d02pfc is a one-step function for solving an initial value problem for a first-order system of ordinary differential equations using Runge–Kutta methods.
The function may be called by the names: d02pfc or nag_ode_ivp_rkts_onestep.
3Description
d02pfc and its associated functions
(d02pqc,d02prc,d02psc,d02ptcandd02puc) solve an initial value problem for a first-order system of ordinary differential equations. The functions, based on Runge–Kutta methods and derived from RKSUITE (see Brankin et al. (1991)), integrate
$${y}^{\prime}=f(t,y)\text{\hspace{1em} given \hspace{1em}}y\left({t}_{0}\right)={y}_{0}$$
where $y$ is the vector of $\mathit{n}$ solution components and $t$ is the independent variable.
d02pfc is designed to be used in complicated tasks when solving systems of ordinary differential equations. You must first call d02pqc to specify the problem and how it is to be solved. Thereafter you (repeatedly) call d02pfc to take one integration step at a time from tstart in the direction of tend (as specified in d02pqc). In this manner d02pfc returns an approximation to the solution ynow and its derivative ypnow at successive points tnow. If d02pfc encounters some difficulty in taking a step, the integration is not advanced and the function returns with the same values of tnow, ynow and ypnow as returned on the previous successful step. d02pfc tries to advance the integration as far as possible subject to passing the test on the local error and not going past tend.
In the call to d02pqc you can specify either the first step size for d02pfc to attempt or that it computes automatically an appropriate value. Thereafter d02pfc estimates an appropriate step size for its next step. This value and other details of the integration can be obtained after any call to d02pfc by a call to d02ptc. The local error is controlled at every step as specified in d02pqc. If you wish to assess the true error, you must set
${\mathbf{errass}}=\mathrm{Nag\_ErrorAssess\_on}$
in the call to d02pqc. This assessment can be obtained after any call to d02pfc by a call to d02puc.
If you want answers at specific points there are two ways to proceed:
(i)The more efficient way is to step past the point where a solution is desired, and then call d02psc to get an answer there. Within the span of the current step, you can get all the answers you want at very little cost by repeated calls to d02psc. This is very valuable when you want to find where something happens, e.g., where a particular solution component vanishes. You cannot proceed in this way with
${\mathbf{method}}=\mathrm{Nag\_RK\_7\_8}$.
(ii)The other way to get an answer at a specific point is to set tend to this value and integrate to tend. d02pfc will not step past tend, so when a step would carry it past, it will reduce the step size so as to produce an answer at tend exactly. After getting an answer there (${\mathbf{tnow}}={\mathbf{tend}}$), you can reset tend to the next point where you want an answer, and repeat. tend could be reset by a call to d02pqc, but you should not do this. You should use d02prc instead because it is both easier to use and much more efficient. This way of getting answers at specific points can be used with any of the available methods, but it is the only way with ${\mathbf{method}}=\mathrm{Nag\_RK\_7\_8}$. It can be inefficient. Should this be the case, the code will bring the matter to your attention.
4References
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: $\mathbf{f}$ – 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 $t$, ${y}_{i}$.
On exit: the values of
${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.
5: $\mathbf{comm}$ – 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 d02pfc you may allocate memory and initialize these pointers with various quantities for use by f when called from d02pfc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note:f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02pfc. If your code inadvertently does return any NaNs or infinities, d02pfc 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: $\mathbf{tnow}$ – double *Output
On exit: $t$, the value of the independent variable at which a solution has been computed.
On exit: an approximation to the solution at tnow. The local error of the step to tnow was no greater than permitted by the specified tolerances (see d02pqc).
On entry: these must be the same arrays supplied in a previous call to d02pqc. They must remain unchanged between calls.
On exit: information about the integration for use on subsequent calls to d02pfc or other associated functions.
9: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MISSING_CALL
On entry, a previous call to the setup function has not been made or the communication arrays have become corrupted.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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
A call to this function cannot be made after it has returned an error. The setup function must be called 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_POINTS
More than $100$ output points have been obtained by integrating to tend (as specified in the setup function). They have been so clustered that it would probably be (much) more efficient to use the interpolation function
(if ${\mathbf{method}}=\mathrm{Nag\_RK\_7\_8}$, switch to ${\mathbf{method}}=\mathrm{Nag\_RK\_4\_5}$ at setup).
However, you can continue integrating the 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, as specified in the setup function, has already been reached. To start a new problem, you will need to call the setup function. To continue integration beyond tend then d02prc must first be called to reset tend to a new end value.
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.
7Accuracy
The accuracy of integration is determined by the arguments tol and thresh in a prior call to d02pqc (see the function document for 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.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
d02pfc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
If d02pfc 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 ynow should be monitored 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.
If d02pfc 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. d02pfc 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.
over the range $[0,2\pi ]$ with initial conditions ${y}_{1}=0.0$ and ${y}_{2}=1.0$. We use relative error control with threshold values of $\text{1.0e\u22128}$ for each solution component and print the solution at each integration step across the range. We use a medium order Runge–Kutta method
(${\mathbf{method}}=\mathrm{Nag\_RK\_4\_5}$) with tolerances ${\mathbf{tol}}=\text{1.0e\u22124}$ and ${\mathbf{tol}}=\text{1.0e\u22125}$
in turn so that we may compare the solutions.