NAG Library Function Document
nag_ode_ivp_adams_roots (d02qfc)
1 Purpose
nag_ode_ivp_adams_roots (d02qfc) is a function for integrating a nonstiff system of first order ordinary differential equations using a variableorder variablestep Adams' method. A rootfinding facility is provided.
2 Specification
#include <nag.h> 
#include <nagd02.h> 
void 
nag_ode_ivp_adams_roots (Integer neqf,
void 
(*fcn)(Integer neqf,
double x,
const double y[],
double f[],
Nag_User *comm),


double *t,
double y[],
double tout,
double 
(*g)(Integer neqf,
double x,
const double y[],
const double yp[],
Integer k,
Nag_User *comm),


Nag_User *comm,
Nag_ODE_Adams *opt,
NagError *fail) 

3 Description
Given the initial values
$x,{y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neqf}}}$ the function integrates a nonstiff system of first order ordinary differential equations of the type,
${y}_{\mathit{i}}^{\prime}={f}_{\mathit{i}}\left(x,{y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neqf}}}\right)$, for
$\mathit{i}=1,2,\dots ,{\mathbf{neqf}}$, from
$x={\mathbf{t}}$ to
$x={\mathbf{tout}}$ using a variableorder variablestep Adams' method. The system is defined by
fcn, which evaluates
${f}_{i}$ in terms of
$x$ and
${y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neqf}}}$, and
${y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neqf}}}$ are supplied at
$x={\mathbf{t}}$. The function is capable of finding roots (values of
$x$) of prescribed event functions of the form
(See
nag_ode_ivp_adams_setup (d02qwc) for the specification of
neqg).
Each
${g}_{j}$ is considered to be independent of the others so that roots are sought of each
${g}_{j}$ individually. The root reported by the function will be the first root encountered by any
${g}_{j}$. Two techniques for determining the presence of a root in an integration step are available: the sophisticated method described in
Watts (1985) and a simplified method whereby sign changes in each
${g}_{j}$ are looked for at the ends of each integration step. The event functions are defined by
g, which evaluates
${g}_{j}$ in terms of
$x,{y}_{1},\dots ,{y}_{{\mathbf{neqf}}}$ and
${y}_{1}^{\prime},\dots ,{y}_{{\mathbf{neqf}}}^{\prime}$. In onestep mode the function returns an approximation to the solution at each integration point. In interval mode this value is returned at the end of the integration range. If a root is detected this approximation is given at the root. You need to select the mode of operation, the error control, the rootfinding technique and various integration inputs with a prior call of the setup function
nag_ode_ivp_adams_setup (d02qwc).
For a description of the practical implementation of an Adams' formula see
Shampine and Gordon (1975) and
Shampine and Watts (1979).
4 References
Shampine L F and Gordon M K (1975) Computer Solution of Ordinary Differential Equations – The Initial Value Problem W H Freeman & Co., San Francisco
Shampine L F and Watts H A (1979) DEPAC – design of a user oriented package of ODE solvers Report SAND792374 Sandia National Laboratory
Watts H A (1985) RDEAM – An Adams ODE code with root solving capability Report SAND851595 Sandia National Laboratory
5 Arguments
 1:
neqf – IntegerInput
On entry: the number of differential equations.
Constraint:
${\mathbf{neqf}}\ge 1$.
 2:
fcn – function, supplied by the userExternal Function

fcn must evaluate the functions
${f}_{i}$ (that is the first derivatives
${y}_{i}^{\prime}$) for given values of its arguments
$x,{y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neqf}}}$.
The specification of
fcn is:
void 
fcn (Integer neqf,
double x,
const double y[],
double f[],
Nag_User *comm)


 1:
neqf – IntegerInput

On entry: the number of differential equations.
 2:
x – doubleInput

On entry: the current value of the argument $x$.
 3:
y[neqf] – const doubleInput

On entry: ${\mathbf{y}}\left[\mathit{i}1\right]$ contains the current value of the argument ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{neqf}}$.
 4:
f[neqf] – doubleOutput

