is the AD Library version of the primal routine
Based (in the C++ interface) on overload resolution,
c05ay can be used for primal, tangent and adjoint
evaluation. It supports tangents and adjoints of first and second order.
The parameter ad_handle can be used to choose whether adjoints are computed using a symbolic adjoint or straightforward algorithmic differentiation.
In addition, the routine has further optimisations when expert symbolic mode is selected.
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:
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.
is the AD Library version of the primal routine
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.
c05ay can provide 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
is a placeholder for any user variable either passed via ruser or via global variables.
Symbolic mode may be selected by calling x10ac with mode set to nagad_symbolic prior
to calling c05ay. No further
changes are needed compared to using algorithmic mode.
3.1.2Symbolic Expert Mode
Symbolic expert mode may be selected by calling x10ac with mode set to
nagad_symbolic_expert prior to calling c05ay. In comparison to the symbolic
mode, in nagad_symbolic_expert mode the
user-supplied primal callback needs a specific
implementation to support symbolic computation, but this can improve
overall performance. See the example
c05ay_a1_sym_expert_dcoe.cpp for details.
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.
You can set or access the adjoints of output argument x. The adjoints of all other output arguments are ignored.
c05ay increments the adjoints of the variable , where is given by the argument ruser or by use of globals (see (3)).
The adjoints of all other input parameters are not referenced.
Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall
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
In addition to the arguments present in the interface of the primal routine,
c05ay 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.
c05ay preserves all error codes from c05ayf and in addition can return:
An unexpected AD error has been triggered by this routine. Please
See Section 4.8.2 in the NAG AD Library Introduction for further information.
The routine was called using a mode that has not yet been implemented.
On entry: ad_handle is nullptr.
This check is only made if the overloaded C++ interface is used with arguments not of type double.
A C++ exception was thrown.
The error message will show the details of the C++ exception text.
Dynamic memory allocation failed for AD.
See Section 4.8.1 in the NAG AD Library Introduction for further information.
8Parallelism and Performance
is not threaded in any implementation.
Please note that the algorithmic adjoint of Brent's method may be ill-conditioned. This means that derivatives of the zero returned in x, with respect to function parameters stored in ruser, may have limited accuracy when computed in algorithmic mode. This routine can be used in symbolic mode which will compute accurate derivatives.
The following examples are variants of the example for
modified to demonstrate calling the NAG AD Library.
Description of the primal example.
This example calculates an approximation to the zero of within the interval using a tolerance of .