It is difficult to estimate reliably the true error at a single point. For this reason the RMS (root-mean-square) average of the estimated global error in each solution component is computed. This average is taken over all steps from the beginning of the integration through to the current integration point. If all has gone well, the average errors reported will be comparable to tol (see nag_ode_ivp_rkts_setup (d02pqc)). The maximum error seen in any component in the integration so far and the point where the maximum error first occurred are also reported.
Brankin R W, Gladwell I and Shampine L F (1991) RKSUITE: A suite of Runge–Kutta codes for the initial value problems for ODEs SoftReport 91-S1 Southern Methodist University
n – IntegerInput
On entry: , the number of ordinary differential equations in the system to be solved by the integration function.
On exit: approximates the RMS average of the true error of the numerical solution for the th solution component, for . The average is taken over all steps from the beginning of the integration to the current integration point.
errmax – double *Output
On exit: the maximum weighted approximate true error taken over all solution components and all steps.
terrmx – double *Output
On exit: the first value of the independent variable where an approximate true error attains the maximum value, errmax.
The NAG error argument (see Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
On entry, argument had an illegal value.
No error assessment is available since you did not ask for it in your call to the setup function.
On entry, , but the value passed to the setup function was .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
You cannot call this function before you have called the integrator.
On entry, a previous call to the setup function has not been made or the communication arrays have become corrupted, or a catastrophic error has already been detected elsewhere. You cannot continue integrating the problem.
You have already made one call to this function after the the integrator could not achieve specified accuracy. You cannot call this function again.
No error assessment is available since the integrator has not actually taken any successful steps.
8 Parallelism and Performance
9 Further Comments
If the integration has proceeded ‘well’ and the problem is smooth enough, stable and not too difficult then the values returned in the arguments rmserr and errmax should be comparable to the value of tol specified in the prior call to nag_ode_ivp_rkts_setup (d02pqc).
This example integrates a two body problem. The equations for the coordinates of one body as functions of time in a suitable frame of reference are
The initial conditions
lead to elliptic motion with . is selected and the system of ODEs is reposed as
over the range . Relative error control is used with threshold values of for each solution component and a high-order Runge–Kutta method () with tolerance .