NAG Library Routine Document
D02PEF
1 Purpose
D02PEF solves an initial value problem for a firstorder system of ordinary differential equations using Runge–Kutta methods.
2 Specification
SUBROUTINE D02PEF ( 
F, N, TWANT, TGOT, YGOT, YPGOT, YMAX, IUSER, RUSER, IWSAV, RWSAV, IFAIL) 
INTEGER 
N, IUSER(*), IWSAV(130), IFAIL 
REAL (KIND=nag_wp) 
TWANT, TGOT, YGOT(N), YPGOT(N), YMAX(N), RUSER(*), RWSAV(32*N+350) 
EXTERNAL 
F 

3 Description
D02PEF and its associated routines (
D02PQF,
D02PTF and
D02PUF) solve an initial value problem for a firstorder system of ordinary differential equations. The routines, based on Runge–Kutta methods and derived from RKSUITE (see
Brankin et al. (1991)), integrate
where
$y$ is the vector of
$\mathit{n}$ solution components and
$t$ is the independent variable.
D02PEF is designed for the usual task, namely to compute an approximate solution at a sequence of points. You must first call
D02PQF to specify the problem and how it is to be solved. Thereafter you call D02PEF repeatedly with successive values of
TWANT, the points at which you require the solution, in the range from
TSTART to
TEND (as specified in
D02PQF). In this manner D02PEF returns the point at which it has computed a solution
TGOT (usually
TWANT), the solution there (
YGOT) and its derivative (
YPGOT). If D02PEF encounters some difficulty in taking a step toward
TWANT, then it returns the point of difficulty (
TGOT) and the solution and derivative computed there (
YGOT and
YPGOT, respectively).
In the call to
D02PQF you can specify either the first step size for D02PEF to attempt or that it computes automatically an appropriate value. Thereafter D02PEF estimates an appropriate step size for its next step. This value and other details of the integration can be obtained after any call to D02PEF 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 D02PEF by a call to
D02PUF.
For more complicated tasks, you are referred to routines
D02PFF,
D02PRF and
D02PSF, all of which are used by D02PEF.
4 References
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 91S1 Southern Methodist University
5 Parameters
 1: F – SUBROUTINE, supplied by the user.External Procedure
F must evaluate the functions
${f}_{i}$ (that is the first derivatives
${y}_{i}^{\prime}$) for given values of the arguments
$t$,
${y}_{i}$.
The specification of
F is:
INTEGER 
N, IUSER(*) 
REAL (KIND=nag_wp) 
T, Y(N), YP(N), RUSER(*) 

 1: T – REAL (KIND=nag_wp)Input
On entry: $t$, the current value of the independent variable.
 2: N – INTEGERInput
On entry: $n$, the number of ordinary differential equations in the system to be solved.
 3: Y(N) – REAL (KIND=nag_wp) arrayInput
On entry: the current values of the dependent variables,
${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.
 4: YP(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the values of
${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.
 5: IUSER($*$) – INTEGER arrayUser Workspace
 6: RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

F is called with the parameters
IUSER and
RUSER as supplied to D02PEF. You are free to use the arrays
IUSER and
RUSER to supply information to
F as an alternative to using COMMON global variables.
F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02PEF is called. Parameters denoted as
Input must
not be changed by this procedure.
 2: N – INTEGERInput
On entry: $n$, the number of ordinary differential equations in the system to be solved.
Constraint:
${\mathbf{N}}\ge 1$.
 3: TWANT – REAL (KIND=nag_wp)Input
On entry: $t$, the next value of the independent variable where a solution is desired.
Constraint:
TWANT must be closer to
TEND than the previous value of
TGOT (or
TSTART on the first call to D02PEF); see
D02PQF for a description of
TSTART and
TEND.
TWANT must not lie beyond
TEND in the direction of integration.
 4: TGOT – REAL (KIND=nag_wp)Output
On exit:
$t$, the value of the independent variable at which a solution has been computed. On successful exit with
${\mathbf{IFAIL}}={\mathbf{0}}$,
TGOT will equal
TWANT. On exit with
${\mathbf{IFAIL}}>{\mathbf{1}}$, a solution has still been computed at the value of
TGOT but in general
TGOT will not equal
TWANT.
 5: YGOT(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: on the first call to D02PEF,
YGOT need not be set. On all subsequent calls
YGOT must remain unchanged.
On exit: an approximation to the true solution at the value of
TGOT. At each step of the integration to
TGOT, the local error has been controlled as specified in
D02PQF. The local error has still been controlled even when
${\mathbf{TGOT}}\ne {\mathbf{TWANT}}$, that is after a return with
${\mathbf{IFAIL}}>{\mathbf{1}}$.
 6: YPGOT(N) – REAL (KIND=nag_wp) arrayOutput
On exit: an approximation to the first derivative of the true solution at
TGOT.
 7: YMAX(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: on the first call to D02PEF,
YMAX need not be set. On all subsequent calls
YMAX must remain unchanged.
On exit: ${\mathbf{YMAX}}\left(i\right)$ contains the largest value of $\left{y}_{i}\right$ computed at any step in the integration so far.
 8: IUSER($*$) – INTEGER arrayUser Workspace
 9: RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and
RUSER are not used by D02PEF, but are passed directly to
F and may be used to pass information to this routine as an alternative to using COMMON global variables.
 10: IWSAV($130$) – INTEGER arrayCommunication Array
 11: RWSAV($32\times {\mathbf{N}}+350$) – 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 D02PEF or other associated routines.
 12: IFAIL – INTEGERInput/Output
On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if
${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is
$1$.
When the value $\mathbf{1}\text{ or}1$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

