d02puc and its associated functions (d02pec, d02pfc, d02pgc, d02phc, d02pjc, d02pqc, d02prc, d02psc and 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^{'} = f (t, y) given y (t_{0}) = y_{0}

where

y

is the vector of

n

solution components and

t

is the independent variable.

After a call to d02pec, d02pfc or d02pgc, d02puc can be called for information about error assessment, if this assessment was specified in the setup function d02pqc. A more accurate ‘true’ solution

\hat{y}

is computed in a secondary integration. The error is measured as specified in d02pqc for local error control. At each step in the primary integration, an average magnitude

μ_{i}

of component

y_{i}

is computed, and the error in the component is

\frac{| y_{i} - {\hat{y}}_{i} |}{\max (μ_{i}, 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 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.

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: $n$ – Integer Input: On entry: $n$ , the number of ordinary differential equations in the system to be solved by the integration function.

Constraint: $n \geq 1$ .
2: $rmserr [n]$ – double Output: On exit: $rmserr [i - 1]$ approximates the RMS average of the true error of the numerical solution for the $i$ th solution component, for $i = 1, 2, \dots, n$ . The average is taken over all steps from the beginning of the integration to the current integration point.
3: $errmax$ – double * Output: On exit: the maximum weighted approximate true error taken over all solution components and all steps.
4: $terrmx$ – double * Output: On exit: the first value of the independent variable where an approximate true error attains the maximum value, errmax.
5: $iwsav [130]$ – Integer Communication Array
6: $rwsav [32 \times n + 350]$ – double Communication Array: On entry: these must be the same arrays supplied in a previous call to d02pec, d02pfc or d02pgc. They must remain unchanged between calls.

On exit: information about the integration for use on subsequent calls to d02pec, d02pfc or d02pgc or other associated functions.
7: $fail$ – 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_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, $n = ⟨ value ⟩$ , but the value passed to the setup function was $n = ⟨ value ⟩$ .
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 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_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.

7 Accuracy

Not applicable.

8 Parallelism and Performance

d02puc is not threaded in any implementation.

9 Further Comments

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 d02pqc.

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, 3 π]

. Relative error control is used with threshold values of

1.0e−10

for each solution component and a high-order Runge–Kutta method (

method = Nag_RK_7_8

) with tolerance

tol = 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 d02pec with

fail . code =

NE_RK_POINTS, NE_STIFF_PROBLEM or NW_RK_TOO_MANY).

d02pu: FL CL CPP AD

NAG CL Interfaced02puc (ivp_​rkts_​errass)

▸▿ 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
d02puc (ivp_rkts_errass)