d02pe is the AD Library version of the primal routine d02pef. Based (in the C++ interface) on overload resolution, d02pe can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first and second order.

2 Specification

Fortran Interface

Subroutine d02pe_AD_f (

f, n, twant, tgot, ygot, ypgot, ymax, iuser, ruser, iwsav, rwsav, ifail)

Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	iuser(*), iwsav(130), ifail
ADTYPE, Intent (In)	::	twant
ADTYPE, Intent (Inout)	::	ygot(n), ymax(n), ruser(), rwsav(32n+350)
ADTYPE, Intent (Out)	::	tgot, ypgot(n)
Type (c_ptr), Intent (Inout)	::	ad_handle
External	::	f

Corresponding to the overloaded C++ function, the Fortran interface provides five routines with names reflecting the type used for active real arguments. The actual subroutine and type names are formed by replacing AD and ADTYPE in the above as follows:

when ADTYPE is	Real(kind=nag_wp)	then AD is p0w
when ADTYPE is	Type(nagad_a1w_w_rtype)	then AD is a1w
when ADTYPE is	Type(nagad_t1w_w_rtype)	then AD is t1w
when ADTYPE is	Type(nagad_a1t1w_w_rtype)	then AD is a1t1w
when ADTYPE is	Type(nagad_t2w_w_rtype)	then AD is t2w

C++ Interface

#include <dco.hpp>
#include <nagad.h>
namespace nag {
namespace ad {
template <typename F_T>

void d02pe (

handle_t &ad_handle, F_T &&f, const Integer &n, const ADTYPE &twant, ADTYPE &tgot, ADTYPE ygot[], ADTYPE ypgot[], ADTYPE ymax[], Integer iwsav[], ADTYPE rwsav[], Integer &ifail)

}
}

The function is overloaded on ADTYPE which represents the type of active arguments. ADTYPE may be any of the following types:
double,
dco::ga1s<double>::type,
dco::gt1s<double>::type,
dco::gt1s<dco::gt1s<double>::type>::type,
dco::ga1s<dco::gt1s<double>::type>::type,

Note: this function can be used with AD tools other than dco/c++. For details, please contact NAG.

3 Description

d02pe is the AD Library version of the primal routine d02pef.

d02pef solves an initial value problem for a first-order system of ordinary differential equations using Runge–Kutta methods. For further information see Section 3 in the documentation for d02pef.

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

In addition to the arguments present in the interface of the primal routine, d02pe includes some arguments specific to AD.

A brief summary of the AD specific arguments is given below. For the remainder, links are provided to the corresponding argument from the primal routine. A tooltip popup for all arguments can be found by hovering over the argument name in Section 2 and in this section.

1: ad_handle – nag::ad::handle_t Input/Output

On entry: a configuration object that holds information on the differentiation strategy. Details on setting the AD strategy are described in AD handle object in the NAG AD Library Introduction.

2: f – Callable Input

f needs to be callable with the specification listed below. This can be a C++ lambda, a functor or a (static member) function pointer. If using a lambda, parameters can be captured safely by reference. No copies of the callable are made internally.

The specification of f is:

Fortran Interface

Subroutine f (

t, n, y, yp, iuser, ruser)

Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	iuser(*)
ADTYPE, Intent (In)	::	t, y(n)
ADTYPE, Intent (Inout)	::	ruser(*)
ADTYPE, Intent (Out)	::	yp(n)
Type (c_ptr), Intent (Inout)	::	ad_handle

C++ Interface

auto f = [&](

const handle_t &ad_handle, const ADTYPE &t, const Integer &n, const ADTYPE y[], ADTYPE yp[])

1: ad_handle – nag::ad::handle_t Input/Output: On entry: a handle to the AD handle object.
2: t – ADTYPE Input
3: n – Integer Input
4: y – ADTYPE array Input
5: yp – ADTYPE array Output
*: iuser – Integer array User Workspace
*: ruser – ADTYPE array User Workspace

3: n – Integer Input

4: twant – ADTYPE Input

5: tgot – ADTYPE Output

6: ygot(n) – ADTYPE array Input/Output

Please consult Overwriting of Inputs in the NAG AD Library Introduction.

7: ypgot(n) – ADTYPE array Output

8: ymax(n) – ADTYPE array Input/Output

Please consult Overwriting of Inputs in the NAG AD Library Introduction.

*: iuser( $*$ ) – Integer array User Workspace

*: ruser( $*$ ) – ADTYPE array User Workspace

Please consult Overwriting of Inputs in the NAG AD Library Introduction.

9: iwsav( $130$ ) – Integer array Communication Array

10: rwsav( $32 \times n + 350$ ) – ADTYPE array Communication Array

Please consult Overwriting of Inputs in the NAG AD Library Introduction.

11: ifail – Integer Input/Output

6 Error Indicators and Warnings

d02pe preserves all error codes from d02pef and in addition can return:

$ifail = - 89$: An unexpected AD error has been triggered by this routine. Please contact NAG.
See Error Handling in the NAG AD Library Introduction for further information.

$ifail = - 199$: The routine was called using a strategy that has not yet been implemented.
See AD Strategies in the NAG AD Library Introduction for further information.

$ifail = - 444$: A C++ exception was thrown.
The error message will show the details of the C++ exception text.

$ifail = - 899$: Dynamic memory allocation failed for AD.
See Error Handling in the NAG AD Library Introduction for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

d02pe is not threaded in any implementation.

9 Further Comments

None.

10 Example

The following examples are variants of the example for d02pef, modified to demonstrate calling the NAG AD Library.

Description of the primal example.

This example solves the equation

y^{''} = - y, y (0) = 0, y^{'} (0) = 1

reposed as

y_{1}^{'} = y_{2}

y_{2}^{'} = - y_{1}

over the range

[0, 2 π]

with initial conditions

y_{1} = 0.0

and

y_{2} = 1.0

. Relative error control is used with threshold values of

1.0E−8

for each solution component and compute the solution at intervals of length

π / 4

across the range. A low-order Runge–Kutta method (see d02pq) is also used with tolerances

tol = 1.0E−3

and

tol = 1.0E−4

in turn so that the solutions can be compared.

10.1 Adjoint modes

Language	Source File	Data	Results
Fortran	d02pe_a1t1w_fe.f90	d02pe_a1t1w_fe.d	d02pe_a1t1w_fe.r
Fortran	d02pe_a1w_fe.f90	d02pe_a1w_fe.d	d02pe_a1w_fe.r
C++	d02pe_a1_algo_dcoe.cpp	None	d02pe_a1_algo_dcoe.r
C++	d02pe_a1t1_algo_dcoe.cpp	None	d02pe_a1t1_algo_dcoe.r

10.2 Tangent modes

Language	Source File	Data	Results
Fortran	d02pe_t1w_fe.f90	d02pe_t1w_fe.d	d02pe_t1w_fe.r
Fortran	d02pe_t2w_fe.f90	d02pe_t2w_fe.d	d02pe_t2w_fe.r
C++	d02pe_t1_algo_dcoe.cpp	None	d02pe_t1_algo_dcoe.r
C++	d02pe_t2_algo_dcoe.cpp	None	d02pe_t2_algo_dcoe.r