d02prc and its associated functions (d02pfc, d02pgc, d02phc, d02pjc, d02pqc, d02psc, d02ptc and d02puc) solve the 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^{'} = f (t, y) given y (t_{0}) = y_{0}

where

y

is the vector of

n

solution components and

t

is the independent variable.

d02prc is used to reset the final value of the independent variable,

t_{f}

, when the integration is already underway. It can be used to extend or reduce the range of integration. The new value must be beyond the current value of the independent variable (as returned in tnow by d02pfc or d02pgc) in the current direction of integration. It is much more efficient to use d02prc for this purpose than to use d02pqc which involves the overhead of a complete restart of the integration.

If you want to change the direction of integration then you must restart by a call to d02pqc.

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 91-S1 Southern Methodist University

5 Arguments

1: $tendnu$ – double Input: On entry: the new value for $t_{f}$ .

Constraint: $sign (tendnu - tnow) = sign (tend - tstart)$ , where tstart and tend are as supplied in the previous call to d02pqc and tnow is returned by the preceding call to d02pfc or d02pgc (i.e., integration must proceed in the same direction as before). tendnu must be distinguishable from tnow for the method and the machine precision being used.
2: $iwsav [130]$ – Integer Communication Array
3: $rwsav [350]$ – double Communication Array: Note: the communication array rwsav used by the other functions in the suite must be used here however, only the first $350$ elements will be referenced.

On entry: these must be the same arrays supplied in a previous call to d02pfc or d02pgc. They must remain unchanged between calls.

On exit: information about the integration for use on subsequent calls to d02pfc or d02pgc or other associated functions.
4: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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 $⟨ value ⟩$ had an illegal value.
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: You cannot call this function before you have called the step integrator.
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, a previous call to the setup function has not been made or 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 the integrator has returned an error.
NE_RK_DIRECTION_NEG: On entry, tendnu is not beyond tnow (step integrator) in the direction of integration.
The direction is negative, $tendnu = ⟨ value ⟩$ and $tnow = ⟨ value ⟩$ .
NE_RK_DIRECTION_POS: On entry, tendnu is not beyond tnow (step integrator) in the direction of integration.
The direction is positive, $tendnu = ⟨ value ⟩$ and $tnow = ⟨ value ⟩$ .
NE_RK_INVALID_CALL: You cannot call this function when the range integrator has been used.
NE_RK_STEP: On entry, tendnu is too close to tnow (step integrator). Their difference is $⟨ value ⟩$ , but this quantity must be at least $⟨ value ⟩$ .

7 Accuracy

Not applicable.

8 Parallelism and Performance

d02prc is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example integrates a two body problem. The equations for the coordinates

(x (t), y (t))

of one body as functions of time

t

in a suitable frame of reference are

x^{''} = - \frac{x}{r^{3}}

y^{''} = - \frac{y}{r^{3}}, r = \sqrt{x^{2} + y^{2}} .

The initial conditions

\begin{array}{l} x (0) = 1 - ε, & x^{'} (0) = 0 \\ y (0) = 0, & y^{'} (0) = \sqrt{\frac{1 + ε}{1 - ε}} \end{array}

lead to elliptic motion with

0 < ε < 1

ε = 0.7

is selected and the system of ODEs is reposed as

\begin{array}{l} y_{1}^{'} = y_{3} \\ y_{2}^{'} = y_{4} \\ y_{3}^{'} = - \frac{y_{1}}{r^{3}} \\ y_{4}^{'} = - \frac{y_{2}}{r^{3}} \end{array}

over the range

[0, 6 π]

. Relative error control is used with threshold values of

1.0e−10

for each solution component and compute the solution at intervals of length

π

across the range using d02prc to reset the end of the integration range. A high-order Runge–Kutta method (

method = Nag_RK_7_8

) is also used with tolerances

tol = 1.0e−4

and

tol = 1.0e−5

in turn so that the solutions may be compared.

d02pr: FL CL CPP AD

NAG CL Interfaced02prc (ivp_​rkts_​reset_​tend)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
d02prc (ivp_rkts_reset_tend)