On exit: ${\mathbf{f}}\left[\mathit{i}1\right]$ must contain the value of ${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{neqf}}$.
 5:
comm – Nag_User *

Pointer to a structure of type Nag_User with the following member:
 p – Pointer

On entry/exit: the pointer $\mathbf{comm}\mathbf{\to}\mathbf{p}$ should be cast to the required type, e.g., struct user *s = (struct user *)comm → p, to obtain the original object's address with appropriate type.
 3:
t – double *Input/Output

On entry: after a call to
nag_ode_ivp_adams_setup (d02qwc) with
${\mathbf{state}}=\mathrm{Nag\_NewStart}$ (i.e., an initial entry),
t must be set to the initial value of the independent variable
$x$.
On exit: the value of
$x$ at which
$y$ has been computed. This may be an intermediate output point, a root,
tout, or a point at which an error has occurred. If the integration is to be continued, possibly with a new value for
tout,
t must not be changed.
 4:
y[neqf] – doubleInput/Output

On entry: the initial values of the solution ${y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neqf}}}$.
On exit: the computed values of the solution at the exit value of
t. If the integration is to be continued, possibly with a new value for
tout, these values must not be changed.
 5:
tout – doubleInput

On entry: the next value of
$x$ at which a computed solution is required. For the initial
t, the input value of
tout is used to determine the direction of integration. Integration is permitted in either direction. If
${\mathbf{tout}}={\mathbf{t}}$ on exit,
tout must be reset beyond
t in the direction of integration, before any continuation call.
 6:
g – function, supplied by the userExternal Function

g must evaluate a given component of
$g\left(x,y,{y}^{\prime}\right)$ at a specified point.
If rootfinding is not required the actual argument for
g must be the NAG defined null double function pointer
NULLDFN.
The specification of
g is:
double 
g (Integer neqf,
double x,
const double y[],
const double yp[],
Integer k,
Nag_User *comm)


 1:
neqf – IntegerInput

On entry: the number of differential equations.
 2:
x – doubleInput

On entry: the current value of the independent variable.
 3:
y[neqf] – const doubleInput

On entry: the current values of the dependent variables.
 4:
yp[neqf] – const doubleInput

On entry: the current values of the derivatives of the dependent variables.
 5:
k – IntegerInput

On entry: the component of $g$ which must be evaluated.
 6:
comm – Nag_User *

Pointer to a structure of type Nag_User with the following member:
 p – Pointer

On entry/exit: the pointer $\mathbf{comm}\mathbf{\to}\mathbf{p}$ should be cast to the required type, e.g., struct user *s = (struct user *)comm → p, to obtain the original object's address with appropriate type.
 7:
comm – Nag_User *

Pointer to a structure of type Nag_User with the following member:
 p – Pointer

On entry/exit: the pointer
$\mathbf{comm}\mathbf{\to}\mathbf{p}$, of type Pointer, allows you to communicate information to and from
fcn and
g. An object of the required type should be declared, e.g., a structure , and its address assigned to the pointer
$\mathbf{comm}\mathbf{\to}\mathbf{p}$ by means of a cast to Pointer in the calling program. E.g.
comm.p = (Pointer)&s.
 8:
opt – Nag_ODE_Adams *

Pointer to a structure of type Nag_ODE_Adams as initialized by the setup function
nag_ode_ivp_adams_setup (d02qwc) with the following members:
 root – Nag_BooleanOutput

