NAG Library Function Document
nag_ode_ivp_bdf_gen (d02ejc)
1 Purpose
nag_ode_ivp_bdf_gen (d02ejc) integrates a stiff system of firstorder ordinary differential equations over an interval with suitable initial conditions, using a variableorder, variablestep method implementing the Backward Differentiation Formulae (BDF), until a userspecified function, if supplied, of the solution is zero, and returns the solution at specified points, if desired.
2 Specification
#include <nag.h> 
#include <nagd02.h> 
void 
nag_ode_ivp_bdf_gen (Integer neq,
void 
(*fcn)(Integer neq,
double x,
const double y[],
double f[],
Nag_User *comm),


void 
(*pederv)(Integer neq,
double x,
const double y[],
double pw[],
Nag_User *comm),


double *x,
double y[],
double xend,
double tol,
Nag_ErrorControl err_c,
double 
(*g)(Integer neq,
double x,
const double y[],
Nag_User *comm),


Nag_User *comm,
NagError *fail) 

3 Description
nag_ode_ivp_bdf_gen (d02ejc) advances the solution of a system of ordinary differential equations
from
$x={\mathbf{x}}$ to
$x={\mathbf{xend}}$ using a variableorder, variablestep method implementing the BDF. The system is defined by
fcn, which evaluates
${f}_{i}$ in terms of
$x$ and
${y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neq}}}$ (see
Section 5). The initial values of
${y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neq}}}$ must be given at
$x={\mathbf{x}}$.
The solution is returned via
output at specified points, if desired: this solution is obtained by
${C}^{1}$ interpolation on solution values produced by the method. As the integration proceeds a check can be made on the userspecified function
$g\left(x,y\right)$ to determine an interval where it changes sign. The position of this sign change is then determined accurately. It is assumed that
$g\left(x,y\right)$ is a continuous function of the variables, so that a solution of
$g\left(x,y\right)=0.0$ can be determined by searching for a change in sign in
$g\left(x,y\right)$. The accuracy of the integration, the interpolation and, indirectly, of the determination of the position where
$g\left(x,y\right)=0.0$, is controlled by the arguments
tol and
err_c. The Jacobian of the system
${y}^{\prime}=f\left(x,y\right)$ may be supplied in function
pederv, if it is available.
For a description of BDF and their practical implementation see
Hall and Watt (1976).
4 References
Hall G and Watt J M (ed.) (1976) Modern Numerical Methods for Ordinary Differential Equations Clarendon Press, Oxford
5 Arguments
 1:
neq – IntegerInput

On entry: the number of differential equations.
Constraint:
${\mathbf{neq}}\ge 1$.
 2:
fcn – function, supplied by the userExternal Function

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


 1:
neq – IntegerInput

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

On entry: the value of the independent variable $x$.
 3:
y[neq] – const doubleInput

On entry: $y\left[i1\right]$ holds the value of the variable ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
 4:
f[neq] – doubleOutput

On exit: $f\left[\mathit{i}1\right]$ must contain the value of ${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
 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. (See the argument
comm below.)
 3:
pederv – function, supplied by the userExternal Function

pederv must evaluate the Jacobian of the system (that is, the partial derivatives
$\frac{\partial {f}_{i}}{\partial {y}_{j}}$) for given values of the variables
$x,{y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neq}}}$.
The specification of
pederv is:
void 
pederv (Integer neq,
double x,
const double y[],
double pw[],
Nag_User *comm)


 1:
neq – IntegerInput

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

On entry: the value of the independent variable $x$.
 3:
y[neq] – const doubleInput

On entry: $y\left[\mathit{i}1\right]$ holds the value of the variable ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
 4:
pw[${\mathbf{neq}}\times {\mathbf{neq}}$] – doubleOutput

On exit: ${\mathbf{pw}}\left[\left(i1\right)\times {\mathbf{neq}}+j1\right]$ must contain the value of $\frac{\partial {f}_{i}}{\partial {y}_{j}}$, for $i,j=1,2,\dots ,{\mathbf{neq}}$.
 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. (See the argument
comm below.)
If you do not wish to supply the Jacobian, the actual argument
pederv must be the NAG defined null function pointer
NULLFN.
 4:
x – double *Input/Output

On entry: the value of the independent variable $x$.
Constraint:
${\mathbf{x}}\ne {\mathbf{xend}}$.
On exit: if
$g$ is supplied,
x contains the point where
$g\left(x,y\right)=0.0$, unless
$g\left(x,y\right)\ne 0.0$ anywhere on the range
x to
xend, in which case,
x will contain
xend. If
$g$ is not supplied
x contains
xend, unless an error has occurred, when it contains the value of
$x$ at the error.
 5:
y[neq] – doubleInput/Output

