# NAG C Library Function Document

## 1Purpose

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.

## 2Specification

 #include #include
 void nag_ode_ivp_rk_step_revcomm (Integer *irevcm, Integer n, double *t, double y[], const double yp[], Integer iwsav[], double rwsav[], NagError *fail)

## 3Description

nag_ode_ivp_rk_step_revcomm (d02pgc) and its associated functions (nag_ode_ivp_rk_interp_setup (d02phc), nag_ode_ivp_rk_interp_eval (d02pjc), nag_ode_ivp_rkts_setup (d02pqc), nag_ode_ivp_rkts_reset_tend (d02prc), nag_ode_ivp_rkts_diag (d02ptc) and nag_ode_ivp_rkts_errass (d02puc)) solve an initial value problem for a first-order system of ordinary differential equations. The functions, based on Runge–Kutta methods and derived from RKSUITE (see Brankin et al. (1991)), integrate
 $y′=ft,y given yt0=y0$
where $y$ is the vector of $\mathit{n}$ solution components and $t$ 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 ${\mathbf{errass}}=\mathrm{Nag_ErrorAssess_on}$ 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:
 (i) The more efficient way is to step past the point where a solution is desired, and then call nag_ode_ivp_rk_interp_setup (d02phc) and nag_ode_ivp_rk_interp_eval (d02pjc) to get an answer there. Within the span of the current step, you can get all the answers you want at very little cost by repeated calls to nag_ode_ivp_rk_interp_eval (d02pjc). This is very valuable when you want to find where something happens, e.g., where a particular solution component vanishes. (ii) 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 (${\mathbf{t}}={\mathbf{tend}}$), 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

## 5Arguments

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 than those specified by the value of irevcm must remain unchanged.
1:    $\mathbf{irevcm}$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 $\text{}>0$ to indicate that a function evaluation is required prior to re-entry; the value of the derivatives ${y}^{\prime }=f\left(t,y\right)$ must be returned in yp where the value of $t$ is supplied in t and the values $y\left(t\right)$ are supplied in the array y. The value of irevcm indicates the reason for the function evaluation as follows:
${\mathbf{irevcm}}=1$
For initial entry values of t and y.
${\mathbf{irevcm}}=2$
To determine stiffness of system.
${\mathbf{irevcm}}=3$
For the stages of the primary step.
${\mathbf{irevcm}}=4$
A final stage of the primary step.
${\mathbf{irevcm}}=5$
For the stages of a secondary step (if global error assessment is required).
On final exit:
${\mathbf{irevcm}}=-1$
Successful exit; t, y and yp contain the solution at the end of a successful integration step.
${\mathbf{irevcm}}=-2$
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.
2:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of ordinary differential equations in the system to be solved.
Constraint: ${\mathbf{n}}\ge 1$. This must be the same value as supplied in a previous call to nag_ode_ivp_rkts_setup (d02pqc).
3:    $\mathbf{t}$double *Output
On intermediate exit: t contains the value of the independent variable $t$ at which the derivatives ${y}^{\prime }$ are to be evaluated.
On final exit: the value of $t$ at which a solution has been computed following a successful step.
4:    $\mathbf{y}\left[{\mathbf{n}}\right]$doubleOutput
On intermediate exit: y contains the value of the solution $y$ at which the derivatives ${y}^{\prime }$ are to be evaluated.
On final exit: the approximation to the solution computed following a successful step.
5:    $\mathbf{yp}\left[{\mathbf{n}}\right]$const doubleInput
On initial entry: yp need not be set.
On intermediate re-entry: yp must contain the value of the derivatives ${y}^{\prime }=f\left(t,y\right)$ where $t$ is supplied in t and $y$ is supplied in the array y.
6:    $\mathbf{iwsav}\left[130\right]$IntegerCommunication Array
7:    $\mathbf{rwsav}\left[32×{\mathbf{n}}+350\right]$doubleCommunication Array
On entry: these must be the same arrays supplied in a previous call to nag_ode_ivp_rkts_setup (d02pqc). They must remain unchanged between calls.
On exit: information about the integration for use on subsequent calls to nag_ode_ivp_rk_step_revcomm (d02pgc) or other associated functions.
8:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
${\mathbf{irevcm}}<0$ on entry.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT_CHANGED
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, but the value passed to the setup function was ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
NE_INTERNAL_ERROR
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.
NE_MISSING_CALL
On entry, a previous call to the setup function has not been made or the communication arrays have become corrupted.
NE_NO_LICENCE
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.
NE_PREV_CALL
On entry, the communication arrays have become corrupted, or a catastrophic error has already been detected elsewhere. You cannot continue integrating the problem.
NE_PREV_CALL_INI
A call to this function cannot be made after it has returned an error.
The setup function must be called to start another problem.
NE_RK_GLOBAL_ERROR_S
The global error assessment algorithm failed at start of integration.
The integration is being terminated.
NE_RK_GLOBAL_ERROR_T
The global error assessment may not be reliable for times beyond $〈\mathit{\text{value}}〉$.
The integration is being terminated.
NE_RK_POINTS
More than $100$ 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.
NE_RK_STEP_TOO_SMALL
In order to satisfy your error requirements the solver has to use a step size of $〈\mathit{\text{value}}〉$ at the current time, $〈\mathit{\text{value}}〉$. This step size is too small for the machine precision, and is smaller than $〈\mathit{\text{value}}〉$.
NE_RK_TGOT_EQ_TEND
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.
NE_STIFF_PROBLEM
Approximately $〈\mathit{\text{value}}〉$ 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 $〈\mathit{\text{value}}〉$ 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.
NW_RK_TOO_MANY
Approximately $〈\mathit{\text{value}}〉$ 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.

