d02pgf is a reverse communication one-step routine 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 routine is d02pff. See Section 7 in How to Use the NAG Library for the difference between forward and reverse communication.
The routine may be called by the names d02pgf or nagf_ode_ivp_rk_step_revcomm.
3Description
d02pgf and its associated routines
(d02phf,d02pjf,d02pqf,d02prf,d02ptfandd02puf) solve an initial value problem for a first-order system of ordinary differential equations. The routines, based on Runge–Kutta methods and derived from RKSUITE (see Brankin et al. (1991)), integrate
$${y}^{\prime}=f(t,y)\text{\hspace{1em} given \hspace{1em}}y\left({t}_{0}\right)={y}_{0}$$
where $y$ is the vector of $\mathit{n}$ solution components and $t$ is the independent variable.
d02pgf is designed to be used in complicated tasks when solving systems of ordinary differential equations. You must first call d02pqf to specify the problem and how it is to be solved. Thereafter you (repeatedly) call d02pgf in reverse communication loops to take one integration step at a time from tstart in the direction of tend (as specified in d02pqf). In this manner d02pgf returns an approximation to the solution y and its derivative yp at successive points t. If d02pgf encounters some difficulty in taking a step, the integration is not advanced and the routine returns with the same values of t, y and yp as returned on the previous successful step. d02pgf 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 d02pqf you can specify either the first step size for d02pgf to attempt or it computes automatically an appropriate value. Thereafter d02pgf 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 d02pgf by a call to d02ptf. The local error is controlled at every step as specified in d02pqf. If you wish to assess the true error, you must set
method to a positive value
in the call to d02pqf. This assessment can be obtained after any call to d02pgf by a call to d02puf.
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 d02phfandd02pjf 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 d02pjf. 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. d02pgf 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 d02pqf, but you should not do this. You should use d02prf 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.
4References
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 routine 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.
1: $\mathbf{irevcm}$ – IntegerInput/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 ${y}^{\prime}=f(t,y)$ 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:
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; ifail should be interrogated to determine the nature of the error.
Note: any values you return to d02pgf 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 d02pgf. If your code does inadvertently return any NaNs or infinities, d02pgf 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 d02pqf.
3: $\mathbf{t}$ – Real (Kind=nag_wp)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: $\mathbf{y}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
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: $\mathbf{yp}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On intermediate re-entry: yp must contain the value of the derivatives ${y}^{\prime}=f(t,y)$ where $t$ is supplied in t and $y$ is supplied in the array y.
7: $\mathbf{rwsav}\left(32\times {\mathbf{n}}+350\right)$ – Real (Kind=nag_wp) arrayCommunication Array
On entry: these must be the same arrays supplied in a previous call to d02pqf. They must remain unchanged between calls.
On exit: information about the integration for use on subsequent calls to d02pgf or other associated routines.
8: $\mathbf{ifail}$ – IntegerInput/Output
On initial entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $-1$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On final exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
A call to this routine cannot be made after it has returned an error. The setup routine must be called to start another problem.
${\mathbf{irevcm}}<0$ on entry.
On entry, a previous call to the setup routine has not been made or the communication arrays have become corrupted.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$, but the value passed to the setup routine was ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
On entry, the communication arrays have become corrupted, or a catastrophic error has already been detected elsewhere. You cannot continue integrating the problem.
tend, as specified in the setup routine, has already been reached. To start a new problem, you will need to call the setup routine. To continue integration beyond tend then d02prf must first be called to reset tend to a new end value.
${\mathbf{ifail}}=2$
More than $100$ output points have been obtained by integrating to tend (as specified in the setup routine). They have been so clustered that it would probably be (much) more efficient to use the interpolation routine. However, you can continue integrating the problem.
${\mathbf{ifail}}=3$
Approximately $\u27e8\mathit{\text{value}}\u27e9$ 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.
${\mathbf{ifail}}=4$
Approximately $\u27e8\mathit{\text{value}}\u27e9$ 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 $\u27e8\mathit{\text{value}}\u27e9$ times as much to reach tend (setup) as it has cost to reach the current time. You should probably call routines intended for stiff problems. However, you can continue integrating the problem.
${\mathbf{ifail}}=5$
In order to satisfy your error requirements the solver has to use a step size of $\u27e8\mathit{\text{value}}\u27e9$ at the current time, $\u27e8\mathit{\text{value}}\u27e9$. This step size is too small for the machine precision, and is smaller than $\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=6$
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 $\u27e8\mathit{\text{value}}\u27e9$.
The integration is being terminated.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
The accuracy of integration is determined by the arguments tol and thresh in a prior call to d02pqf (see the routine document for d02pqf 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
d02pgf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
If d02pgf returns with ${\mathbf{ifail}}>{\mathbf{1}}$ 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 d02pgf returns with ${\mathbf{ifail}}={\mathbf{5}}$ 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 d02pgf (except when ${\mathbf{ifail}}={\mathbf{1}}$) by a call to d02ptf. If
${\mathbf{method}}>0$
in the call to d02pqf, global error assessment is available after any return from d02pgf (except when ${\mathbf{ifail}}={\mathbf{1}}$) by a call to d02puf.
After a failure with ${\mathbf{ifail}}={\mathbf{5}}$ or ${\mathbf{6}}$ each of the diagnostic routines d02ptfandd02puf
may be called only once.
If d02pgf returns with ${\mathbf{ifail}}={\mathbf{4}}$ then it is advisable to change to another code more suited to the solution of stiff problems. d02pgf will not return with ${\mathbf{ifail}}={\mathbf{4}}$ if the problem is actually stiff but it is estimated that integration can be completed using less function evaluations than already computed.
over the range $[0,2\pi ]$ with initial conditions ${y}_{1}=0.0$ and ${y}_{2}=1.0$. We use relative error control with threshold values of $\text{1.0E\u22128}$ for each solution component and print the solution at regular intervals using the interpolation routines d02phfandd02pjf 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 routine c05azf and the interpolation routines. We use a medium order Runge–Kutta method
(${\mathbf{method}}=2$) with tolerance ${\mathbf{tol}}=\text{1.0E\u22125}$.