d02bjf integrates a system of first-order ordinary differential equations over an interval with suitable initial conditions, using a fixed order Runge–Kutta method, until a user-specified function, if supplied, of the solution is zero, and returns the solution at points specified by you, if desired.
from $x={\mathbf{x}}$ to $x={\mathbf{xend}}$ using a fixed order Runge–Kutta method. The system is defined by fcn, which evaluates ${f}_{i}$ in terms of $x$ and $y=({y}_{1},{y}_{2},\dots ,{y}_{\mathit{n}})$. The initial values of $y=({y}_{1},{y}_{2},\dots ,{y}_{\mathit{n}})$ must be given at $x={\mathbf{x}}$.
The solution is returned via the output supplied by you and at points specified by you, 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 user-specified function $g(x,y)$ to determine an interval where it changes sign. The position of this sign change is then determined accurately by ${C}^{1}$ interpolation to the solution. It is assumed that $g(x,y)$ is a continuous function of the variables, so that a solution of $g(x,y)=0$ can be determined by searching for a change in sign in $g(x,y)$. The accuracy of the integration, the interpolation and, indirectly, of the determination of the position where $g(x,y)=0$, is controlled by the arguments tol and relabs.
4References
Shampine L F (1994) Numerical solution of ordinary differential equations Chapman and Hall
5Arguments
1: $\mathbf{x}$ – Real (Kind=nag_wp)Input/Output
On entry: the initial value of the independent variable $x$.
On exit: if $g$ is supplied by you, it contains the point where $g(x,y)=0$, unless $g(x,y)\ne 0$ anywhere on the range x to xend, in which case, x will contain xend (and the error indicator ${\mathbf{ifail}}={\mathbf{6}}$ is set); if $g$ is not supplied by you it contains xend. However, if an error has occurred, it contains the value of $x$ at which the error occurred.
2: $\mathbf{xend}$ – Real (Kind=nag_wp)Input
On entry: the final value of the independent variable. If ${\mathbf{xend}}<{\mathbf{x}}$, integration will proceed in the negative direction.
Constraint:
${\mathbf{xend}}\ne {\mathbf{x}}$.
3: $\mathbf{n}$ – IntegerInput
On entry: $\mathit{n}$, the number of equations.
Constraint:
${\mathbf{n}}>0$.
4: $\mathbf{y}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the initial values of the solution ${y}_{1},{y}_{2},\dots ,{y}_{\mathit{n}}$ at $x={\mathbf{x}}$.
On exit: the computed values of the solution at the final point $x={\mathbf{x}}$.
5: $\mathbf{fcn}$ – Subroutine, supplied by the user.External Procedure
fcn must evaluate the functions ${f}_{i}$ (i.e., the derivatives ${y}_{i}^{\prime}$) for given values of its arguments $x,{y}_{1},\dots ,{y}_{\mathit{n}}$.
On entry: $x$, the value of the independent variable.
2: $\mathbf{y}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension, $\mathit{n}$, of y is n as in the call of d02bjf.
On entry: ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$, the value of the variable.
3: $\mathbf{f}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension, $\mathit{n}$, of f is n as in the call of d02bjf.
On exit: the value of
${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.
fcn must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02bjf is called. Arguments denoted as Input must not be changed by this procedure.
Note:fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02bjf. If your code inadvertently does return any NaNs or infinities, d02bjf is likely to produce unexpected results.
6: $\mathbf{tol}$ – Real (Kind=nag_wp)Input
On entry: a positive tolerance for controlling the error in the integration. Hence tol affects the determination of the position where $g(x,y)=0$, if $g$ is supplied.
d02bjf 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 d02bjf 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 d02bjf with ${\mathbf{relabs}}=\text{'D'}$ and with each of ${\mathbf{tol}}={10.0}^{-p}$ and ${\mathbf{tol}}={10.0}^{-p-1}$ where $p$ correct significant digits are required in the solution, $y$. The accuracy of the value $x$ such that $g(x,y)=0$ is indirectly controlled by varying tol. You should experiment to determine this accuracy.
On entry: the type of error control. At each step in the numerical solution an estimate of the local error, $\mathit{est}$, is made. For the current step to be accepted the following condition must be satisfied:
where ${\epsilon}_{r}$ and ${\epsilon}_{a}$ are small machine-dependent numbers and ${e}_{i}$ is an estimate of the local error at ${y}_{i}$, computed internally. If the 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, relabs should be set to 'A' on entry, whereas if the error requirement is in terms of the number of correct significant digits, relabs should be set to 'R'. If you prefer a mixed error test, relabs should be set to 'M', otherwise if you have no preference, relabs should be set to the default 'D'. Note that in this case 'D' is taken to be 'R'.
Constraint:
${\mathbf{relabs}}=\text{'M'}$, $\text{'A'}$, $\text{'R'}$ or $\text{'D'}$.
8: $\mathbf{output}$ – Subroutine, supplied by the NAG Library or the user.External Procedure
output permits access to intermediate values of the computed solution (for example to print or plot them), at successive user-specified points. It is initially called by d02bjf 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, d02bjf will integrate to xend with no further calls to output; if a call to output is required at the point ${\mathbf{xsol}}={\mathbf{xend}}$, xsol must be given precisely the value xend.
1: $\mathbf{xsol}$ – Real (Kind=nag_wp)Input/Output
On entry: the output 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.
2: $\mathbf{y}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension, $\mathit{n}$, of y is n as in the call of d02bjf.
On entry: the computed solution at the point xsol.
output must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02bjf is called. Arguments denoted as Input must not be changed by this procedure.
Note:output should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02bjf. If your code inadvertently does return any NaNs or infinities, d02bjf is likely to produce unexpected results.
If you do not wish to access intermediate output, the actual argument outputmust be the dummy routine d02bjx. (d02bjx is included in the NAG Library.)