## 7Accuracy

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.

## 8Parallelism 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.

## 9Further Comments

If nag_ode_ivp_rk_step_revcomm (d02pgc) returns with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_RK_GLOBAL_ERROR_S, NE_RK_GLOBAL_ERROR_T, NE_RK_POINTS, NE_RK_STEP_TOO_SMALL, NE_STIFF_PROBLEM or NW_RK_TOO_MANY then the values returned in t and y are for the last successful step, or the initial conditions supplied if no successful step has been taken.
If nag_ode_ivp_rk_step_revcomm (d02pgc) returns with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ 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.
Performance statistics are available after any return from nag_ode_ivp_rk_step_revcomm (d02pgc) (except when ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_BAD_PARAM, NE_INT_CHANGED, NE_MISSING_CALL, NE_PREV_CALL, NE_PREV_CALL_INI or NE_RK_TGOT_EQ_TEND) by a call to nag_ode_ivp_rkts_diag (d02ptc). If ${\mathbf{errass}}=\mathrm{Nag_ErrorAssess_on}$ in the call to nag_ode_ivp_rkts_setup (d02pqc), global error assessment is available after any return from nag_ode_ivp_rk_step_revcomm (d02pgc) (except when ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_BAD_PARAM, NE_INT_CHANGED, NE_MISSING_CALL, NE_PREV_CALL, NE_PREV_CALL_INI or NE_RK_TGOT_EQ_TEND) by a call to nag_ode_ivp_rkts_errass (d02puc).
After a failure with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_RK_GLOBAL_ERROR_S, NE_RK_GLOBAL_ERROR_T or NE_RK_STEP_TOO_SMALL each of the diagnostic functions nag_ode_ivp_rkts_diag (d02ptc) and nag_ode_ivp_rkts_errass (d02puc) may be called only once.
If nag_ode_ivp_rk_step_revcomm (d02pgc) returns with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ 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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ 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.

## 10Example

This example solves the equation
 $y′′ = -y , y0 = 0 , y′0 = 1$
reposed as
 $y1′ = y2$
 $y2′ = -y1$
over the range $\left[0,2\pi \right]$ with initial conditions ${y}_{1}=0.0$ and ${y}_{2}=1.0$. We use relative error control with threshold values of $\text{1.0e−8}$ 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 ${y}_{1}$ or ${y}_{2}$ 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 (${\mathbf{method}}=\mathrm{Nag_RK_4_5}$) with tolerance ${\mathbf{tol}}=\text{1.0e−5}$.

### 10.1Program Text

Program Text (d02pgce.c)

### 10.2Program Data

Program Data (d02pgce.d)

### 10.3Program Results

Program Results (d02pgce.r)