c05ay_ad_f : NAG AD Library Routine Document, Mark 26

Fortran Interface

Subroutine c05ay_a1w_f (

ad_handle, a, b, eps, eta, f, x, iuser, ruser, ifail)

Integer, Intent (Inout)	::	iuser(*), ifail
External	::	f
Type (nagad_a1w_w_rtype), Intent (In)	::	a, b, eps, eta
Type (nagad_a1w_w_rtype), Intent (Inout)	::	ruser(*)
Type (nagad_a1w_w_rtype), Intent (Out)	::	x
Type (c_ptr), Intent (In)	::	ad_handle

Subroutine f (

ad_handle, x, ret, iuser, ruser)

Integer, Intent (Inout)	::	iuser(*)
Type (nagad_a1w_w_rtype), Intent (Inout)	::	x, ruser(*)
Type (nagad_a1w_w_rtype), Intent (Out)	::	ret
Type (c_ptr), Intent (In)	::	ad_handle

C++ Header Interface

#include <nagad.h>

void c05ay_a1w_f_ (

void *&ad_handle, const nagad_a1w_w_rtype &a, const nagad_a1w_w_rtype &b, const nagad_a1w_w_rtype &eps, const nagad_a1w_w_rtype &eta,
void (NAG_CALL f)(void *&ad_handle, nagad_a1w_w_rtype &x, nagad_a1w_w_rtype &ret, Integer iuser[], nagad_a1w_w_rtype ruser[]),
nagad_a1w_w_rtype &x, Integer iuser[], nagad_a1w_w_rtype ruser[], Integer &ifail)

Description

c05ayf locates a simple zero of a continuous function in a given interval using Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection. For further information see Section 3 in the documentation for c05ayf .

3.1

Symbolic Adjoints

c05ay_a1w_f can provide symbolic adjoints by setting the symbolic mode as described in Section 3.2.2 in the X10 Chapter Introduction. Please see Section 4 in NAG AD Library Introduction for API description on how to use symbolic adjoints.

The symbolic adjoints assumes

(i)

successful computation of primal problem (

ifail = 0

2

on exit of c05ayf), i.e.,

f (\hat{x}, p) = 0,

(1)

where

\hat{x}

is a solution;

(ii)

the first derivative at the solution is not equal zero

\nabla_{x} f (\hat{x}, p) \neq 0 .

(2)

In comparison to the algorithmic adjoint, the user-supplied primal and adjoint callbacks need specific implementation to support symbolic adjoint computation. Please see Section 4.2.3 in NAG AD Library Introduction and recall what primal and adjoint callbacks need to calculate in case of an algorithmic adjoint.

Assuming the original user-supplied function evaluates

z = f (x, p),

(3)

where

p

is given by ruser or by use of COMMON globals. The variables

x

and

z

correspond to x of user-supplied subroutine f and return the value of f respectively.

The symbolic adjoint of c05ayf then requires in addition the following capability / modes:

(a)

Function value evaluation only

(b)

Function value evaluation and adjoint computation w.r.t. x only (corresponds to

x

in the following equation), i.e.,

x_{(1)} + = {[\nabla_{x} f (x, p)]}^{T} z_{(1)}

(4)

(c)

Function value evaluation and adjoint computation w.r.t.

p

only, i.e.,

p_{(1)} + = {[\nabla_{p} f (x, p)]}^{T} z_{(1)} .

(5)

Here

p

is a placeholder for any user variable either passed via ruser or via COMMON global variables.

The library routine communicates via ad_config to the primal callback which version to provide, given the following API.

3.1.1

Mathematical Background

The symbolic adjoint computes

z = - \frac{x_{(1)}}{\nabla_{x} f (\hat{x}, p)}

followed by an adjoint projection through the user-supplied adjoint routine

p_{(1)} = p_{(1)} + \nabla_{p} f (\hat{x}, p) \cdot z .

Both

\nabla_{x} f (x, p)

as well as

\nabla_{p} f (x, p)

are computed using the user-supplied adjoint routine.

