tailieunhanh - Lập Trình C# all Chap "NUMERICAL RECIPES IN C" part 116

Tham khảo tài liệu 'lập trình c# all chap "numerical recipes in c" part 116', công nghệ thông tin phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 710 Chapter 16. Integration of Ordinary Differential Equations CITED REFERENCES AND FURTHER READING Gear . 1971 Numerical Initial Value Problems in Ordinary Differential Equations Englewood Cliffs NJ Prentice-Hall . Acton . 1970 Numerical Methods That Work 1990 corrected edition Washington Mathematical Association of America Chapter 5. Stoer J. and Bulirsch R. 1980 Introduction to NumericalAnalysis New York Springer-Verlag Chapter 7. Lambert J. 1973 Computational Methods in Ordinary Differential Equations New York Wiley . Lapidus L. and Seinfeld J. 1971 Numerical Solution of Ordinary Differential Equations New York Academic Press . Runge-Kutta Method The formula for the Euler method is yn 1 yn hf xn yn which advances a solutionfrom xn to xn 1 xn h. The formula is unsymmetrical It advances the solution through an interval h but uses derivative information only at the beginning of that interval see Figure . That means and you can verify by expansion in power series that the step s error is only one power of h smaller than the correction O h2 added to . There are several reasons that Euler s method is not recommended for practical use among them i the method is not very accurate when compared to other fancier methods run at the equivalent stepsize and ii neither is it very stable see below . Consider however the use of a step like to take a trial step to the midpoint of the interval. Then use the value of both x and y at that midpoint to compute the real step across the whole interval. Figure illustrates the idea. In equations ki hf Xn yn k-2 hf xn 2 h yn 1 kJ yn 1 yn k2 O h 31 As indicated in the error term this symmetrization cancels out the first-order error term making the method second order. A method is conventionally called nth order if its error term is O hn 1 . In fact is called the second-order Runge-Kutta or midpoint method. We needn t stop there. There are many ways to evaluate the .

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