On entry: $y\left[\mathit{i}1\right]$ holds the value of the variable ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
On exit: the computed values of the solution at the final point $x={\mathbf{x}}$.
 6:
xend – doubleInput

On entry: the final value of the independent variable.
 ${\mathbf{xend}}<{\mathbf{x}}$
 Iintegration proceeds in the negative direction.
Constraint:
${\mathbf{xend}}\ne {\mathbf{x}}$.
 7:
tol – doubleInput

On entry: a
positive tolerance for controlling the error in the integration. Hence
tol affects the determination of the position where
$g\left(x,y\right)=0.0$, if
$g$ is supplied.
nag_ode_ivp_bdf_gen (d02ejc) has been designed so that, for most problems, a reduction in
tol leads to an approximately proportional reduction in the error in the solution. However, the actual relation between
tol and the accuracy achieved cannot be guaranteed. You are strongly recommended to call nag_ode_ivp_bdf_gen (d02ejc) with more than one value for
tol and to compare the results obtained to estimate their accuracy. In the absence of any prior knowledge, you might compare the results obtained by calling nag_ode_ivp_bdf_gen (d02ejc) with
${\mathbf{tol}}={10}^{p}$ and
${\mathbf{tol}}={10}^{p1}$ if
$p$ correct decimal digits are required in the solution.
Constraint:
${\mathbf{tol}}>0.0$.
 8:
err_c – Nag_ErrorControlInput
On entry: the type of error control. At each step in the numerical solution an estimate of the local error,
est, is made. For the current step to be accepted the following condition must be satisfied:
where
${\tau}_{r}$ and
${\tau}_{a}$ are defined by
err_c 
${\tau}_{r}$ 
${\tau}_{a}$ 
Nag_Relative 
tol 
$\epsilon $ 
Nag_Absolute 
0.0 
tol 
Nag_Mixed 
tol 
tol 
where
$\epsilon $ is a small machinedependent number and
${e}_{i}$ is an estimate of the local error at
${y}_{i}$, computed internally. If the appropriate condition is not satisfied, the step size is reduced and the solution is recomputed on the current step. If you wish to measure the error in the computed solution in terms of the number of correct decimal places, then
err_c should be set to Nag_Absolute on entry, whereas if the error requirement is in terms of the number of correct significant digits, then
err_c should be set to Nag_Relative. If you prefer a mixed error test, then
err_c should be set to Nag_Mixed. The recommended value for
err_c is Nag_Relative.
Constraint:
${\mathbf{err\_c}}=\mathrm{Nag\_Absolute}$, $\mathrm{Nag\_Mixed}$ or $\mathrm{Nag\_Relative}$.
 9:
output – function, supplied by the userExternal Function

output permits access to intermediate values of the computed solution (for example to print or plot them), at successive userspecified points. It is initially called by nag_ode_ivp_bdf_gen (d02ejc) with
${\mathbf{xsol}}={\mathbf{x}}$ (the initial value of
$x$). You must reset
xsol to the next point (between the current
xsol and
xend) where
output is to be called, and so on at each call to
output. If, after a call to
output, the reset point
xsol is beyond
xend, nag_ode_ivp_bdf_gen (d02ejc) will integrate to
xend with no further calls to
output; if a call to
output is required at the point
${\mathbf{xsol}}={\mathbf{xend}}$, then
xsol must be given precisely the value
xend.
The specification of
output is:
void 
output (Integer neq,
double *xsol,
const double y[],
Nag_User *comm)


 1:
neq – IntegerInput

On entry: the number of differential equations.
 2:
xsol – double *Input/Output

On entry: the value of the independent variable $x$.
On exit: you must set
xsol to the next value of
$x$ at which
output is to be called.
 3:
y[neq] – const doubleInput

On entry: $y\left[\mathit{i}1\right]$ holds the value of the variable ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
 4:
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. (See the argument
comm below.)
If you do not wish to access intermediate output, the actual argument
output must be the NAG defined null function pointer
NULLFN.
 10:
g – function, supplied by the userExternal Function

g must evaluate
$g\left(x,y\right)$ for specified values
$x,y$. It specifies the function
$g$ for which the first position
$x$ where
$g\left(x,y\right)=0$ is to be found.
The specification of
g is:
double 
g (Integer neq,
double x,
const double y[],
Nag_User *comm)


 1:
neq – IntegerInput

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

On entry: the value of the independent variable $x$.
 3:
y[neq] – const doubleInput

