# NAG Library Routine Document

## 1Purpose

d02pef solves an initial value problem for a first-order system of ordinary differential equations using Runge–Kutta methods.

## 2Specification

Fortran Interface
 Subroutine d02pef ( f, n, tgot, ygot, ymax,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: iuser(*), iwsav(130), ifail Real (Kind=nag_wp), Intent (In) :: twant Real (Kind=nag_wp), Intent (Inout) :: ygot(n), ymax(n), ruser(*), rwsav(32*n+350) Real (Kind=nag_wp), Intent (Out) :: tgot, ypgot(n) External :: f
#include nagmk26.h
 void d02pef_ (void (NAG_CALL *f)(const double *t, const Integer *n, const double y[], double yp[], Integer iuser[], double ruser[]),const Integer *n, const double *twant, double *tgot, double ygot[], double ypgot[], double ymax[], Integer iuser[], double ruser[], Integer iwsav[], double rwsav[], Integer *ifail)

## 3Description

d02pef and its associated routines (d02pqf, d02ptf and d02puf) 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
 $y′=ft,y given yt0=y0$
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.
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

1:     $\mathbf{f}$ – 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:
Fortran Interface
 Subroutine f ( t, n, y, yp,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: t, y(n) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: yp(n)
#include nagmk26.h
 void f (const double *t, const Integer *n, const double y[], double yp[], Integer iuser[], double ruser[])
1:     $\mathbf{t}$ – Real (Kind=nag_wp)Input
On entry: $t$, the current value of the independent variable.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of ordinary differential equations in the system to be solved.
3:     $\mathbf{y}\left({\mathbf{n}}\right)$ – 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:     $\mathbf{yp}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the values of ${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.
5:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
6:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
f is called with the arguments iuser and ruser as supplied to d02pef. You should use the arrays iuser and ruser to supply information to f.
f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02pef is called. Arguments denoted as Input must not be changed by this procedure.
Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02pef. If your code inadvertently does return any NaNs or infinities, d02pef 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: ${\mathbf{n}}\ge 1$.
3:     $\mathbf{twant}$ – 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:     $\mathbf{tgot}$ – 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:     $\mathbf{ygot}\left({\mathbf{n}}\right)$ – 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:     $\mathbf{ypgot}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: an approximation to the first derivative of the true solution at tgot.
7:     $\mathbf{ymax}\left({\mathbf{n}}\right)$ – 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:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
9:     $\mathbf{ruser}\left(*\right)$ – 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.
10:   $\mathbf{iwsav}\left(130\right)$ – Integer arrayCommunication Array
11:   $\mathbf{rwsav}\left(32×{\mathbf{n}}+350\right)$ – 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 d02pef or other associated routines.
12:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 arguments 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).

## 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:
${\mathbf{ifail}}=1$
On entry, a previous call to the setup routine has not been made or the communication arrays have become corrupted.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, but the value passed to the setup routine was ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
On entry, the communication arrays have become corrupted, or a catastrophic error has already been detected elsewhere. You cannot continue integrating the problem.
To start a new problem, you will need to call the setup routine.
twant does not lie in the direction of integration. ${\mathbf{twant}}=〈\mathit{\text{value}}〉$.
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 $〈\mathit{\text{value}}〉$. Their absolute difference is $〈\mathit{\text{value}}〉$.
twant lies beyond tend (setup) in the direction of integration, but is very close to tend.
You may have intended ${\mathbf{twant}}={\mathbf{tend}}$.
$\left|{\mathbf{twant}}-{\mathbf{tend}}\right|=〈\mathit{\text{value}}〉$.
twant lies beyond tend (setup) in the direction of integration.
${\mathbf{twant}}=〈\mathit{\text{value}}〉$ and ${\mathbf{tend}}=〈\mathit{\text{value}}〉$.
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 $〈\mathit{\text{value}}〉$ 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.
${\mathbf{ifail}}=4$
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 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 $〈\mathit{\text{value}}〉$ at the current time, $〈\mathit{\text{value}}〉$. This step size is too small for the machine precision, and is smaller than $〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=6$
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 $〈\mathit{\text{value}}〉$.
The integration is being terminated.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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

d02pef 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.

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 a return from d02pef with ${\mathbf{ifail}}={\mathbf{0}}$, ${\mathbf{2}}$, ${\mathbf{3}}$, ${\mathbf{4}}$, ${\mathbf{5}}$ or ${\mathbf{6}}$ 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.

## 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 ${y}_{2}=1.0$. Relative error control is used with threshold values of $\text{1.0E−8}$ for each solution component and compute the solution at intervals of length $\pi /4$ across the range. A low-order Runge–Kutta method (see d02pqf) is also used with tolerances ${\mathbf{tol}}=\text{1.0E−3}$ and ${\mathbf{tol}}=\text{1.0E−4}$ in turn so that the solutions can be compared.

### 10.1Program Text

Program Text (d02pefe.f90)

### 10.2Program Data

Program Data (d02pefe.d)

### 10.3Program Results

Program Results (d02pefe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017