NAG Library Function Document

1Purpose

nag_ode_ivp_rkts_reset_tend (d02prc) resets the end point in an integration performed by nag_ode_ivp_rkts_onestep (d02pfc) and nag_ode_ivp_rk_step_revcomm (d02pgc).

2Specification

 #include #include
 void nag_ode_ivp_rkts_reset_tend (double tendnu, Integer iwsav[], double rwsav[], NagError *fail)

3Description

nag_ode_ivp_rkts_reset_tend (d02prc) and its associated functions (nag_ode_ivp_rkts_onestep (d02pfc), nag_ode_ivp_rk_step_revcomm (d02pgc), nag_ode_ivp_rk_interp_setup (d02phc), nag_ode_ivp_rk_interp_eval (d02pjc), nag_ode_ivp_rkts_setup (d02pqc), nag_ode_ivp_rkts_interp (d02psc), nag_ode_ivp_rkts_diag (d02ptc) and nag_ode_ivp_rkts_errass (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
 $y′=ft,y given yt0=y0$
where $y$ is the vector of $n$ solution components and $t$ is the independent variable.
nag_ode_ivp_rkts_reset_tend (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 nag_ode_ivp_rkts_onestep (d02pfc) or nag_ode_ivp_rk_step_revcomm (d02pgc)) 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).

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:    $\mathbf{tendnu}$doubleInput
On entry: the new value for ${t}_{f}$.
Constraint: $\mathrm{sign}\left({\mathbf{tendnu}}-{\mathbf{tnow}}\right)=\mathrm{sign}\left({\mathbf{tend}}-{\mathbf{tstart}}\right)$, 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) or nag_ode_ivp_rk_step_revcomm (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:    $\mathbf{iwsav}\left[130\right]$IntegerCommunication Array
3:    $\mathbf{rwsav}\left[350\right]$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) or nag_ode_ivp_rk_step_revcomm (d02pgc). 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 nag_ode_ivp_rk_step_revcomm (d02pgc) or other associated functions.
4:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ 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 2.7.6 in How to Use the NAG Library and its Documentation 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 2.7.5 in How to Use the NAG Library and its Documentation 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, ${\mathbf{tendnu}}=〈\mathit{\text{value}}〉$ and ${\mathbf{tnow}}=〈\mathit{\text{value}}〉$.
NE_RK_DIRECTION_POS
On entry, tendnu is not beyond tnow (step integrator) in the direction of integration.
The direction is positive, ${\mathbf{tendnu}}=〈\mathit{\text{value}}〉$ and ${\mathbf{tnow}}=〈\mathit{\text{value}}〉$.
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 $〈\mathit{\text{value}}〉$, but this quantity must be at least $〈\mathit{\text{value}}〉$.

Not applicable.

8Parallelism and Performance

nag_ode_ivp_rkts_reset_tend (d02prc) is not threaded in any implementation.

None.

10Example

This example integrates a two body problem. The equations for the coordinates $\left(x\left(t\right),y\left(t\right)\right)$ of one body as functions of time $t$ in a suitable frame of reference are
 $x′′=-xr3$
 $y′′=-yr3, r=x2+y2.$
The initial conditions
 $x0=1-ε, x′0=0 y0=0, y′0= 1+ε 1-ε$
lead to elliptic motion with $0<\epsilon <1$. $\epsilon =0.7$ is selected and the system of ODEs is reposed as
 $y1′=y3 y2′=y4 y3′=- y1r3 y4′=- y2r3$
over the range $\left[0,6\pi \right]$. Relative error control is used with threshold values of $\text{1.0e−10}$ for each solution component and compute the solution at intervals of length $\pi$ 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 (${\mathbf{method}}=\mathrm{Nag_RK_7_8}$) is also used with tolerances ${\mathbf{tol}}=\text{1.0e−4}$ and ${\mathbf{tol}}=\text{1.0e−5}$ in turn so that the solutions may be compared.

10.1Program Text

Program Text (d02prce.c)

10.2Program Data

Program Data (d02prce.d)

10.3Program Results

Program Results (d02prce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017