lines 681-784 of file: lib/cplusplus/fun.cpp

{xrst_begin cpp_fun_forward}
{xrst_spell
   xp
   yp
}


Forward Mode AD
###############

Syntax
******
*yp* = *f*\ ``.forward`` ( *p* , *xp* )

Taylor Coefficient
******************
For a function :math:`g(t)` of a scalar argument :math:`t \in \B{R}`,
the *p*-th order Taylor coefficient is its
``p`` -th order derivative divided by *p* factorial
and evaluated at :math:`t = 0`; i.e.,

.. math::

   g^{(p)} (0) /  p !

f
*
This is either a
:ref:`d_fun<cpp_fun_ctor@Syntax@d_fun>` or
:ref:`a_fun<cpp_fun_ctor@Syntax@a_fun>` function object.
Note that its state is changed by this operation because
all the Taylor coefficient that it calculates for every
variable in recording are stored.
See more discussion of this fact under the heading
:ref:`p<cpp_fun_forward@p>` below.

f(x)
****
We use the notation :math:`f: \B{R}^n \rightarrow \B{R}^m`
for the function corresponding to *f* .
Note that *n* is the size of :ref:`ax<cpp_fun_ctor@ax>`
and *m* is the size of :ref:`ay<cpp_fun_ctor@ay>`
in to the constructor for *f* .

X(t)
****
We use the notation :math:`X : \B{R} \rightarrow \B{R}^n`
for a function that the calling routine chooses.

Y(t)
****
We define the function :math:`Y : \B{R} \rightarrow \B{R}^n`
by :math:`Y(t) = f(X(t))`.

p
*
This argument has prototype

| |tab| ``int`` *p*

and is non-negative.
It is the order of the Taylor coefficient being calculated.
If there was no call to ``forward`` for this *f* ,
the value of *p* must be zero.
Otherwise, it must be between zero and one greater that its
value for the previous call using this *f* .
After this call, the Taylor coefficients for orders zero though *p* ,
and for every variable in the recording, will be stored in *f* .

size_order
==========
After this call,
:ref:`f_size_order()<cpp_fun_property@size_order>` is *p* +1 .

xp
**
If *f* is a ``d_fun`` or ``a_fun`` ,
this argument has prototype

| |tab| ``const vec_double&`` *xp*
| |tab| ``const vec_a_double&`` *xp*

respectively and its size must be *n* .
It specifies the *p*-th order Taylor coefficients for *X(t* ) .

yp
**
If *f* is a ``d_fun`` or ``a_fun`` ,
the result has prototype

| |tab| ``vec_double&`` *yp*
| |tab| ``vec_a_double&`` *yp*

respectively and its size is *m* .
It is the *p*-th order Taylor coefficients for :math:`Y(t)`.

{xrst_toc_hidden
   example/cplusplus/fun_forward_xam.cpp
}
Example
*******
:ref:`fun_forward_xam.cpp-name`

{xrst_end cpp_fun_forward}