# NAG C Library Function Document

## 1Purpose

nag_ode_ivp_rk_interp_eval (d02pjc) evaluates the interpolant calculated by nag_ode_ivp_rk_interp_setup (d02phc), following an integration step performed by nag_ode_ivp_rk_step_revcomm (d02pgc) to solve an initial value problem.

## 2Specification

 #include #include
 void nag_ode_ivp_rk_interp_eval (Nag_Boolean icheck, Integer n, Integer nwant, double t, Integer ideriv, double sol[], double wcomm[], Integer lwcomm, Integer iwsav[], double rwsav[], NagError *fail)

## 3Description

When integrating using the reverse communication Runge–Kutta integrator nag_ode_ivp_rk_step_revcomm (d02pgc), the solution or its derivatives can be obtained inexpensively between steps by interpolation. nag_ode_ivp_rk_interp_setup (d02phc) is called after a step by nag_ode_ivp_rk_step_revcomm (d02pgc) from a previous value of $t$ ($={t}_{k-1}$) to its current value, $t={t}_{k}$ (i.e., a $k$th successful time-step has been taken). nag_ode_ivp_rk_interp_eval (d02pjc) can then be called to evaluate interpolated approximations of the function or its derivatives at any value of $t$ in the interval $\left({t}_{k-1},{t}_{k}\right)$.
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:    $\mathbf{icheck}$Nag_BooleanInput
On entry: indicates whether consistency checks on input arguments should be performed
${\mathbf{icheck}}=\mathrm{Nag_FALSE}$
Don't perform checks on input arguments.
${\mathbf{icheck}}=\mathrm{Nag_TRUE}$
Perform consistency checks on input arguments.
It is recommended to use ${\mathbf{icheck}}=\mathrm{Nag_TRUE}$ on the first call following a call to nag_ode_ivp_rk_interp_setup (d02phc) and to set ${\mathbf{icheck}}=\mathrm{Nag_FALSE}$ on subsequent calls within the last step to avoid the overhead of argument checking.
2:    $\mathbf{n}$IntegerInput
On entry: $n$, the dimension of the system of ODEs being integrated.
Constraint: this must be the same value as supplied in a previous call to nag_ode_ivp_rkts_setup (d02pqc).
3:    $\mathbf{nwant}$IntegerInput
On entry: only the first nwant system components to be computed. This should be the same value as passed to nag_ode_ivp_rk_interp_setup (d02phc) when computing the interpolant.
Constraint: ${\mathbf{nwant}}={\mathbf{nwant}}$ passed to nag_ode_ivp_rk_interp_setup (d02phc).
4:    $\mathbf{t}$doubleInput
On entry: $t$, the value of the independent variable where a solution is desired. Although any value of $t$ can be supplied, accurate solutions can only be obtained for values in the range of the last time-step taken by nag_ode_ivp_rk_step_revcomm (d02pgc).
5:    $\mathbf{ideriv}$IntegerInput
On entry:
${\mathbf{ideriv}}=0$
Compute approximations to the first nwant components of the solution $y\left(t\right)$.
${\mathbf{ideriv}}=1$
Compute approximations to the first nwant components of the first derivatives of the solution ${y}^{\prime }\left(t\right)$.
Constraint: ${\mathbf{ideriv}}=0$ or $1$.
6:    $\mathbf{sol}\left[{\mathbf{nwant}}\right]$doubleOutput
On exit:
${\mathbf{ideriv}}=0$
The first nwant components of the solution $y\left(t\right)$.
${\mathbf{ideriv}}=1$
The first nwant components of the first derivatives of the solution ${y}^{\prime }\left(t\right)$.
7:    $\mathbf{wcomm}\left[{\mathbf{lwcomm}}\right]$doubleCommunication Array
On entry: this must be the same array supplied in a previous call to nag_ode_ivp_rk_interp_setup (d02phc). It must remain unchanged between calls.
8:    $\mathbf{lwcomm}$IntegerInput
On entry: length of wcomm. This should be the same value as supplied in a previous call to nag_ode_ivp_rk_interp_setup (d02phc).
If in a previous call to nag_ode_ivp_rkts_setup (d02pqc):
• ${\mathbf{method}}=\mathrm{Nag_RK_2_3}$, lwcomm must be at least $1$.
• ${\mathbf{method}}=\mathrm{Nag_RK_4_5}$, lwcomm must be at least ${\mathbf{n}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},5×{\mathbf{nwant}}\right)$.
• ${\mathbf{method}}=\mathrm{Nag_RK_7_8}$, ${\mathbf{lwcomm}}\ge 8×{\mathbf{nwant}}$.
9:    $\mathbf{iwsav}\left[130\right]$IntegerCommunication Array
10:  $\mathbf{rwsav}\left[32×{\mathbf{n}}+350\right]$doubleCommunication Array
On entry: these must be the same arrays supplied in a previous call 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_rk_step_revcomm (d02pgc), nag_ode_ivp_rk_interp_setup (d02phc) or other associated functions.
11:  $\mathbf{fail}$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_INT
On entry, ${\mathbf{ideriv}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ideriv}}=0$ or $1$.
On entry, ${\mathbf{lwcomm}}=〈\mathit{\text{value}}〉$.
Constraint: for ${\mathbf{method}}=\mathrm{Nag_RK_2_3}$, ${\mathbf{lwcomm}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{lwcomm}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nwant}}=〈\mathit{\text{value}}〉$.
Constraint: for ${\mathbf{method}}=\mathrm{Nag_RK_7_8}$, ${\mathbf{lwcomm}}\ge 8×{\mathbf{nwant}}$.
NE_INT_3
On entry, ${\mathbf{lwcomm}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nwant}}=〈\mathit{\text{value}}〉$.
Constraint: for ${\mathbf{method}}=\mathrm{Nag_RK_4_5}$, ${\mathbf{lwcomm}}\ge {\mathbf{n}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},5×{\mathbf{nwant}}\right)$.
NE_INT_CHANGED
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, but the value passed to the setup routine was ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{nwant}}=〈\mathit{\text{value}}〉$, but on interpolation setup ${\mathbf{nwant}}=〈\mathit{\text{value}}〉$.
Constraint: nwant must be unchanged from setup.
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
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.
You cannot call this function before you have called the interpolation setup.
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_INI
The previous call to the interpolation setup function returned an error.

## 7Accuracy

The computed values will be of a similar accuracy to that computed by nag_ode_ivp_rk_step_revcomm (d02pgc).

## 8Parallelism and Performance

nag_ode_ivp_rk_interp_eval (d02pjc) is not threaded in any implementation.