Please see Du Toit and Naumann (2017), Naumann et al. (2017) and Giles (2017) for reference.

3.1.2

Usable Adjoints

You can set or access the adjoints of output argument x. The adjoints of all other output arguments are ignored.

c05ay_a1w_f increments the adjoints of the variable

p

, where

p

is given by the argument ruser or by use of COMMON globals (see (3)).

The adjoints of all other input parameters are not referenced.

References

Du Toit J, Naumann U (2017) Adjoint Algorithmic Differentiation Tool Support for Typical Numerical Patterns in Computational Finance

Giles M (2017) Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation

Naumann U, Lotz J, Leppkes K and Towara M (2017) Algorithmic Differentiation of Numerical Methods: Tangent and Adjoint Solvers for Parameterized Systems of Nonlinear Equations

Arguments

c05ay_a1w_f provides access to all the arguments available in the primal routine. There are also additional arguments specific to AD. A tooltip popup for each argument can be found by hovering over the argument name in Section 2 and a summary of the arguments are provided below:

ad_handle – a handle to the AD configuration data object, as created by x10aa_a1w_f. Symbolic adjoint mode may be selected by calling x10ac_a1w_f with this handle.
a – $a$ , the lower bound of the interval.
b – $b$ , the upper bound of the interval.
eps – the termination tolerance on $x$ (see [Description]).
eta – a value such that if $| f (x) | \leq eta$ , $x$ is accepted as the zero.
f – this argument must evaluate the function $f$ whose zero is to be determined.
x – on exit: if the function exits successfully or ifail = 2, this argument is the final approximation to the zero.
iuser – may be used to pass information to user-supplied argument(s).
ruser – may be used to pass information to user-supplied argument(s).
ifail – on entry: ifail must be set to $0$ , $- 1 or 1$ . on exit: ifail = 0 unless the routine detects an error or a warning has been flagged (see Section 6).

Error Indicators and Warnings

c05ay_a1w_f preserves all error codes from c05ayf and in addition can return:

$ifail = - 89$: An unexpected AD error has been triggered by this routine. Please contact NAG.
See Section 5.2 in the NAG AD Library Introduction for further information.

$ifail = - 899$: Dynamic memory allocation failed for AD.
See Section 5.1 in the NAG AD Library Introduction for further information.

In symbolic mode the following may be returned:

Accuracy

Parallelism and Performance

c05ay_a1w_f is not threaded in any implementation.

Further Comments

Example

The following examples are variants of the example for c05ayf , modified to demonstrate calling the NAG AD Library.

We assume the original user-supplied routine to compute

z = f (x, p) = e^{- x} - x \cdot p .

Thus, in symbolic mode, the callback modes need to calculate

(a)	$z = f (x, p) = e^{- x} - x \cdot p$
(b)	$x_{(1)} + = {[\nabla_{x} f (x, p)]}^{T} \cdot z_{(1)} = ({- e}^{- x} - p) \cdot z_{(1)}$ .
(c)	$p_{(1)} + = {[\nabla_{p} f (x, p)]}^{T} \cdot z_{(1)} = - x \cdot z_{(1)}$ .

Language	Source File	Data	Results
Fortan	c05ay_a1w_fe.f90	c05ay_a1w_fe.d	c05ay_a1w_fe.r
C++	c05ay_a1w_hcppe.cpp	c05ay_a1w_hcppe.d	c05ay_a1w_hcppe.r

NAG AD Library Routine Document

c05ay_a1w_f (contfn_brent_a1w)

▸▿ Contents

1

Purpose

2

Specification

3

Description

3.1

Symbolic Adjoints

3.1.1

Mathematical Background

3.1.2

Usable Adjoints

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

NAG AD Library Routine Document

c05ay_a1w_f (contfn_brent_a1w)

▸▿ Contents

1 Purpose

2 Specification

3 Description

3.1 Symbolic Adjoints

3.1.1 Mathematical Background

3.1.2 Usable Adjoints

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example