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x(t)
and
\varepsilon(t) ]
near
t = 0
by the first order expansions
\[
\begin{array}{rcl}
x(t) & = & x^{(0)} + x^{(1)} * t
\\
\varepsilon(t) & = & \varepsilon^{(0)} + \varepsilon^{(1)} * t
\end{array}
\]
It follows that
x^{(0)}
(
\varepsilon^{(0)}
) is the zero,
and
x^{(1)}
(
\varepsilon^{(1)}
) the first,
order derivative of
x(t)
at
t = 0
(
\varepsilon (t)
)
at
t = 0
.
exp_eps(
x,
epsilon)
with x is equal to .5
and epsilon is equal to .2.
For this case, the mathematical function for the operation sequence
corresponding to exp_eps
is
\[
f ( x , \varepsilon ) = 1 + x + x^2 / 2
\]
The corresponding partial derivative with respect to
x
,
and the value of the derivative, are
\[
\partial_x f ( x , \varepsilon ) = 1 + x = 1.5
\]
t
,
at
t = 0
, for each variable in the sequence.
\[
v_j (t) = v_j^{(0)} + v_j^{(1)} t
\]
We use
x^{(1)} = 1
and
\varepsilon^{(1)} = 0
,
so that differentiation with respect to
t
,
at
t = 0
,
is the same partial differentiation with respect to
x
at
x = x^{(0)}
.
Index |
| Operation |
| Zero Order |
| Derivative |
| First Order |
1 |
|
v_1 = x
| 0.5 |
v_1^{(1)} = x^{(1)}
|
v_1^{(1)} = 1
| |||
2 |
|
v_2 = 1 * v_1
| 0.5 |
v_2^{(1)} = 1 * v_1^{(1)}
|
v_2^{(1)} = 1
| |||
3 |
|
v_3 = v_2 / 1
| 0.5 |
v_3^{(1)} = v_2^{(1)} / 1
|
v_3^{(1)} = 1
| |||
4 |
|
v_4 = 1 + v_3
| 1.5 |
v_4^{(1)} = v_3^{(1)}
|
v_4^{(1)} = 1
| |||
5 |
|
v_5 = v_3 * v_1
| 0.25 |
v_5^{(1)} = v_3^{(1)} * v_1^{(0)} + v_3^{(0)} * v_1^{(1)}
|
v_5^{(1)} = 1
| |||
6 |
|
v_6 = v_5 / 2
| 0.125 |
v_6^{(1)} = v_5^{(1)} / 2
|
v_6^{(1)} = 0.5
| |||
7 |
|
v_7 = v_4 + v_6
| 1.625 |
v_7^{(1)} = v_4^{(1)} + v_6^{(1)}
|
v_7^{(1)} = 1.5
|
\[
\begin{array}{rcl}
1.5
& = &
v_7^{(1)} =
\left[ \D{v_7}{t} \right]_{t=0} =
\left[ \D{}{t} f( x^{(0)} + x^{(1)} * t , \varepsilon^{(0)} ) \right]_{t=0}
\\
& = &
\partial_x f ( x^{(0)} , \varepsilon^{(0)} ) * x^{(1)} =
\partial_x f ( x^{(0)} , \varepsilon^{(0)} )
\end{array}
\]
(We have used the fact that
x^{(1)} = 1
and
\varepsilon^{(1)} = 0
.)
x = .1
,
what are the results of a zero and first order forward mode sweep for
the operation sequence above;
i.e., what are the corresponding values for
v_1^{(0)}, v_2^{(0)}, \cdots , v_7^{(0)}
and
v_1^{(1)}, v_2^{(1)}, \cdots , v_7^{(1)}
?
x = .1
and
\epsilon = .2
,
what is the operation sequence corresponding to
exp_eps(
x,
epsilon)