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y_p = f.Forward(p, x_p )
y_p = f.Forward(p, x_p, s)
F : B^n \rightarrow B^m
to denote the
AD function
corresponding to
f
.
Given a function
X : B \rightarrow B^n
,
defined by its
Taylor coefficients
,
forward mode computes the Taylor coefficients for the function
\[
Y (t) = F [ X(t) ]
\]
.
F(x)
,
ForwardZero
is simpler to understand
than this explanation of the general case.
F^{(1)} (x) * dx
,
ForwardOne
is simpler to understand
than this explanation of the general case.
X : B \rightarrow B^n
is defined using
a sequence of Taylor coefficients
x^{(k)} \in B^n
:
\[
X(t) = x^{(0)} + x^{(1)} * t + \cdots + x^{(p)} * t^p
\]
For
k = 0, \ldots , p
,
the vector
x^{(k)}
above is defined as the value of
x_k
in the previous call (counting this call) of the form
f.Forward(k, x_k)
k = 0
,
x^{(0)}
is the value of the independent variables when the
corresponding
AD of
Base
operation sequence
was recorded.
Note that
x^{(k)}
is related to the k-th derivative of
X(t)
by
\[
x^{(k)} = \frac{1}{k !} X^{(k)} (0)
\]
Y : B \rightarrow B^m
is defined by
Y(t) = F[ X(t) ]
.
We use
y^{(k)} \in B^m
to denote the k-th order Taylor coefficient of
Y(t)
; i.e.,
\[
Y(t) = y^{(0)} + y^{(1)} * t + \cdots, + y^{(p)} * t^p + o( t^p )
\]
where
o( t^p ) * t^{-p} \rightarrow 0
as
t \rightarrow 0
.
Note that
y^{(k)}
is related to
the k-th derivative of
Y(t)
by
\[
y^{(k)} = \frac{1}{k !} Y^{(k)} (0)
\]
f
has prototype
ADFun<Base> f
f
is not const.
Before this call to Forward, the value returned by
f.size_taylor()
p
.
After this call it will be will be
p+1
(see size_taylor
).
p
has prototype
size_t p
x_p
has prototype
const Vector& x_p
n
, the dimension of the
domain
space for
f
.
The p-th order Taylor coefficient for
X(t)
is defined by this value; i.e.,
x^{(p)} = x\_p
.
(The lower order Taylor coefficients for
X(t)
are
defined by previous calls to Forward.)
s
is not present, std::cout
is used in its place.
Otherwise, this argument has prototype
std::ostream& s
p == 0
,
s
specifies where the output corresponding to PrintFor
will be written.
If
p != 0
,
s
is not used.
y_p
has prototype
Vector y_p
Y(t)
; i.e.,
y^{(p)} = y\_p
.
The size of
y_p
is equal to
m
, the dimension of the
range
space for
f
.
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.
p = 0
,
the result
y_p
is given by
\[
\begin{array}{rcl}
y^{(0)} & = & (F \circ X) ( 0 ) \\
& = & F[ x^{(0)} ]
\end{array}
\]
The agrees with the simplification where
p
,
x^{(0)}
, and
y^{(0)}
above are replaced by
0,
x
, and
y
in ForwardZero
.
p = 1
,
the result
y_p
is given by
\[
\begin{array}{rcl}
y^{(1)} & = & (F \circ X)^{(1)} ( 0 ) \\
& = & F^{(1)} [ X(0) ] * X^{(1)} (0) \\
& = & F^{(1)} ( x^{(0)} ) * x^{(1)}
\end{array}
\]
The agrees with the simplification where
p
,
x^{(0)}
,
x^{(1)}
, and
y^{(1)}
above are replaced by
1,
x
,
dx
, and
dy
in ForwardOne
.
x^{(1)}
is the j-th
elementary vector
\[
y^{(1)} = \D{F}{x_j} ( x^{(0)} )
\]
p = 2
,
the i-th element of
the result
y_p
is given by
\[
\begin{array}{rcl}
y_i^{(2)}
& = & \frac{1}{2} (F_i \circ X)^{(2)} ( 0 )
\\
& = & \frac{1}{2} \left[ F_i^{(1)} [ X(0) ] * X^{(2)} (0)
+ X^{(1)} (0)^T * F_i^{(2)} [ X(0) ] * X^{(1)} (0) \right]
\\
& = & \frac{1}{2} \left[
2 * F_i^{(1)} ( x^{(0)} ) * x^{(2)}
+
( x^{(1)} )^T * F_i^{(2)} ( x^{(0)} ) * x^{(1)}
\right ]
\end{array}
\]
Note that if
x^{(1)}
is the j-th
elementary vector
and
x^{(2)}
is zero,
\[
\begin{array}{rcl}
\DD{F_i}{x_j}{x_j} ( x^{(0)} ) = 2 y_i^{(2)}
\end{array}
\]
If
x^{(1)}
is the sum of the j-th and l-th
elementary vectors
and
x^{(2)}
is zero,
\[
\begin{array}{rcl}
y_i^{(2)}
& = & \frac{1}{2} \left[
\DD{F_i}{x_j}{x_j} ( x^{(0)} )
+
\DD{F_i}{x_j}{x_\ell} ( x^{(0)} )
+
\DD{F_i}{x_\ell}{x_j} ( x^{(0)} )
+
\DD{F_i}{x_\ell}{x_\ell} ( x^{(0)} )
\right]
\\
\DD{F_i}{x_\ell}{x_j} ( x^{(0)} )
& = &
y_i^{(2)}
-
\frac{1}{2} \DD{F_i}{x_j}{x_j} ( x^{(0)} )
-
\frac{1}{2} \DD{F_i}{x_\ell}{x_\ell} ( x^{(0)} )
\end{array}
\]