On exit: if rootfinding was required (
${\mathbf{neqg}}>0$ in a call to the setup function
nag_ode_ivp_adams_setup (d02qwc) ), then
$\mathbf{root}$ specifies whether or not the output value of the argument
t is a root of one of the event functions. If
$\mathbf{root}=\mathrm{Nag\_FALSE}$, then no root was detected, whereas
$\mathbf{root}=\mathrm{Nag\_TRUE}$ indicates a root.
If rootfinding was not required (${\mathbf{neqg}}=0$) then on exit $\mathbf{root}=\mathrm{Nag\_FALSE}$.
If $\mathbf{root}=\mathrm{Nag\_FALSE}$, then $\mathbf{opt}\mathbf{\to}\mathbf{index}$, $\mathbf{opt}\mathbf{\to}\mathbf{type}$, $\mathbf{opt}\mathbf{\to}\mathbf{events}$ and $\mathbf{opt}\mathbf{\to}\mathbf{resids}$ are indeterminate.
 index – IntegerOutput

On exit: the index $k$ of the event equation ${g}_{k}\left(x,y,{y}^{\prime}\right)=0$ for which the root has been detected.
 type – IntegerOutput

On exit: information about the root detected for the event equation defined by
$\mathbf{opt}\mathbf{\to}\mathbf{index}$. The possible values of
$\mathbf{type}$ with their interpretations are as follows:
 If $\mathbf{type}=1$, a simple root, or lack of distinguishing information available.
 If $\mathbf{type}=2$, a root of even multiplicity is believed to have been detected, that is no change in sign of the event function was found.
 If $\mathbf{type}=3$, a high order root of odd multiplicity.
 If $\mathbf{type}=4$, a possible root, but due to high multiplicity or a clustering of roots accurate evaluation of the event function was prohibited by roundoff error and/or cancellation.
In general, the accuracy of the root is less reliable for values of $\mathbf{type}>1$.
 events – Integer *Output

On exit: array pointer containing information about the
$k$th event function on a very small interval containing the root,
t. All roots lying in this interval are considered indistinguishable numerically and therefore should be regarded as defining a root at
t. The possible values of
$\mathbf{events}\left[j1\right]$,
$j=1,2,\dots ,$neqg, with their interpretations are as follows:
 $\mathbf{events}\left[j1\right]=0$, the $j$th event function did not have a root;
 $\mathbf{events}\left[j1\right]=1$, the $j$th event function changed sign from positive to negative about a root, in the direction of integration;
 $\mathbf{events}\left[j1\right]=1$, the $j$th event function changed sign from negative to positive about a root, in the direction of integration;
 $\mathbf{events}\left[j1\right]=2$, a root was identified, but no change in sign was observed.
 resids – doubleOutput

On exit: array pointer,
$\mathbf{opt}\mathbf{\to}\mathbf{resids}\left[j1\right]$,
$j=1,2,\dots ,$neqg, contains value of the
$j$th event function computed at the root,
t.
 yp – doubleOutput

On exit: array pointer to the approximate derivative of the solution component
${y}_{i}$ at the output value of
t. These values are obtained by the evaluation of
${y}^{\prime}=f\left(x,y\right)$ except when the output value of the argument
t is
tout and
$\mathbf{opt}\mathbf{\to}\mathbf{tcurr}\ne {\mathbf{tout}}$, in which case they are obtained by interpolation.
 tcurr – doubleOutput

On exit: the value of the independent variable which the integrator has actually reached.
$\mathbf{tcurr}$ will always be at least as far as the output value of the argument
t in the direction of integration, but may be further.
 hlast – doubleOutput

On exit: the last successful step size used in the integration.
 hnext – doubleOutput

On exit: the next step size which the integration would attempt.
 ord_last – IntegerOutput

On exit: the order of the method last used (successfully) in the integration.
 ord_next – IntegerOutput

On exit: the order of the method which the integration would attempt on the next step.
 nsuccess – IntegerOutput

On exit: the number of integration steps attempted that have been successful since the start of the current problem.
 nfail – IntegerOutput

On exit: the number of integration steps attempted that have failed since the start of the current problem.
 tolfac – doubleOutput

