NAG AD Library
c05ay_a1w_f (contfn_brent_a1w)
Note: _a1w_ denotes that first order adjoints are computed in working precision; this has the corresponding argument type
nagad_a1w_w_rtype. Further implementations, for example for higher order differentiation or using the tangent linear approach, may become available at later marks of the NAG AD Library.
The method of codifying AD implementations in the routine name and corresponding argument types is described in the
NAG AD Library Introduction.
1
Purpose
c05ay_a1w_f
is the adjoint version of the primal routine
c05ayf.
Depending on the value of
ad_handle,
c05ay_a1w_f uses algorithmic differentiation or symbolic adjoints to compute adjoints of the primal.
2
Specification
Fortran Interface
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 |
|
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 &retval, Integer iuser[], nagad_a1w_w_rtype ruser[]), nagad_a1w_w_rtype &x, Integer iuser[], nagad_a1w_w_rtype ruser[], Integer &ifail) |
|
The routine may be called by the names c05ay_a1w_f or nagf_roots_contfn_brent_a1w.
3
Description
c05ay_a1w_f
is the adjoint version of the primal routine
c05ayf.
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 the Introduction to the NAG AD Library for API description on how to use symbolic adjoints.
The symbolic adjoints assumes
-
(i)successful computation of primal problem ( or on exit of c05ayf), i.e.,
where is a solution;
-
(ii)the first derivative at the solution is not equal zero
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 the Introduction to the NAG AD Library and recall what primal and adjoint callbacks need to calculate in case of an algorithmic adjoint.
Assuming the original user-supplied function evaluates
where
is given by
ruser or by use of COMMON globals. The variables
and
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 in the following equation), i.e.,
-
(c)Function value evaluation and adjoint computation w.r.t. only, i.e.,
Here 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
followed by an adjoint projection through the user-supplied adjoint routine
Both as well as are computed using the user-supplied adjoint routine.
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
, where
is given by the argument
ruser or by use of COMMON globals (see
(3)).
The adjoints of all other input parameters are not referenced.
4
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
5
Arguments
In addition to the arguments present in the interface of the primal routine,
c05ay_a1w_f 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 – Type (c_ptr)
Input
-
On entry: a handle to the AD configuration data object, as created by x10aa_a1w_f. Symbolic adjoint mode may be selected by calling x10aa_a1w_f with this handle.
-
2:
a – Type (nagad_a1w_w_rtype)
Input
-
3:
b – Type (nagad_a1w_w_rtype)
Input
-
4:
eps – Type (nagad_a1w_w_rtype)
Input
-
5:
eta – Type (nagad_a1w_w_rtype)
Input
-
6:
f – Subroutine
External Procedure
-
Note that f is a subroutine in this interface, returning the function value via the additional output parameter retval.
The specification of f is:
Fortran Interface
Integer, Intent (Inout) |
:: |
iuser(*) |
Type (nagad_a1w_w_rtype), Intent (In) |
:: |
x |
Type (nagad_a1w_w_rtype), Intent (Inout) |
:: |
ruser(*) |
Type (nagad_a1w_w_rtype), Intent (Out) |
:: |
retval |
Type (c_ptr), Intent (In) |
:: |
ad_handle |
|
-
1:
ad_handle – Type (c_ptr)
Input
-
On entry: a handle to the AD configuration data object.
-
2:
x – Type (nagad_a1w_w_rtype)
Input
-
3:
retval – Type (nagad_a1w_w_rtype)
Output
-
On exit: the value of evaluated at x.
-
4:
iuser – Integer array
User Workspace
-
5:
ruser – Type (nagad_a1w_w_rtype) array
User Workspace
-
7:
x – Type (nagad_a1w_w_rtype)
Output
-
8:
iuser() – Integer array
User Workspace
-
9:
ruser() – Type (nagad_a1w_w_rtype) array
User Workspace
-
10:
ifail – Integer
Input/Output
6
Error Indicators and Warnings
c05ay_a1w_f preserves all error codes from
c05ayf and in addition can return:
An unexpected AD error has been triggered by this routine. Please
contact
NAG.
See
Section 4.5.2 in the NAG AD Library Introduction for further information.
Dynamic memory allocation failed for AD.
See
Section 4.5.1 in the NAG AD Library Introduction for further information.
In symbolic mode the following may be returned:
-
is zero. The adjoints have not been computed.
7
Accuracy
Not applicable.
8
Parallelism and Performance
c05ay_a1w_f
is not threaded in any implementation.
None.
10
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
Thus, in symbolic mode, the callback modes need to calculate
-
(a)
-
(b).
-
(c).
Please refer to
Section 4.3 in the Introduction to the NAG AD Library for details on implementation of the callback.