# NAG C Library Function Document

## 1Purpose

nag_ode_ivp_rkts_errass (d02puc) provides details about global error assessment computed during an integration with either nag_ode_ivp_rkts_range (d02pec), nag_ode_ivp_rkts_onestep (d02pfc) or nag_ode_ivp_rk_step_revcomm (d02pgc).

## 2Specification

 #include #include
 void nag_ode_ivp_rkts_errass (Integer n, double rmserr[], double *errmax, double *terrmx, Integer iwsav[], double rwsav[], NagError *fail)

## 3Description

nag_ode_ivp_rkts_errass (d02puc) and its associated functions (nag_ode_ivp_rkts_range (d02pec), 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_reset_tend (d02prc), nag_ode_ivp_rkts_interp (d02psc) and nag_ode_ivp_rkts_diag (d02ptc)) 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 $\mathit{n}$ solution components and $t$ is the independent variable.
After a call to nag_ode_ivp_rkts_range (d02pec), nag_ode_ivp_rkts_onestep (d02pfc) or nag_ode_ivp_rk_step_revcomm (d02pgc), nag_ode_ivp_rkts_errass (d02puc) can be called for information about error assessment, if this assessment was specified in the setup function nag_ode_ivp_rkts_setup (d02pqc). A more accurate ‘true’ solution $\stackrel{^}{y}$ is computed in a secondary integration. The error is measured as specified in nag_ode_ivp_rkts_setup (d02pqc) for local error control. At each step in the primary integration, an average magnitude ${\mu }_{i}$ of component ${y}_{i}$ is computed, and the error in the component is
 $yi-y^i maxμi,thresh[i-1] .$
It is difficult to estimate reliably the true error at a single point. For this reason the RMS (root-mean-square) average of the estimated global error in each solution component is computed. This average is taken over all steps from the beginning of the integration through to the current integration point. If all has gone well, the average errors reported will be comparable to tol (see nag_ode_ivp_rkts_setup (d02pqc)). The maximum error seen in any component in the integration so far and the point where the maximum error first occurred are also reported.

## 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:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of ordinary differential equations in the system to be solved by the integration function.
Constraint: ${\mathbf{n}}\ge 1$.
2:    $\mathbf{rmserr}\left[{\mathbf{n}}\right]$doubleOutput
On exit: ${\mathbf{rmserr}}\left[\mathit{i}-1\right]$ approximates the RMS average of the true error of the numerical solution for the $\mathit{i}$th solution component, for $\mathit{i}=1,2,\dots ,\mathit{n}$. The average is taken over all steps from the beginning of the integration to the current integration point.
3:    $\mathbf{errmax}$double *Output
On exit: the maximum weighted approximate true error taken over all solution components and all steps.
4:    $\mathbf{terrmx}$double *Output
On exit: the first value of the independent variable where an approximate true error attains the maximum value, errmax.
5:    $\mathbf{iwsav}\left[130\right]$IntegerCommunication Array
6:    $\mathbf{rwsav}\left[32×{\mathbf{n}}+350\right]$doubleCommunication Array
On entry: these must be the same arrays supplied in a previous call to nag_ode_ivp_rkts_range (d02pec), 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_range (d02pec), nag_ode_ivp_rkts_onestep (d02pfc) or nag_ode_ivp_rk_step_revcomm (d02pgc) or other associated functions.
7:    $\mathbf{fail}$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_ERRASS_REQ
No error assessment is available since you did not ask for it in your call to the setup function.
NE_INT_2
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, but the value passed to the setup function was ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
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 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_RK_INVALID_CALL
You have already made one call to this function after the integrator could not achieve specified accuracy.
You cannot call this function again.
NE_RK_NOSTEP
No error assessment is available since the integrator has not actually taken any successful steps.

Not applicable.

## 8Parallelism and Performance

nag_ode_ivp_rkts_errass (d02puc) is not threaded in any implementation.

If the integration has proceeded ‘well’ and the problem is smooth enough, stable and not too difficult then the values returned in the arguments rmserr and errmax should be comparable to the value of tol specified in the prior call to nag_ode_ivp_rkts_setup (d02pqc).

## 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 $\left[0,3\pi \right]$. Relative error control is used with threshold values of $\text{1.0e−10}$ for each solution component and a high-order Runge–Kutta method (${\mathbf{method}}=\mathrm{Nag_RK_7_8}$) with tolerance ${\mathbf{tol}}=\text{1.0e−6}$.
Note that for illustration purposes since it is not necessary for this problem, this example integrates to the end of the range regardless of efficiency concerns (i.e., returns from nag_ode_ivp_rkts_range (d02pec) with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_RK_POINTS, NE_STIFF_PROBLEM or NW_RK_TOO_MANY).

### 10.1Program Text

Program Text (d02puce.c)

### 10.2Program Data

Program Data (d02puce.d)

### 10.3Program Results

Program Results (d02puce.r)