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@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@

*dy* = *f*.ForOne(*x*, *j*)

We use @(@ F : B^n \rightarrow B^m @)@ to denote the AD function corresponding to

*f*

.
The syntax above sets
*dy*

to the
partial of @(@
F
@)@ with respect to @(@
x_j
@)@; i.e.,
@[@
dy
= \D{F}{ x_j } (x)
= \left[
\D{ F_0 }{ x_j } (x) , \cdots , \D{ F_{m-1} }{ x_j } (x)
\right]
@]@
The object

*f*

has prototype

ADFun<*Base*> *f*

Note that the ADFun
object
*f*

is not `const`

(see ForOne Uses Forward
below).
The argument

*x*

has prototype

const *Vector* &*x*

(see Vector
below)
and its size
must be equal to
*n*

, the dimension of the
domain
space for
*f*

.
It specifies
that point at which to evaluate the partial derivative.
The argument

*j*

has prototype

size_t *j*

an is less than
*n*

,
domain
space for
*f*

.
It specifies the component of
*F*

for which we are computing the partial derivative.
The result

*dy*

has prototype

*Vector* *dy*

(see Vector
below)
and its size is @(@
m
@)@, the dimension of the
range
space for
*f*

.
The value of
*dy*

is the partial of @(@
F
@)@ with respect to
@(@
x_j
@)@ evaluated at
*x*

; i.e.,
for @(@
i = 0 , \ldots , m - 1
@)@
@[@
.
dy[i] = \D{ F_i }{ x_j } ( x )
@]@
The type

*Vector*

must be a SimpleVector
class with
elements of type
*Base*

.
The routine CheckSimpleVector
will generate an error message
if this is not the case.
After each call to Forward , the object

*f*

contains the corresponding
Taylor coefficients
.
After a call to `ForOne`

,
the zero order Taylor coefficients correspond to
*f*.Forward(0,*x*)

and the other coefficients are unspecified.
The routine ForOne is both an example and test. It returns

`true`

, if it succeeds and `false`

otherwise.
Input File: cppad/core/for_one.hpp