The
Cash-Karp formula is a 5(4) Runge-Kutta formula for
solving non-stiff systems of initial value ordinary differential equations. It
is fully described in the book

*Numerical
Recipes in C (and also Numerical Recipes in Fortran):
The Art of Scientific Computing (second edition)*

*by** Press, Flannery, Teukolsky and Vetterling.*

It
is also described in

*A
variable order Runge-Kutta method for initial value
problems with rapidly varying right hand sides*

*J.R.
Cash and A.H. Karp, ACM Transactions on Mathematical Software, Vol 16, No.3, Pages 201-222*

The
MATLAB code can be found in

*Numerical
Methods in Engineering with Matlab*

*Jaan
Kiusalaas, CUP, August 2005, pages 280-285*

Programs
which implement this method can be found in the ODE section of the above
Numerical Recipes books.

Then
starting from compute and define

A
solution at an off step point can be computed at
virtually zero extra cost using

,

,

,

,

,

,

,

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