NAG Library Function Document

nag_ode_ivp_rkts_reset_tend (d02prc)

, 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 nag_ode_ivp_rkts_onestep (d02pfc)) in the current direction of integration. It is much more efficient to use nag_ode_ivp_rkts_reset_tend (d02prc) for this purpose than to use nag_ode_ivp_rkts_setup (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 nag_ode_ivp_rkts_setup (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 – doubleInput: 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 nag_ode_ivp_rkts_setup (d02pqc) and tnow is returned by the preceding call to nag_ode_ivp_rkts_onestep (d02pfc) (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[ $130$ ] – IntegerCommunication Array
3: rwsav[ $350$ ] – doubleCommunication 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 nag_ode_ivp_rkts_onestep (d02pfc). They must remain unchanged between calls.

On exit: information about the integration for use on subsequent calls to nag_ode_ivp_rkts_onestep (d02pfc) or other associated functions.
4: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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.
NE_MISSING_CALL: You cannot call this function before you have called the step integrator.
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

Not applicable.

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 nag_ode_ivp_rkts_reset_tend (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.

NAG Library Function Documentnag_ode_ivp_rkts_reset_tend (d02prc)

+− Contents

1 Purpose

2 Specification

3 Description