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X(t)
,
Y(t)
, and Z(t) are scalar valued functions
and the corresponding p-th order Taylor coefficients row vectors are
x
,
y
and
z
; i.e.,
\[
\begin{array}{lcr}
X(t) & = & x^{(0)} + x^{(1)} * t + \cdots + x^{(p)} * t^p + o( t^p ) \\
Y(t) & = & y^{(0)} + y^{(1)} * t + \cdots + y^{(p)} * t^p + o( t^p ) \\
Z(t) & = & z^{(0)} + z^{(1)} * t + \cdots + z^{(p)} * t^p + o( t^p )
\end{array}
\]
For the purposes of this section, we are given
x
and
y
and need to determine
z
.
\[
\begin{array}{rcl}
Z(t)
& = & X(t) + Y(t)
\\
\sum_{j=0}^p z^{(j)} * t^j
& = & \sum_{j=0}^p x^{(j)} * t^j + \sum_{j=0}^p y^{(j)} * t^j + o( t^p )
\\
z^{(j)} & = & x^{(j)} + y^{(j)}
\end{array}
\]
\[
\begin{array}{rcl}
Z(t)
& = & X(t) - Y(t)
\\
\sum_{j=0}^p z^{(j)} * t^j
& = & \sum_{j=0}^p x^{(j)} * t^j - \sum_{j=0}^p y^{(j)} * t^j + o( t^p )
\\
z^{(j)} & = & x^{(j)} - y^{(j)}
\end{array}
\]
\[
\begin{array}{rcl}
Z(t)
& = & X(t) * Y(t)
\\
\sum_{j=0}^p z^{(j)} * t^j
& = & \left( \sum_{j=0}^p x^{(j)} * t^j \right)
*
\left( \sum_{j=0}^p y^{(j)} * t^j \right) + o( t^p )
\\
z^{(j)} & = & \sum_{k=0}^j x^{(j-k)} * y^{(k)}
\end{array}
\]
\[
\begin{array}{rcl}
Z(t)
& = & X(t) / Y(t)
\\
x
& = & z * y
\\
\sum_{j=0}^p x^{(j)} * t^j
& = &
\left( \sum_{j=0}^p z^{(j)} * t^j \right)
*
\left( \sum_{j=0}^p y^{(j)} * t^j \right)
+
o( t^p )
\\
x^{(j)} & = & \sum_{k=0}^j z^{(j-k)} y^{(k)}
\\
z^{(j)} & = & \frac{1}{y^{(0)}} \left( x^{(j)} - \sum_{k=1}^j z^{(j-k)} y^{(k)} \right)
\end{array}
\]
F
is a standard math function and
\[
Z(t) = F[ X(t) ]
\]
\[
B(u) * F^{(1)} (u) - A(u) * F (u) = D(u)
\]
We use
a
,
b
and
d
to denote the
p-th order Taylor coefficient row vectors for
A [ X (t) ]
,
B [ X (t) ]
and
D [ X (t) ]
respectively.
We assume that these coefficients are known functions of
x
,
the p-th order Taylor coefficients for
X(t)
.
z
,
the p-th order Taylor coefficient row vector for
Z(t)
,
in terms of these other known coefficients.
It follows from the formulas above that
\[
\begin{array}{rcl}
Z^{(1)} (t)
& = & F^{(1)} [ X(t) ] * X^{(1)} (t)
\\
B[ X(t) ] * Z^{(1)} (t)
& = & \{ D[ X(t) ] + A[ X(t) ] * Z(t) \} * X^{(1)} (t)
\\
B[ X(t) ] * Z^{(1)} (t) & = & E(t) * X^{(1)} (t)
\end{array}
\]
where we define
\[
E(t) = D[X(t)] + A[X(t)] * Z(t)
\]
We can compute the value of
z^{(0)}
using the formula
\[
z^{(0)} = F ( x^{(0)} )
\]
Suppose by induction (on
j
) that we are given the
Taylor coefficients of
E(t)
up to order
j-1
; i.e.,
e^{(k)}
for
k = 0 , \ldots , j-1
and the coefficients
z^{(k)}
for
k = 0 , \ldots , j
.
We can compute
e^{(j)}
using the formula
\[
e^{(j)} = d^{(j)} + \sum_{k=0}^j a^{(j-k)} * z^{(k)}
\]
We need to complete the induction by finding formulas for
z^{(j+1)}
.
It follows for the formula for the
multiplication
operator that
\[
\begin{array}{rcl}
\left( \sum_{k=0}^j b^{(k)} t^k \right)
*
\left( \sum_{k=1}^{j+1} k z^{(k)} * t^{k-1} \right)
& = &
\left( \sum_{k=0}^j e^{(k)} * t^k \right)
*
\left( \sum_{k=1}^{j+1} k x^{(k)} * t^{k-1} \right)
+
o( t^p )
\\
z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
\left(
\sum_{k=0}^j e^{(k)} (j+1-k) x^{(j+1-k)}
- \sum_{k=1}^j b^{(k)} (j+1-k) z^{(j+1-k)}
\right)
\\
z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
\left(
\sum_{k=1}^{j+1} k x^{(k)} e^{(j+1-k)}
- \sum_{k=1}^j k z^{(k)} b^{(j+1-k)}
\right)
\end{array}
\]
This completes the induction that computes
e^{(j)}
and
z^{(j+1)}
.