the method finds its most important application as a part of the more powerful Bulirsch-Stoer technique treated in 16.4. You can therefore consider this section as a preamble to 16.4. The number of right-hand side evaluations required by the modified midpoint method is n 1. The formulas for the method are zo y x Z1 zo hf x zo Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 zm i zm-i 2hf x mh zm for m 1 2 . n 1 y x H yn 2 z zn-i hf x H zn 16.3.2 16.3 Modified Midpoint Method 723 Here the z s are intermediate approximations which march along in steps of h while yn is the final approximation to y x H . The method