9: $\mathbf{g}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure
g must evaluate the function $g(x,y)$ for specified values $x,y$. It specifies the function $g$ for which the first position $x$ where $g(x,y)=0$ is to be found.
On entry: $x$, the value of the independent variable.
2: $\mathbf{y}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension, $\mathit{n}$, of y is n as in the call of d02bjf.
On entry: ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$, the value of the variable.
g must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02bjf is called. Arguments denoted as Input must not be changed by this procedure.
Note:g should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02bjf. If your code inadvertently does return any NaNs or infinities, d02bjf is likely to produce unexpected results.
If you do not require the root-finding option, the actual argument gmust be the dummy routine d02bjw. (d02bjw is included in the NAG Library.)
10: $\mathbf{w}\left(20\times {\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
11: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ 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 $\mathrm{-1}$, 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, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{relabs}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{relabs}}=\text{'M'}$, $\text{'A'}$, $\text{'R'}$ or $\text{'D'}$.
On entry, ${\mathbf{tol}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: $\u27e8\mathit{\text{value}}\u27e9\le {\mathbf{tol}}<0.01$.
On entry, ${\mathbf{x}}={\mathbf{xend}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{x}}\ne {\mathbf{xend}}$.
${\mathbf{ifail}}=2$
With the given value of tol, no further progress can be made across the integration range from the current point $x={\mathbf{x}}$.
The solution is returned at this point, ${\mathbf{x}}=\u27e8\mathit{\text{value}}\u27e9$.
With the given value of tol, no further progress can be made across the integration range from the current point $x={\mathbf{x}}$. (See Section 9 for a discussion of this error exit.) The components ${\mathbf{y}}\left(1\right),{\mathbf{y}}\left(2\right),\dots ,{\mathbf{y}}\left({\mathbf{n}}\right)$ contain the computed values of the solution at the current point $x={\mathbf{x}}$. If you have supplied $g$, no point at which $g(x,y)$ changes sign has been located up to the point $x={\mathbf{x}}$.
${\mathbf{ifail}}=3$
With the given value of tol, no initial progress can be made from the starting point of integration provided.
tol is too small for d02bjf to take an initial step. x and ${\mathbf{y}}\left(1\right),{\mathbf{y}}\left(2\right),\dots ,{\mathbf{y}}\left({\mathbf{n}}\right)$ retain their initial values.
${\mathbf{ifail}}=4$
On the first call to output, xsol remained unchanged.
On the first call to output, xsol was returned as $\u27e8\mathit{\text{value}}\u27e9$, which is inconsistent with $({\mathbf{x}},{\mathbf{xend}})$ — $(\u27e8\mathit{\text{value}}\u27e9,\u27e8\mathit{\text{value}}\u27e9)$.
On a call to output, xsol was returned as $\u27e8\mathit{\text{value}}\u27e9$, which is inconsistent with previous xsol and xend — $(\u27e8\mathit{\text{value}}\u27e9,\u27e8\mathit{\text{value}}\u27e9)$.
${\mathbf{ifail}}=6$
No change in sign of the function $g(x,y)$ was detected in the integration range.
${\mathbf{ifail}}=7$
Unexpected internal error in call to interpolation routine.
The interpolation routine returned error flag $\u27e8\mathit{\text{value}}\u27e9$.
Unexpected internal error in call to zero-finding routine.
The zero-finding routine returned error flag $\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=8$
Unexpected internal error in call to step integrator.
The step integrator returned error flag $\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
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 ${\mathbf{relabs}}=\text{'D'}$ unless you have a good reason for a different choice.
If the problem is a root-finding one, then the accuracy of the root determined will depend on the properties of $g(x,y)$ and on the values of tol and relabs. You should try to code g without introducing any unnecessary cancellation errors.
8Parallelism and Performance
d02bjf 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.
9Further Comments
If more than one root is required, then to determine the second and later roots d02bjf may be called again starting a short distance past the previously determined roots. Alternatively you may construct your own root-finding code using c05azf,d02pffandd02psf.
If d02bjf fails with ${\mathbf{ifail}}={\mathbf{3}}$, then it can be called again with a larger value of tol if this has not already been tried. If the accuracy requested is really needed and cannot be obtained with this routine, the system may be very stiff (see below) or so badly scaled that it cannot be solved to the required accuracy.
If d02bjf fails with ${\mathbf{ifail}}={\mathbf{2}}$, it is probable that it has been called with a value of tol which is so small that a solution cannot be obtained on the range x to xend. This can happen for well-behaved systems and very small values of tol. You should, however, consider whether there is a more fundamental difficulty. For example:
(a)in the region of a singularity (infinite value) of the solution, the routine will usually stop with ${\mathbf{ifail}}={\mathbf{2}}$, unless overflow occurs first. Numerical integration cannot be continued through a singularity, and analytic treatment should be considered;
(b)for ‘stiff’ equations where the solution contains rapidly decaying components, the routine will use very small steps in $x$ (internally to d02bjf) to preserve stability. This will exhibit itself by making the computing time excessively long, or occasionally by an exit with ${\mathbf{ifail}}={\mathbf{2}}$. Runge–Kutta methods are not efficient in such cases, and you should try d02ejf.
10Example
This example illustrates the solution of four different problems. In each case the differential system (for a projectile) is
over an interval ${\mathbf{x}}=0.0$ to ${\mathbf{xend}}=10.0$ starting with values $y=0.5$, $v=0.5$ and $\varphi =\pi /5$. We solve each of the following problems with local error tolerances $\text{1.0E\u22124}$ and $\text{1.0E\u22125}$.
(i)To integrate to $x=10.0$ producing intermediate output at intervals of $2.0$ until a root is encountered where $y=0.0$.