numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...

, the Cash–Karp method is a method for solving

ordinary differential equations In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...

(ODEs). It was proposed by Professor Jeff R. Cash Jeff R. Cash, Professor of Numerical Analysis, Imperial College London
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Imperial College London Imperial College London (legally Imperial College of Science, Technology and Medicine) is a public research university in London, United Kingdom. Its history began with Prince Albert, consort of Queen Victoria, who developed his vision for a ...

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 In mathematics and numerical analysis, an adaptive step size is used in some methods for the numerical solution of ordinary differential equations (including the special case of numerical integration) in order to control the errors of the method ...

integration algorithms. Other similar integration methods are Fehlberg (RKF) and Dormand–Prince (RKDP). The

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is: The first row of ''b'' coefficients gives the fifth-order accurate solution, and the second row gives the fourth-order solution.

Notes

References

* 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. . {{DEFAULTSORT:Cash-Karp Method Numerical differential equations Runge–Kutta methods

See also

Notes

References