NAG Library Function Document

nag_ode_ivp_rkts_range (d02pec)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{f}$ – function, supplied by the userExternal Function

f must evaluate the functions $f_i$ (that is the first derivatives $y_i^{\prime }$) for given values of the arguments $t$, $y_i$.

The specification of f is:

void  f (double t, Integer n, const double y[], double yp[], Nag_Comm *comm)

1:    $\mathbf{t}$ – doubleInput

On entry: $t$, the current value of the independent variable.

2:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of ordinary differential equations in the system to be solved.

3:    $\mathbf{y}\left[\mathbf{n}\right]$ – const doubleInput

On entry: the current values of the dependent variables, $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.

4:    $\mathbf{yp}\left[\mathbf{n}\right]$ – doubleOutput

On exit: the values of $f_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.

5:    $\mathbf{comm}$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to f.

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_ode_ivp_rkts_range (d02pec) you may allocate memory and initialize these pointers with various quantities for use by f when called from nag_ode_ivp_rkts_range (d02pec) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_ode_ivp_rkts_range (d02pec). If your code inadvertently does return any NaNs or infinities, nag_ode_ivp_rkts_range (d02pec) is likely to produce unexpected results.

2:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of ordinary differential equations in the system to be solved.

Constraint: $\mathbf{n}\ge 1$.

3:    $\mathbf{twant}$ – doubleInput

On entry: $t$, the next value of the independent variable where a solution is desired.

Constraint: twant must be closer to tend than the previous value of tgot (or tstart on the first call to nag_ode_ivp_rkts_range (d02pec)); see nag_ode_ivp_rkts_setup (d02pqc) for a description of tstart and tend. twant must not lie beyond tend in the direction of integration.

4:    $\mathbf{tgot}$ – double *Output

On exit: $t$, the value of the independent variable at which a solution has been computed. On successful exit with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR, tgot will equal twant. On exit with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_RK_GLOBAL_ERROR_S, NE_RK_GLOBAL_ERROR_T, NE_RK_POINTS, NE_RK_STEP_TOO_SMALL, NE_STIFF_PROBLEM or NW_RK_TOO_MANY, a solution has still been computed at the value of tgot but in general tgot will not equal twant.

5:    $\mathbf{ygot}\left[\mathbf{n}\right]$ – doubleInput/Output

On entry: on the first call to nag_ode_ivp_rkts_range (d02pec), ygot need not be set. On all subsequent calls ygot must remain unchanged.

On exit: an approximation to the true solution at the value of tgot. At each step of the integration to tgot, the local error has been controlled as specified in nag_ode_ivp_rkts_setup (d02pqc). The local error has still been controlled even when $\mathbf{tgot}\ne \mathbf{twant}$, that is after a return with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_RK_GLOBAL_ERROR_S, NE_RK_GLOBAL_ERROR_T, NE_RK_POINTS, NE_RK_STEP_TOO_SMALL, NE_STIFF_PROBLEM or NW_RK_TOO_MANY.

6:    $\mathbf{ypgot}\left[\mathbf{n}\right]$ – doubleOutput

On exit: an approximation to the first derivative of the true solution at tgot.

7:    $\mathbf{ymax}\left[\mathbf{n}\right]$ – doubleInput/Output

On entry: on the first call to nag_ode_ivp_rkts_range (d02pec), ymax need not be set. On all subsequent calls ymax must remain unchanged.

On exit: $\mathbf{ymax}\left[i-1\right]$ contains the largest value of $\left|y_i\right|$ computed at any step in the integration so far.

8:    $\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

9:    $\mathbf{iwsav}\left[130\right]$ – IntegerCommunication Array

10:  $\mathbf{rwsav}\left[32\times \mathbf{n}+350\right]$ – doubleCommunication Array

On entry: these must be the same arrays supplied in a previous call to nag_ode_ivp_rkts_setup (d02pqc). They must remain unchanged between calls.

On exit: information about the integration for use on subsequent calls to nag_ode_ivp_rkts_range (d02pec) 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).

6

Error 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.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_INT_CHANGED

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

On entry, a previous call to the setup function has not been made or the communication arrays have become corrupted.

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, 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 it has returned an error.
You must call the setup function to start another problem.

NE_RK_GLOBAL_ERROR_S

The global error assessment algorithm failed at start of integration.
The integration is being terminated.

NE_RK_GLOBAL_ERROR_T

The global error assessment may not be reliable for times beyond $〈\mathit{\text{value}}〉$.
The integration is being terminated.

NE_RK_INVALID_CALL

You cannot call this function when you have specified, in the setup function, that the step integrator will be used.

NE_RK_POINTS

This function is being used inefficiently because the step size has been reduced drastically many times to obtain answers at many points. Using the order $4$ and $5$ pair method at setup is more appropriate here. You can continue integrating this problem.

NE_RK_STEP_TOO_SMALL

In order to satisfy your error requirements the solver has to use a step size of $〈\mathit{\text{value}}〉$ at the current time, $〈\mathit{\text{value}}〉$. This step size is too small for the machine precision, and is smaller than $〈\mathit{\text{value}}〉$.

NE_RK_TGOT_EQ_TEND

tend (setup) had already been reached in a previous call.
To start a new problem, you will need to call the setup function.

NE_RK_TGOT_RANGE_TEND

twant does not lie in the direction of integration. $\mathbf{twant}=〈\mathit{\text{value}}〉$.

twant lies beyond tend (setup) in the direction of integration.
$\mathbf{twant}=〈\mathit{\text{value}}〉$ and $\mathbf{tend}=〈\mathit{\text{value}}〉$.

NE_RK_TGOT_RANGE_TEND_CLOSE

twant lies beyond tend (setup) in the direction of integration, but is very close to tend.
You may have intended $\mathbf{twant}=\mathbf{tend}$.
$\left|\mathbf{twant}-\mathbf{tend}\right|=〈\mathit{\text{value}}〉$.

NE_RK_TWANT_CLOSE_TGOT

twant is too close to the last value of tgot (tstart on setup).
When using the method of order $8$ at setup, these must differ by at least $〈\mathit{\text{value}}〉$. Their absolute difference is $〈\mathit{\text{value}}〉$.

NE_STIFF_PROBLEM

Approximately $〈\mathit{\text{value}}〉$ function evaluations have been used to compute the solution since the integration started or since this message was last printed. Your problem has been diagnosed as stiff. If the situation persists, it will cost roughly $〈\mathit{\text{value}}〉$ times as much to reach tend (setup) as it has cost to reach the current time. You should probably call functions intended for stiff problems. However, you can continue integrating the problem.

NW_RK_TOO_MANY

Approximately $〈\mathit{\text{value}}〉$ function evaluations have been used to compute the solution since the integration started or since this message was last printed. However, you can continue integrating the problem.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (d02pece.c)

10.2

Program Data

Program Data (d02pece.d)

10.3

Program Results

Program Results (d02pece.r)