NAG Library Function Document
nag_ode_ivp_rkts_range (d02pec)
1 Purpose
nag_ode_ivp_rkts_range (d02pec) solves an initial value problem for a firstorder system of ordinary differential equations using Runge–Kutta methods.
2 Specification
#include <nag.h> 
#include <nagd02.h> 
void 
nag_ode_ivp_rkts_range (
void 
(*f)(double t,
Integer n,
const double y[],
double yp[],
Nag_Comm *comm),


Integer n,
double twant,
double *tgot,
double ygot[],
double ypgot[],
double ymax[],
Nag_Comm *comm, Integer iwsav[],
double rwsav[],
NagError *fail) 

3 Description
nag_ode_ivp_rkts_range (d02pec) and its associated functions (
nag_ode_ivp_rkts_setup (d02pqc),
nag_ode_ivp_rkts_diag (d02ptc) and
nag_ode_ivp_rkts_errass (d02puc)) solve an initial value problem for a firstorder system of ordinary differential equations. The functions, 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.
nag_ode_ivp_rkts_range (d02pec) is designed for the usual task, namely to compute an approximate solution at a sequence of points. You must first call
nag_ode_ivp_rkts_setup (d02pqc) to specify the problem and how it is to be solved. Thereafter you call nag_ode_ivp_rkts_range (d02pec) repeatedly with successive values of
twant, the points at which you require the solution, in the range from
tstart to
tend (as specified in
nag_ode_ivp_rkts_setup (d02pqc)). In this manner nag_ode_ivp_rkts_range (d02pec) returns the point at which it has computed a solution
tgot (usually
twant), the solution there (
ygot) and its derivative (
ypgot). If nag_ode_ivp_rkts_range (d02pec) 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
nag_ode_ivp_rkts_setup (d02pqc) you can specify either the first step size for nag_ode_ivp_rkts_range (d02pec) to attempt or that it computes automatically an appropriate value. Thereafter nag_ode_ivp_rkts_range (d02pec) estimates an appropriate step size for its next step. This value and other details of the integration can be obtained after any call to nag_ode_ivp_rkts_range (d02pec) 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_rkts_range (d02pec) by a call to
nag_ode_ivp_rkts_errass (d02puc).
For more complicated tasks, you are referred to functions
nag_ode_ivp_rkts_onestep (d02pfc),
nag_ode_ivp_rkts_reset_tend (d02prc) and
nag_ode_ivp_rkts_interp (d02psc), all of which are used by nag_ode_ivp_rkts_range (d02pec).
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 Arguments
 1:
f – function, supplied by the userExternal Function
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:
void 
f (double t,
Integer n,
const double y[],
double yp[],
Nag_Comm *comm)


 1:
t – doubleInput
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] – const doubleInput
On entry: the current values of the dependent variables,
${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.
 4:
yp[n] – doubleOutput
On exit: the values of
${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.
 5:
comm – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
f.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling nag_ode_ivp_rkts_range (d02pec) you may allocate memory and initialize these pointers with various quantities for use by
f when called from nag_ode_ivp_rkts_range (d02pec) (see
Section 3.2.1.1 in the Essential Introduction).
 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 – doubleInput
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 nag_ode_ivp_rkts_range (d02pec)); see
nag_ode_ivp_rkts_setup (d02pqc) for a description of
tstart and
tend.
twant must not lie beyond
tend in the direction of integration.
 4:
tgot – double *Output
On exit:
$t$, the value of the independent variable at which a solution has been computed. On successful exit with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR,
tgot will equal
twant. On exit 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, a solution has still been computed at the value of
tgot but in general
tgot will not equal
twant.
 5:
ygot[n] – doubleInput/Output
On entry: on the first call to nag_ode_ivp_rkts_range (d02pec),
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
nag_ode_ivp_rkts_setup (d02pqc). The local error has still been controlled even when
${\mathbf{tgot}}\ne {\mathbf{twant}}$, that is after a return 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.
 6:
ypgot[n] – doubleOutput
On exit: an approximation to the first derivative of the true solution at
tgot.
 7:
ymax[n] – doubleInput/Output
On entry: on the first call to nag_ode_ivp_rkts_range (d02pec),
ymax need not be set. On all subsequent calls
ymax must remain unchanged.
On exit: ${\mathbf{ymax}}\left[i1\right]$ contains the largest value of $\left{y}_{i}\right$ computed at any step in the integration so far.
 8:
comm – Nag_Comm *Communication Structure

The NAG communication argument (see
Section 3.2.1.1 in the Essential Introduction).
 9:
iwsav[$130$] – IntegerCommunication Array
 10:
rwsav[$32\times {\mathbf{n}}+350$] – 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_rkts_range (d02pec) or other associated functions.
 11:
fail – NagError *Input/Output

The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
 NE_BAD_PARAM

On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
 NE_INT_CHANGED

On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$, but the value passed to the setup function was ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
 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.
 NE_MISSING_CALL

On entry, a previous call to the setup function has not been made or the communication arrays have become corrupted.
 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

You cannot call this function after it has returned an error.
You must call the setup function 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 $\u27e8\mathit{\text{value}}\u27e9$.
The integration is being terminated.
 NE_RK_INVALID_CALL

You cannot call this function when you have specified, in the setup function, that the step integrator will be used.
 NE_RK_POINTS

This function 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.
 NE_RK_STEP_TOO_SMALL

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 $\u27e8\mathit{\text{value}}\u27e9$.
 NE_RK_TGOT_EQ_TEND

tend (setup) had already been reached in a previous call.
To start a new problem, you will need to call the setup function.
 NE_RK_TGOT_RANGE_TEND

twant does not lie in the direction of integration.
${\mathbf{twant}}=\u27e8\mathit{\text{value}}\u27e9$.
twant lies beyond
tend (setup) in the direction of integration.
${\mathbf{twant}}=\u27e8\mathit{\text{value}}\u27e9$ and
${\mathbf{tend}}=\u27e8\mathit{\text{value}}\u27e9$.
 NE_RK_TGOT_RANGE_TEND_CLOSE

twant lies beyond
tend (setup) in the direction of integration, but is very close to
tend.
You may have intended
${\mathbf{twant}}={\mathbf{tend}}$.
$\left{\mathbf{twant}}{\mathbf{tend}}\right=\u27e8\mathit{\text{value}}\u27e9$.
 NE_RK_TWANT_CLOSE_TGOT

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
$\u27e8\mathit{\text{value}}\u27e9$.
 NE_STIFF_PROBLEM

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 functions intended for stiff problems. However, you can continue integrating the problem.
 NW_RK_TOO_MANY

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.
7 Accuracy
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.
8 Parallelism and Performance
nag_ode_ivp_rkts_range (d02pec) is not threaded by NAG in any implementation.
nag_ode_ivp_rkts_range (d02pec) 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
Users' Note for your implementation for any additional implementationspecific information.
If nag_ode_ivp_rkts_range (d02pec) 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
ygot and
ymax should be monitored (or
nag_ode_ivp_rkts_onestep (d02pfc) 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 nag_ode_ivp_rkts_range (d02pec) 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 a return from nag_ode_ivp_rkts_range (d02pec) with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR,
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 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_rkts_range (d02pec) 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_rkts_range (d02pec) 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.
10 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
nag_ode_ivp_rkts_setup (d02pqc)) 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 10 in nag_ode_ivp_rkts_errass (d02puc).
10.1 Program Text
Program Text (d02pece.c)
10.2 Program Data
Program Data (d02pece.d)
10.3 Program Results
Program Results (d02pece.r)