Computation and Properties of Maps
133
5.2.3 Functions on Differential Algebras
In this subsection we will generalize standard functions like exponentials,
logarithmic and trigonometric functions to Differential Algebras. As we will
see below, virtually all functions existing on a computer can be generalized in
a straightforward way.
We start our discussion by noting that for any Differential Algebra (DA)
vector of the form (0, a 1 , . . . , a N ) ∈ n D v , i.e., with a zero in the component
belonging to the zeroth order monomial, we have the following property:
(0, a 1 , . . . , a N )
i = (0, 0, . . . , 0) for i > n,
(5.20)
which follows directly from the definition of the multiplication in n D v defined
in eq. (5.17).
Let us begin our discussion of special functions with the exponential function exp(x). Assume we have to compute the exponential of a DA vector
that has already been created by previous operations. First we note that the
functional equation
exp(x + y) = exp(x) · exp(y)
also holds for DA vectors. As we will see, this facilitates the computation of
the exponential considerably.
exp[(a 0 , a 1 , a 2 , . . . , a N )] = exp(a 0 ) · exp[(0, a 1 , a 2 , . . . , a N )]
= exp(a 0 ) ·
∞
i=0
(0, a 1 , a 2 , . . . , a N )
i
i!
= exp(a 0 ) ·
n
i=0
(0, a 1 , a 2 , . . . , a N )
i
i!
.
In the last step use has been made of eq. (5.20) which entails that the
sum has to be taken only through order n and thus allows the computation
of the root in finitely many steps. Hence the evaluation of the real number
exponential exp(a 0 ), which internally on a computer requires a power series
summation and hence cannot be done accurately, is more subtle than the rest
of the operations in Differential Algebra.
A logarithm of a DA vector exists if and only if a 0 > 0. In this case one
obtains
log [(a 0 , a 1 , a 2 , . . . , a N )] = log
a 0 ·
1 +
0,
a 1
a 0
,
a 2
a 0
, · · · ,
a N
a 0
= (log(a 0 ), 0, . . . , 0) +
∞
i=1
(−1)
i+1 1
i
0,
a 1
a 0
,
a 2
a 0
, · · · ,
a N
a 0
i
= (log(a 0 ), 0, . . . , 0) +
n
i=1
(−1)
i+1 1
i
0,
a 1
a 0
,
a 2
a 0
, · · · ,
a N
a 0
i
.
Again use has been made of the fundamental property of the logarithm
log(x · y) = log(x) + log(y)
133
5.2.3 Functions on Differential Algebras
In this subsection we will generalize standard functions like exponentials,
logarithmic and trigonometric functions to Differential Algebras. As we will
see below, virtually all functions existing on a computer can be generalized in
a straightforward way.
We start our discussion by noting that for any Differential Algebra (DA)
vector of the form (0, a 1 , . . . , a N ) ∈ n D v , i.e., with a zero in the component
belonging to the zeroth order monomial, we have the following property:
(0, a 1 , . . . , a N )
i = (0, 0, . . . , 0) for i > n,
(5.20)
which follows directly from the definition of the multiplication in n D v defined
in eq. (5.17).
Let us begin our discussion of special functions with the exponential function exp(x). Assume we have to compute the exponential of a DA vector
that has already been created by previous operations. First we note that the
functional equation
exp(x + y) = exp(x) · exp(y)
also holds for DA vectors. As we will see, this facilitates the computation of
the exponential considerably.
exp[(a 0 , a 1 , a 2 , . . . , a N )] = exp(a 0 ) · exp[(0, a 1 , a 2 , . . . , a N )]
= exp(a 0 ) ·
∞
i=0
(0, a 1 , a 2 , . . . , a N )
i
i!
= exp(a 0 ) ·
n
i=0
(0, a 1 , a 2 , . . . , a N )
i
i!
.
In the last step use has been made of eq. (5.20) which entails that the
sum has to be taken only through order n and thus allows the computation
of the root in finitely many steps. Hence the evaluation of the real number
exponential exp(a 0 ), which internally on a computer requires a power series
summation and hence cannot be done accurately, is more subtle than the rest
of the operations in Differential Algebra.
A logarithm of a DA vector exists if and only if a 0 > 0. In this case one
obtains
log [(a 0 , a 1 , a 2 , . . . , a N )] = log
a 0 ·
1 +
0,
a 1
a 0
,
a 2
a 0
, · · · ,
a N
a 0
= (log(a 0 ), 0, . . . , 0) +
∞
i=1
(−1)
i+1 1
i
0,
a 1
a 0
,
a 2
a 0
, · · · ,
a N
a 0
i
= (log(a 0 ), 0, . . . , 0) +
n
i=1
(−1)
i+1 1
i
0,
a 1
a 0
,
a 2
a 0
, · · · ,
a N
a 0
i
.
Again use has been made of the fundamental property of the logarithm
log(x · y) = log(x) + log(y)