Cash–Karp method
Encyclopedia
In numerical analysis
, the Cash–Karp method is a method for solving ordinary differential equations (ODEs). It was proposed by Professor Jeff R. Cash from Imperial College London
and Alan H. Karp from IBM
Scientific Center. The method is a member of the Runge–Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Fehlberg (RKF) and Dormand–Prince (RKDP).
The Butcher tableau is:
The first row of b coefficients gives the fifth-order accurate solution, and the second row gives the fourth order solution.
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
, the Cash–Karp method is a method for solving ordinary differential equations (ODEs). It was proposed by Professor Jeff R. Cash from Imperial College London
Imperial College London
Imperial College London is a public research university located in London, United Kingdom, specialising in science, engineering, business and medicine...
and Alan H. Karp from IBM
IBM
International Business Machines Corporation or IBM is an American multinational technology and consulting corporation headquartered in Armonk, New York, United States. IBM manufactures and sells computer hardware and software, and it offers infrastructure, hosting and consulting services in areas...
Scientific Center. The method is a member of the Runge–Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Fehlberg (RKF) and Dormand–Prince (RKDP).
The Butcher tableau is:
| 0 | ||||||
| | 1/5 | 1/5 | |||||
| | 3/10 | 3/40 | 9/40 | ||||
| | 3/5 | 3/10 | −9/10 | 6/5 | |||
| | 1 | −11/54 | 5/2 | −70/27 | 35/27 | ||
| | 7/8 | 1631/55296 | 175/512 | 575/13824 | 44275/110592 | 253/4096 | |
| | | 37/378 | 0 | 250/621 | 125/594 | 0 | 512/1771 |
| | | 2825/27648 | 0 | 18575/48384 | 13525/55296 | 277/14336 | 1/4 |
The first row of b coefficients gives the fifth-order accurate solution, and the second row gives the fourth order solution.
External links
- J. R. Cash, A. H. Karp. "A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides", ACM Transactions on Mathematical Software 16: 201-222, 1990. .