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t
(see Section 10.2 of
Evaluating Derivatives
).
Here and below
,
, and
Z(t)
are scalar valued functions
and the corresponding p
-th order Taylor coefficients row vectors are
,
and
; i.e.,
For the purposes of this section, we are given
and
and need to determine
.
p
-th order Taylor coefficient row vectors for
,
and
respectively.
We assume that these coefficients are known functions of
,
the p
-th order Taylor coefficients for
.
p
-th order Taylor coefficient row vector for
,
in terms of these other known coefficients.
It follows from the formulas above that
where we define
We can compute the value of
using the formula
Suppose by induction (on
) that we are given the
Taylor coefficients of
up to order
; i.e.,
for
and the coefficients
for
.
We can compute
using the formula
We need to complete the induction by finding formulas for
.
It follows for the formula for the
multiplication
operator that
This completes the induction that computes
and
.
exp_forward | Exponential Function Forward Mode Theory |
log_forward | Logarithm Function Forward Mode Theory |
sqrt_forward | Square Root Function Forward Mode Theory |
sin_cos_forward | Trigonometric and Hyperbolic Sine and Cosine Forward Theory |
atan_forward | Inverse Tangent and Hyperbolic Tangent Forward Mode Theory |
asin_forward | Inverse Sine and Hyperbolic Sine Forward Mode Theory |
acos_forward | Inverse Cosine and Hyperbolic Cosine Forward Mode Theory |
tan_forward | Tangent and Hyperbolic Tangent Forward Taylor Polynomial Theory |