nag_ode_ivp_rk_step_revcomm (d02pgc) is a reverse communication one-step function for solving an initial value problem for a first-order system of ordinary differential equations using Runge–Kutta methods. The direct communication version of this function is nag_ode_ivp_rkts_onestep (d02pfc). See Section 3.3.2 in How to Use the NAG Library and its Documentation for the difference between forward and reverse communication.
where is the vector of solution components and is the independent variable.
nag_ode_ivp_rk_step_revcomm (d02pgc) is designed to be used in complicated tasks when solving systems of ordinary differential equations. You must first call nag_ode_ivp_rkts_setup (d02pqc) to specify the problem and how it is to be solved. Thereafter you (repeatedly) call nag_ode_ivp_rk_step_revcomm (d02pgc) in reverse communication loops to take one integration step at a time from tstart in the direction of tend (as specified in nag_ode_ivp_rkts_setup (d02pqc)). In this manner nag_ode_ivp_rk_step_revcomm (d02pgc) returns an approximation to the solution y and its derivative yp at successive points t. If nag_ode_ivp_rk_step_revcomm (d02pgc) encounters some difficulty in taking a step, the integration is not advanced and the function returns with the same values of t, y and yp as returned on the previous successful step. nag_ode_ivp_rk_step_revcomm (d02pgc) tries to advance the integration as far as possible subject to passing the test on the local error and not going past tend.
In the call to nag_ode_ivp_rkts_setup (d02pqc) you can specify either the first step size for nag_ode_ivp_rk_step_revcomm (d02pgc) to attempt or it computes automatically an appropriate value. Thereafter nag_ode_ivp_rk_step_revcomm (d02pgc) estimates an appropriate step size for its next step. This value and other details of the integration can be obtained after a completed step by nag_ode_ivp_rk_step_revcomm (d02pgc) by a call to nag_ode_ivp_rkts_diag (d02ptc). The local error is controlled at every step as specified in nag_ode_ivp_rkts_setup (d02pqc). If you wish to assess the true error, you must set
in the call to nag_ode_ivp_rkts_setup (d02pqc). This assessment can be obtained after any call to nag_ode_ivp_rk_step_revcomm (d02pgc) by a call to nag_ode_ivp_rkts_errass (d02puc).
If you want answers at specific points there are two ways to proceed:
Alternatively, set tend to the desired value and integrate to tend. nag_ode_ivp_rk_step_revcomm (d02pgc) will not step past tend, so when a step would carry it past, it will reduce the step size so as to produce an answer at tend exactly. After getting an answer there (), you can reset tend to the next point where you want an answer, and repeat. tend could be reset by a call to nag_ode_ivp_rkts_setup (d02pqc), but you should not do this. You should use nag_ode_ivp_rkts_reset_tend (d02prc) instead because it is both easier to use and much more efficient. This way of getting answers at specific points can be used with any of the available methods, but it can be inefficient. Should this be the case, the code will bring the matter to your attention.
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
Note: this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument irevcm. Between intermediate exits and re-entries, all arguments other thanthose specified by the value of irevcm must remain unchanged.
– Integer *Input/Output
On initial entry: irevcm must be set to zero to indicate that a new step is being taken.
On intermediate re-entry: irevcm should remain unchanged.
On intermediate exit:
irevcm returns a value to indicate that a function evaluation is required prior to re-entry; the value of the derivatives must be returned in yp where the value of is supplied in t and the values are supplied in the array y. The value of irevcm indicates the reason for the function evaluation as follows:
For the stages of a secondary step (if global error assessment is required).
On final exit:
Successful exit; t, y and yp contain the solution at the end of a successful integration step.
Error exit; fail should be interrogated to determine the nature of the error.
Note: any values you return to nag_ode_ivp_rk_step_revcomm (d02pgc) as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_ode_ivp_rk_step_revcomm (d02pgc). If your code inadvertently does return any NaNs or infinities, nag_ode_ivp_rk_step_revcomm (d02pgc) is likely to produce unexpected results.
On entry: , the number of ordinary differential equations in the system to be solved.
On exit: information about the integration for use on subsequent calls to nag_ode_ivp_rk_step_revcomm (d02pgc) or other associated functions.
– NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).
Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 18.104.22.168 in How to Use the NAG Library and its Documentation for further information.
On entry, argument had an illegal value.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
On entry, a previous call to the setup function has not been made or the communication arrays have become corrupted.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, the communication arrays have become corrupted, or a catastrophic error has already been detected elsewhere. You cannot continue integrating the problem.
A call to this function cannot be made after it has returned an error. The setup function must be called to start another problem.
The global error assessment algorithm failed at start of integration.
The integration is being terminated.
The global error assessment may not be reliable for times beyond .
The integration is being terminated.
More than output points have been obtained by integrating to tend (as specified in the setup function). They have been so clustered that it would probably be (much) more efficient to use the interpolation function. However, you can continue integrating the problem.
In order to satisfy your error requirements the solver has to use a step size of at the current time, . This step size is too small for the machine precision, and is smaller than .
tend, as specified in the setup function, has already been reached. To start a new problem, you will need to call the setup function. To continue integration beyond tend then nag_ode_ivp_rkts_reset_tend (d02prc) must first be called to reset tend to a new end value.
Approximately function evaluations have been used to compute the solution since the integration started or since this message was last printed. Your problem has been diagnosed as stiff. If the situation persists, it will cost roughly times as much to reach tend (setup) as it has cost to reach the current time. You should probably call functions intended for stiff problems. However, you can continue integrating the problem.
Approximately function evaluations have been used to compute the solution since the integration started or since this message was last printed. However, you can continue integrating the problem.
The accuracy of integration is determined by the arguments tol and thresh in a prior call to nag_ode_ivp_rkts_setup (d02pqc) (see the function document for nag_ode_ivp_rkts_setup (d02pqc) for further details and advice). Note that only the local error at each step is controlled by these arguments. The error estimates obtained are not strict bounds but are usually reliable over one step. Over a number of steps the overall error may accumulate in various ways, depending on the properties of the differential system.
Parallelism and Performance
nag_ode_ivp_rk_step_revcomm (d02pgc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
If nag_ode_ivp_rk_step_revcomm (d02pgc) returns with NE_RK_STEP_TOO_SMALL and the accuracy specified by tol and thresh is really required then you should consider whether there is a more fundamental difficulty. For example, the solution may contain a singularity. In such a region the solution components will usually be large in magnitude. Successive output values of y should be monitored with the aim of trapping the solution before the singularity. In any case, numerical integration cannot be continued through a singularity and analytical treatment may be necessary.
If nag_ode_ivp_rk_step_revcomm (d02pgc) returns with NE_STIFF_PROBLEM then it is advisable to change to another code more suited to the solution of stiff problems. nag_ode_ivp_rk_step_revcomm (d02pgc) will not return with NE_STIFF_PROBLEM if the problem is actually stiff but it is estimated that integration can be completed using less function evaluations than already computed.
This example solves the equation
over the range with initial conditions and . We use relative error control with threshold values of for each solution component and print the solution at regular intervals using the interpolation functions nag_ode_ivp_rk_interp_setup (d02phc) and nag_ode_ivp_rk_interp_eval (d02pjc) within integration steps across the range; points on the range at which or change sign are also evaluated using a combination of the root finding function nag_zero_cont_func_brent_rcomm (c05azc) and the interpolation functions. We use a medium order Runge–Kutta method
() with tolerance .