In
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
, 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 contrast w ...
(ODEs). It was proposed by Professor Jeff R. Cash
Jeff R. Cash, Professor of Numerical Analysis, Imperial College London
/ref> from 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 cu ...
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 Butcher tableau
A butcher is a person who may slaughter animals, dress their flesh, sell their meat, or participate within any combination of these three tasks. They may prepare standard cuts of meat and poultry for sale in retail or wholesale food establishm ...
is:
The first row of ''b'' coefficients gives the fifth-order accurate solution, and the second row gives the fourth-order solution.
See also
* Adaptive Runge–Kutta methods
* List of Runge–Kutta methods
Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation
:\frac = f(t, y).
Explicit Runge–Kutta methods take the form
:\begin
y_ &= y_n + h \sum_^s b_i k_i \\
k_1 &= f(t_n, y_n), \\
k_2 &= f(t_n+c_2h ...
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