c05au
is the AD Library version of the primal routine
c05auf.
Based (in the C++ interface) on overload resolution,
c05au can be used for primal, tangent and adjoint
evaluation. It supports tangents and adjoints of first and second order.
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.
3Description
c05au
is the AD Library version of the primal routine
c05auf.
c05auf locates a simple zero of a continuous function from a given starting value. It uses a binary search to locate an interval containing a zero of the function, then Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection, to locate the zero precisely.
For further information see Section 3 in the documentation for c05auf.
4References
Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall
5Arguments
In addition to the arguments present in the interface of the primal routine,
c05au 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.
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.
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.
Note that f is a subroutine in this interface, returning the function value via the additional output parameter retval.
c05au preserves all error codes from c05auf and in addition can return:
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.
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.
A C++ exception was thrown.
The error message will show the details of the C++ exception text.
Dynamic memory allocation failed for AD.
See Error Handling in the NAG AD Library Introduction for further information.
7Accuracy
Not applicable.
8Parallelism and Performance
c05au
is not threaded in any implementation.
9Further Comments
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 passed in the callable f, may have limited accuracy when computed in algorithmic mode. The routine c05ay (which requires an initial interval containing the zero) can be used in symbolic mode and will compute accurate derivatives.
10Example
The following examples are variants of the example for
c05auf,
modified to demonstrate calling the NAG AD Library.
Description of the primal example.
This example calculates an approximation to the zero of using a tolerance of starting from and using an initial search step .