On exit: a tolerance scale factor,
$\mathbf{tolfac}\ge 1.0$, returned when nag_ode_ivp_adams_roots (d02qfc) exits with
${\mathbf{fail}}\mathbf{.}\mathbf{code}={\mathbf{NE\_ODE\_TOL}}$. If
rtol and
atol are uniformly scaled up by a factor of
$\mathbf{tolfac}$ and
nag_ode_ivp_adams_setup (d02qwc) is called, the next call to nag_ode_ivp_adams_roots (d02qfc) is deemed likely to succeed.
 9:
fail – NagError *Input/Output

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

The value of
tout,
$\u27e8\mathit{\text{value}}\u27e9$, indicates a change in the integration direction. This is not permitted on a continuation call.
 NE_MAX_STEP

The maximum number of steps have been attempted.
If integration is to be continued then the function may be called again and a further
max_step steps will be attempted (see
nag_ode_ivp_adams_setup (d02qwc) for details of
max_step ).
 NE_NEQF

The value of
neqf supplied is not the same as that given to the setup function
nag_ode_ivp_adams_setup (d02qwc).
${\mathbf{neqf}}=\u27e8\mathit{\text{value}}\u27e9$ but the value given to
nag_ode_ivp_adams_setup (d02qwc) was
$\u27e8\mathit{\text{value}}\u27e9$.
 NE_NO_G_FUN

Root finding has been requested by setting
${\mathbf{neqg}}>0$,
${\mathbf{neqg}}=\u27e8\mathit{\text{value}}\u27e9$, but argument
g is a null function.
 NE_NO_SETUP

The setup function
nag_ode_ivp_adams_setup (d02qwc) has not been called.
 NE_ODE_TOL

The error tolerances are too stringent.
rtol and
atol should be scaled up by the factor
$\mathbf{opt}\mathbf{\to}\mathbf{tolfac}$ and the integration function reentered.
$\mathbf{opt}\mathbf{\to}\mathbf{tolfac}=\u27e8\mathit{\text{value}}\u27e9$ (see
Section 9).
 NE_SETUP_ERROR

The call to setup function
nag_ode_ivp_adams_setup (d02qwc) produced an error.
 NE_SINGULAR_POINT

A change in sign of an event function has been detected but the rootfinding process appears to have converged to a singular point of
t rather than a root.
Integration may be continued by calling the function again.
 NE_STIFF_PROBLEM

The problem appears to be stiff.
(See the
d02 Chapter Introduction for a discussion of the term ‘stiff’). Although it is inefficient to use this integrator to solve stiff problems, integration may be continued by resetting
fail and calling the function again.
 NE_T_CHANGED

The value of
t has been changed from
$\u27e8\mathit{\text{value}}\u27e9$ to
$\u27e8\mathit{\text{value}}\u27e9$. This is not permitted on a continuation call.
 NE_T_SAME_TOUT

On entry,
${\mathbf{tout}}={\mathbf{t}}$,
t is
$\u27e8\mathit{\text{value}}\u27e9$.
 NE_TOUT_TCRIT

${\mathbf{tout}}=\u27e8\mathit{\text{value}}\u27e9$ but
crit was set Nag_TRUE in setup call and integration cannot be attempted beyond
${\mathbf{tcrit}}=\u27e8\mathit{\text{value}}\u27e9$.
 NE_WEIGHT_ZERO

