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\[
B(u) * F^{(1)} (u) - A(u) * F (u) = D(u)
\]
In this sections we consider forward mode for the following choices:
|
F(u)
|
\sin(u)
|
\cos(u)
|
\sinh(u)
|
\cosh(u)
| |||||
A(u)
|
0
|
0
|
0
|
0
| ||||||
B(u)
|
1
|
1
|
1
|
1
| ||||||
D(u)
|
\cos(u)
|
- \sin(u)
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\cosh(u)
|
\sinh(u)
|
a
,
b
,
d
and
f
for the
Taylor coefficients of
A [ X (t) ]
,
B [ X (t) ]
,
D [ X (t) ]
,
and
F [ X(t) ]
respectively.
It now follows from the general
Taylor coefficients recursion formula
that for
j = 0 , 1, \ldots
,
\[
\begin{array}{rcl}
f^{(0)} & = & D ( x^{(0)} )
\\
e^{(j)}
& = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * f^{(k)}
\\
& = & d^{(j)}
\\
f^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
\left(
\sum_{k=1}^{j+1} k x^{(k)} e^{(j+1-k)}
- \sum_{k=1}^j k f^{(k)} b^{(j+1-k)}
\right)
\\
& = & \frac{1}{j+1}
\sum_{k=1}^{j+1} k x^{(k)} d^{(j+1-k)}
\end{array}
\]
The formula above generates the
order
j+1
coefficient of
F[ X(t) ]
from the
lower order coefficients for
X(t)
and
D[ X(t) ]
.