8th Order Runge-Kutta for Integrating System of ODEs (ODE86)

by admin in , , on April 4, 2019

ODE86 Integrates a system of ordinary differential equations using  a 12-stages, 8th and 6th order Runge-Kutta formulas. Better than ODE45 for tolerances stringent than 1e-6.

Forms of using the function ODE86 :

  • [t,y] = ODE86(‘yprime’, t0, tfinal, y0) – integrates the system of ordinary differential equations described by the M-file main.m over the interval T0 to Tfinal and using initial conditions Y0.
  • [t, y] = ODE86(f, t0, tfinal, y0, tol) uses tolerance TOL.


  • f – String containing name of user-supplied problem description.
  • t – Time (scalar).
  • y – Solution column-vector.
  • yprime – Returned derivative column-vector; yprime(i) = dy(i)/dt.
  • t0 – Initial value of t.
  • tfinal- Final value of t.
  • y0 – Initial value column-vector.
  • tol – The desired accuracy. (Default: tol = 1.e-8).


  • t – Returned integration time points (row-vector).
  • y – Returned solution, one solution column-vector per tout-value.
  • ir – Returned number of rejected steps

The result can be displayed by: plot(tout, yout). Example: Solve two-body problem using inline function:
The problem :
y1’=y3, y2’=y4, y3’=-y1/(y1^2+y2^2)^1.5, y4’=-y2/(y1^2+y2^2)^1.5
With initial condtitions y1(0)=.5, y2(0)=0, y3(0)=0, y4(0)=3^0.5
Matlab call :
[x,y]=ode86(inline(‘[y(3);y(4);-y(1)/sqrt(y(1)^2+y(2)^2)^3;-y(2)/sqrt(y(1)^2+y(2)^2)^3]’,’x’,’y’), 0, 20, [.5 0 0 sqrt(3)]’, 1e-11);

The code based on the ODE45 by C.B. Moler, 25-3-1987.

The error control method and coefficients are taken from Ch. Tsitouras and S. N. Papakostas, ‘Cheap Error Estimation for Runge-Kutta methods’, SIAM J. Sci.


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  • Released

    April 4, 2019

  • Last Updated

    May 29, 2019

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