An error weight has become zero during the integration, see d02qwc document; ${\mathbf{atol}}\left[\u27e8\mathit{\text{value}}\u27e9\right]$ was set to 0.0 but ${\mathbf{y}}\left[\u27e8\mathit{\text{value}}\u27e9\right]$ is now 0.0. Integration successful as far as ${\mathbf{t}}=\u27e8\mathit{\text{value}}\u27e9$.
The value of the array index is returned in ${\mathbf{fail}}\mathbf{.}\mathbf{errnum}$.
7 Accuracy
The accuracy of integration is determined by the arguments
vectol,
rtol and
atol in a prior call to
nag_ode_ivp_adams_setup (d02qwc). 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 equation system. The code is designed so that a reduction in the tolerances should lead to an approximately proportional reduction in the error. You are strongly recommended to call nag_ode_ivp_adams_roots (d02qfc) with more than one set of tolerances and to compare the results obtained to estimate their accuracy.
The accuracy obtained depends on the type of error test used. If the solution oscillates around zero a relative error test should be avoided, whereas if the solution is exponentially increasing an absolute error test should not be used. If different accuracies are required for different components of the solution then a componentwise error test should be used. For a description of the error test see the specifications of the arguments
vectol,
atol and
rtol in the function document for
nag_ode_ivp_adams_setup (d02qwc).
The accuracy of any roots located will depend on the accuracy of integration and may also be restricted by the numerical properties of $g\left(x,y,{y}^{\prime}\right)$. When evaluating $g$ you should try to write the code so that unnecessary cancellation errors will be avoided.
8 Parallelism and Performance
Not applicable.
If the function fails with
${\mathbf{fail}}\mathbf{.}\mathbf{code}={\mathbf{NE\_ODE\_TOL}}$, then the combination of
atol and
rtol may be so small that a solution cannot be obtained, in which case the function should be called again using larger values for
rtol and/or
atol when calling the setup function
nag_ode_ivp_adams_setup (d02qwc). If the accuracy requested is really needed then you should consider whether there is a more fundamental difficulty. For example:
(a) 
in the region of a singularity the solution components will usually be of a large magnitude. The function could be used in onestep mode to monitor the size of the solution 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; 
(b) 
for ‘stiff’ equations, where the solution contains rapidly decaying components, the function will require a very small step size to preserve stability. This will usually be exhibited by excessive computing time and sometimes an error exit with ${\mathbf{fail}}\mathbf{.}\mathbf{code}={\mathbf{NE\_ODE\_TOL}}$, but usually an error exit with ${\mathbf{fail}}\mathbf{.}\mathbf{code}={\mathbf{NE\_MAX\_STEP}}$ or NE_STIFF_PROBLEM. The Adams' methods are not efficient in such cases. A high proportion of failed steps (see argument $\mathbf{opt}\mathbf{\to}\mathbf{nfail}$) may indicate stiffness but there may be other reasons for this phenomenon. 
nag_ode_ivp_adams_roots (d02qfc) can be used for producing results at short intervals (for example, for graph plotting); you should set
${\mathbf{crit}}=\mathrm{Nag\_TRUE}$ and
tcrit to the last output point required in a prior call to
nag_ode_ivp_adams_setup (d02qwc) and then set
tout appropriately for each output point in turn in the call to nag_ode_ivp_adams_roots (d02qfc).
The structure
opt will contain pointers which have been allocated memory by calls to
nag_ode_ivp_adams_setup (d02qwc). This allocated memory is then accessed by nag_ode_ivp_adams_roots (d02qfc) and, if required,
nag_ode_ivp_adams_interp (d02qzc). When all calls to these functions have been completed the function
nag_ode_ivp_adams_free (d02qyc) may be called to free memory allocated to the structure.
10 Example
We solve the equation
reposed as
over the range
$\left[0,10.0\right]$ with initial conditions
${y}_{1}=0.0$ and
${y}_{2}=1.0$ using vector error control (
${\mathbf{vectol}}=\mathrm{Nag\_TRUE}$) and computation of the solution at
${\mathbf{tout}}=10.0$ with
${\mathbf{tcrit}}=10.0$ (
${\mathbf{crit}}=\mathrm{Nag\_TRUE}$). Also, we use nag_ode_ivp_adams_roots (d02qfc) to locate the positions where
${y}_{1}=0.0$ or where the first component has a turning point, that is
${y}_{1}^{\prime}=0.0$.
10.1 Program Text
Program Text (d02qfce.c)
10.2 Program Data
None.
10.3 Program Results
Program Results (d02qfce.r)