Tham khảo tài liệu 'lập trình c# all chap "numerical recipes in c" part 112', công nghệ thông tin phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 732 Chapter 16. Integration of Ordinary Differential Equations dy j yest j else for k 1 k iest k fx k 1 x iest-k xest for j 1 j nv j Evaluate next diagonal in tableau. v d j 1 d j 1 yy c yest j for k 2 k iest k b1 fx k v b b1-c if b b c-v b ddy c b c b1 b else Care needed to avoid division by 0. ddy v if k iest v d j k d j k ddy yy ddy dy j ddy yz j yy free_vector fx 1 iest CITED REFERENCES AND FURTHER READING Stoer J. and Bulirsch R. 1980 Introduction to Numerical Analysis New York Springer-Verlag . 1 Gear . 1971 Numerical Initial Value Problems in Ordinary Differential Equations Englewood Cliffs NJ Prentice-Hall . Deuflhard P. 1983 Numerische Mathematik vol. 41 pp. 399-422. 2 Deuflhard P. 1985 SIAM Review vol. 27 pp. 505-535. 3 Second-Order Conservative Equations Usually when you have a system of high-order differential equations to solve it is best to reformulate them as a system of first-order equations as discussed in . There is a particular class of equations that occurs quite frequently in practice where you can gain about a factor of two in efficiency by differencing the equations directly. The equations are second-order systems where the derivative does not appear on the right-hand side y f x y y xo yo y xo zo As usual y can denote a vector of values. Stoermer s rule dating back to 1907 has been a popular method for discretizing such systems. With h H m we have yi yo h zo 1 hf x0 yo yk 1 - 2yk yk-1 h2 f xo kh yk k . m - 1 zm ym - ym-i h 1 hf xo H ym Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 Second-Order Conservative Equations 733 Here zm is y xo H . Henrici showed howto rewrite equations to reduce roundoff error by using the quantities Ak yk i yk . Start with Ao h z0 1 hf xo yo yi yo Ao Then for k 1 . m 1 set Ak Ak-i hh f xo kh yk yk i yk Ak Finally compute the derivative from zm Am-1 h 1 hf xo H ym Gragg again showed that the error series .
