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# include "exp_eps.hpp"
y = exp_eps(x, epsilon)
\[
\exp (x) = 1 + x^1 / 1 ! + x^2 / 2 ! + \cdots
\]
We define
k ( x, \varepsilon )
as the smallest
non-negative integer such that
\varepsilon \geq x^k / k !
; i.e.,
\[
k( x, \varepsilon ) =
\min \{ k \in {\rm Z}_+ \; | \; \varepsilon \geq x^k / k ! \}
\]
The mathematical form for our approximation of the exponential function is
\[
\begin{array}{rcl}
{\rm exp\_eps} (x , \varepsilon ) & = & \left\{
\begin{array}{ll}
\frac{1}{ {\rm exp\_eps} (-x , \varepsilon ) }
& {\rm if} \; x < 0
\\
1 + x^1 / 1 ! + \cdots + x^{k( x, \varepsilon)} / k( x, \varepsilon ) !
& {\rm otherwise}
\end{array}
\right.
\end{array}
\]
cppad-yyyymmdd/introduction/exp_apx
cppad-yyyymmdd
is the distribution directory
created during the beginning steps of the
installation
of CppAD.
x
has prototype
const Type &x
Type
below).
It specifies the point at which to evaluate the
approximation for the exponential function.
epsilon
has prototype
const Type &epsilon
\varepsilon
in the
exponential function approximation defined above.
y
has prototype
Type y
u
and v are Type objects and i
is an int:
| Operation | Result Type | Description |
Type(i)
|
Type
|
object with value equal to
i
|
Type u = v
|
Type
|
construct
u
with value equal to
v
|
u > v
|
bool
|
true,
if
u
greater than v, an false otherwise
|
u = v
|
Type
|
new
u
(and result) is value of v
|
u * v
|
Type
|
result is value of
u * v
|
u / v
|
Type
|
result is value of
u / v
|
u + v
|
Type
|
result is value of
u + v
|
-u
|
Type
|
result is value of
- u
|
k( x, \varepsilon )
above,
what is the value of
k(.5, 1)
,
k(.5, .1)
, and
k(.5, .01)
?
exp_eps:
double x = 1.;
double epsilon = .01;
double y = exp_eps(x, epsilon);
k, temp, term, and sum
the first time through the while loop in exp_eps.hpp
?
k, temp, term, and sum
the second time through the while loop in exp_eps.hpp
?