On entry: $y\left[\mathit{i}1\right]$ holds the value of the variable ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
 4:
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. (See the argument
comm below.)
If you do not require the root finding option, the actual argument
g must be the NAG defined null double function pointer
NULLDFN.
 11:
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,
pederv,
output 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. The type pointer will be
void * with a C compiler that defines
void * and
char * otherwise.
 12:
fail – NagError *Input/Output

The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
 NE_2_REAL_ARG_EQ
On entry, ${\mathbf{x}}=\u2329\mathit{\text{value}}\u232a$ while ${\mathbf{xend}}=\u2329\mathit{\text{value}}\u232a$. These arguments must satisfy ${\mathbf{x}}\ne {\mathbf{xend}}$.
 NE_ALLOC_FAIL
Dynamic memory allocation failed.
 NE_BAD_PARAM
On entry, argument
err_c had an illegal value.
 NE_INT_ARG_LT
On entry, ${\mathbf{neq}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{neq}}\ge 1$.
 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_NO_SIGN_CHANGE
No change in sign of the function $g\left(x,y\right)$ was detected in the integration range.
 NE_REAL_ARG_LE
On entry,
tol must not be less than or equal to 0.0:
${\mathbf{tol}}=\u2329\mathit{\text{value}}\u232a$.
 NE_TOL_PROGRESS
The value of
tol,
$\u2329\mathit{\text{value}}\u232a$, is too small for the function to make any further progress across the integration range. Current value of
${\mathbf{x}}=\u2329\mathit{\text{value}}\u232a$.
 NE_TOL_TOO_SMALL
The value of
tol,
$\u2329\mathit{\text{value}}\u232a$, is too small for the function to take an initial step.
 NE_XSOL_INCONSIST
On call
$\u2329\mathit{\text{value}}\u232a$ to the supplied print function
xsol was set to a value behind the previous value of
xsol in the direction of integration.
Previous
${\mathbf{xsol}}=\u2329\mathit{\text{value}}\u232a$,
${\mathbf{xend}}=\u2329\mathit{\text{value}}\u232a$, new
${\mathbf{xsol}}=\u2329\mathit{\text{value}}\u232a$.
 NE_XSOL_NOT_RESET
On call
$\u2329\mathit{\text{value}}\u232a$ to the supplied print function
xsol was not reset.
 NE_XSOL_SET_WRONG
xsol was set to a value behind
x in the direction of integration by the first call to the supplied print function.
The integration range is
$\left(\u2329\mathit{\text{value}}\u232a,\u2329\mathit{\text{value}}\u232a\right)$,
${\mathbf{xsol}}=\u2329\mathit{\text{value}}\u232a$.
7 Accuracy
The accuracy of the computation of the solution vector
y may be controlled by varying the local error tolerance
tol. In general, a decrease in local error tolerance should lead to an increase in accuracy. You are advised to choose
${\mathbf{err\_c}}=\mathrm{Nag\_Relative}$ unless you have a good reason for a different choice. It is particularly appropriate if the solution decays.
If the problem is a rootfinding one, then the accuracy of the root determined will depend strongly on $\frac{\partial g}{\partial x}$ and $\frac{\partial g}{\partial {y}_{\mathit{i}}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$. Large values for these quantities may imply large errors in the root.
If more than one root is required, then to determine the second and later roots nag_ode_ivp_bdf_gen (d02ejc) may be called again starting a short distance past the previously determined roots.
If it is easy to code, you should supply the function
pederv. However, it is important to be aware that if
pederv is coded incorrectly, a very inefficient integration may result and possibly even a failure to complete the integration (
${\mathbf{fail}}\mathbf{.}\mathbf{code}={\mathbf{NE\_TOL\_PROGRESS}}$).
9 Example
We illustrate the solution of five different problems. In each case the differential system is the wellknown stiff Robertson problem.
with initial conditions
${y}_{1}=1.0$,
${y}_{2}={y}_{3}=0.0$ at
$x=0.0$. We solve each of the following problems with local error tolerances
$\text{1.0e\u22123}$ and
$\text{1.0e\u22124}$.
(i) 
To integrate to $x=10.0$ producing output at intervals of 2.0 until a point is encountered where ${y}_{1}=0.9$. The Jacobian is calculated numerically. 
(ii) 
As (i) but with the Jacobian calculated analytically. 
(iii) 
As (i) but with no intermediate output. 
(iv) 
As (i) but with no rootfinding termination condition. 
(v) 
Integrating the equations as in (i) but with no intermediate output and no rootfinding termination condition. 
9.1 Program Text
Program Text (d02ejce.c)
9.2 Program Data
None.
9.3 Program Results
Program Results (d02ejce.r)