On entry, a previous call to the setup routine 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.
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 ${\mathbf{N}}=\u27e8\mathit{\text{value}}\u27e9$.
TEND (setup) had already been reached in a previous call.
To start a new problem, you will need to call the setup routine.
TWANT does not lie in the direction of integration.
${\mathbf{TWANT}}=\u27e8\mathit{\text{value}}\u27e9$.
TWANT is too close to the last value of
TGOT (
TSTART on setup).
When using the method of order
$8$ at setup, these must differ by at least
$\u27e8\mathit{\text{value}}\u27e9$. Their absolute difference is
$\u27e8\mathit{\text{value}}\u27e9$.
TWANT lies beyond
TEND (setup) in the direction of integration, but is very close to
TEND.
You may have intended
${\mathbf{TWANT}}={\mathbf{TEND}}$.
$\left{\mathbf{TWANT}}{\mathbf{TEND}}\right=\u27e8\mathit{\text{value}}\u27e9$.
TWANT lies beyond
TEND (setup) in the direction of integration.
${\mathbf{TWANT}}=\u27e8\mathit{\text{value}}\u27e9$ and
${\mathbf{TEND}}=\u27e8\mathit{\text{value}}\u27e9$.
You cannot call this routine after it has returned an error.
You must call the setup routine to start another problem.
You cannot call this routine when you have specified, in the setup routine, that the step integrator will be used.
 ${\mathbf{IFAIL}}=2$

This routine is being used inefficiently because the step size has been reduced drastically many times to obtain answers at many points. Using the order $4$ and $5$ pair method at setup is more appropriate here.
You can continue integrating this 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.
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.
7 Accuracy
The accuracy of integration is determined by the parameters
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 parameters. 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.
If D02PEF 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
YGOT and
YMAX should be monitored (or
D02PFF should be used since this takes one integration step at a time) 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 D02PEF by a call to
D02PTF. If
${\mathbf{METHOD}}>0$ in the call to
D02PQF, global error assessment is available after any return from D02PEF (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
D02PTF and
D02PUF may be called only once.
If D02PEF returns with ${\mathbf{IFAIL}}={\mathbf{4}}$ then it is advisable to change to another code more suited to the solution of stiff problems. D02PEF 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.
9 Example
This example solves the equation
reposed as
over the range
$\left[0,2\pi \right]$ with initial conditions
${y}_{1}=0.0$ and
${y}_{2}=1.0$. Relative error control is used with threshold values of
$\text{1.0E\u22128}$ for each solution component and compute the solution at intervals of length
$\pi /4$ across the range. A loworder Runge–Kutta method (see
D02PQF) is also used with tolerances
${\mathbf{TOL}}=\text{1.0E\u22123}$ and
${\mathbf{TOL}}=\text{1.0E\u22124}$ in turn so that the solutions can be compared.
See also
Section 9 in D02PUF.
9.1 Program Text
Program Text (d02pefe.f90)
9.2 Program Data
Program Data (d02pefe.d)
9.3 Program Results
Program Results (d02pefe.r)