d02phc is a reverse communication function that computes the interpolant for evaluation by d02pjc anywhere on an integration step taken by d02pgc. The direct communication version of the d02phc and d02pjc pair is d02psc. A significant difference in functionality between the forward and reverse communication versions is that d02phc and d02pjc can interpolate for the high-order Runge–Kutta method.

2 Specification

#include <nag.h>

void	d02phc (Integer irevcm, Integer n, Integer nwant, double t, double y[], const double yp[], double wcomm[], Integer lwcomm, Integer iwsav[], double rwsav[], NagError *fail)

The function may be called by the names: d02phc or nag_ode_ivp_rk_interp_setup.

3 Description

d02phc and its associated functions (d02pgc, d02pjc, d02pqc, d02prc, d02ptc and 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^{'} = f (t, y) given y (t_{0}) = y_{0}

where

y

is the vector of

n

solution components and

t

is the independent variable.

d02pgc computes the solution at the end of an integration step. Using the information computed on that step d02phc computes the interpolant which can be evaluated at any point on that step by d02pjc. If

method = Nag_RK_2_3

then there is enough information available from the stages of the last step to provide an interpolant of sufficient order of accuracy; no further derivative evaluations will, therefore, be requested. If

method = Nag_RK_4_5

then the interpolant is an order

8

continuous Runge–Kutta process that requires a further

3

stages of derivative evaluations that will be requested in turn before a final exit. If

method = Nag_RK_7_8

was specified in the call to setup function d02pqc then the interpolant is a continuous Runge–Kutta process requiring a further

7

stages of derivative evaluations that will be requested in turn.

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

Note: this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument irevcm. Between intermediate exits and re-entries, all arguments other than those specified by the value of irevcm must remain unchanged.

1: $irevcm$ – Integer * Input/Output

On initial entry: irevcm must be set to zero to indicate that the interpolant for a new step is being taken.

On intermediate re-entry: irevcm should remain unchanged.

On intermediate exit: irevcm returns a value

1

to indicate that a function evaluation is required prior to re-entry; the value of the derivatives must be returned in yp where the value of

t

is supplied in t and the values

y (t)

are supplied in the array y.

On final exit:

$irevcm = −1$: Successful exit; rwsav and wcomm contain details of the interpolant.
$irevcm = −2$: Error exit; fail should be interrogated to determine the nature of the error.

Note: any values you return to d02phc as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by d02phc. If your code inadvertently does return any NaNs or infinities, d02phc is likely to produce unexpected results.

2: $n$ – 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

. This must be the same value as supplied in a previous call to d02pqc.

3: $nwant$ – Integer Input

On entry: the number of components of the solution to be computed. The first nwant components are evaluated.

Constraint:

1 \leq nwant \leq n

4: $t$ – double * Output

On intermediate exit: t contains the value of the independent variable

t

at which the derivatives

y^{'}

are to be evaluated.

On final exit: contains no useful information.

5: $y [n]$ – double Output

On intermediate exit: y contains the value of the solution

y

at which the derivatives

y^{'}

are to be evaluated.

On final exit: contains no useful information.

6: $yp [n]$ – const double Input

On initial entry: need not be set.

On intermediate re-entry: yp must contain the values of the derivatives

y_{i}^{'}

for the given values of the parameters

t

y_{i}

7: $wcomm [lwcomm]$ – double Communication Array

On entry: wcomm need not be set.

On intermediate re-entry: wcomm contains the partial computation of the polynomial coefficients corresponding to a continuous Runge–Kutta process for interpolating medium and high order Runge–Kutta methods.

On final exit: if

method = Nag_RK_4_5

Nag_RK_7_8

, wcomm contains details of the interpolant which must be passed unchanged to d02pjc for evaluation of the interpolant.

8: $lwcomm$ – Integer Input

On entry: length of wcomm.

If in a previous call to d02pqc:

$method = Nag_RK_2_3$ , lwcomm must be at least $1$ .
$method = Nag_RK_4_5$ , lwcomm must be at least $n + \max (n, 5 \times nwant)$ .
$method = Nag_RK_7_8$ , lwcomm must be at least $8 \times nwant$ .

9: $iwsav [130]$ – Integer Communication Array

10: $rwsav [32 \times n + 350]$ – double Communication Array

On entry: these must be the same arrays supplied in a previous call d02pgc. They must remain unchanged between calls.

On exit: information about the integration for use on subsequent calls to d02pgc, d02pjc or other associated functions.

11: $fail$ – 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_INT: On entry, $lwcomm = ⟨ value ⟩$ .
Constraint: for $method = Nag_RK_2_3$ , $lwcomm \geq 1$ .
NE_INT_2: On entry, $lwcomm = ⟨ value ⟩$ and $nwant = ⟨ value ⟩$ .
Constraint: for $method = Nag_RK_7_8$ , $lwcomm \geq 8 \times nwant$ .

On entry, $nwant = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq nwant \leq n$ .
NE_INT_3: On entry, $lwcomm = ⟨ value ⟩$ , $n = ⟨ value ⟩$ and $nwant = ⟨ value ⟩$ .
Constraint: for $method = Nag_RK_4_5$ , $lwcomm \geq n + \max (n, 5 \times nwant)$ .
NE_INT_CHANGED: 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: 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 step 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_INI: You cannot call this function after the integrator has returned an error.
NE_RK_INVALID_CALL: You cannot call this function after the range integrator has been called.

7 Accuracy

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

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

d02phc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.