Algorithmic Differentiation in Python with Application Examples

Univariate Taylor Polynomial Arithmetic (UTP) (cont.)
Problem can be formulated as arithmetic on univariate Taylor
polynomials (UTP)
D−1
∑
[x] D = [x 0 , . . . , x D−1 ] = x d T d ∈ R(T)/(T D ) ,
d=0
T is an indeterminate, i.e. a formal parameter
x d ∈ R is called Taylor coefficient
Define extension of Functions f : R → R, y = f (x):
E D (f ) : R[T]/(T D ) → R[T]/(T D )
[x] D ↦→ [y] D := ∑ 1 d d D−1
d! dt d f ( ∑
x d t d )
T d ,
∣
d=0
k=0
} {{ t=0
}
≡y d
Sebastian F. Walter, Humboldt-Universität zu Berlin Algorithmic () Differentiation in Python with Application Examples Wednesday, 10.07.2010 14 / 27