AD Tutorial at the TU Berlin
AD Tutorial at the TU Berlin
AD Tutorial at the TU Berlin
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Univari<strong>at</strong>e Taylor Polynomial Arithmetic (UTP)<br />
Not<strong>at</strong>ion for UTPs:<br />
D−1<br />
∑<br />
[x] D := [x 0 , x 1 , . . . , x D−1 ] = x d T d ∈ R[T]/(T D ) ,<br />
d=0<br />
T is an indetermin<strong>at</strong>e, i.e. a formal parameter<br />
x d ∈ R is called Taylor coefficient<br />
a UTP is defined by <strong>the</strong> D coefficients x 0 , . . . , x D−1<br />
Definition of Functions on UTPs:<br />
E D (f ) : R[T]/(T D ) → R[T]/(T D )<br />
[x] D ↦→ [y] D := ∑ 1 d d D−1<br />
d! dt d f ( ∑<br />
x d t d )<br />
T d ,<br />
∣<br />
d=0<br />
k=0<br />
} {{ t=0<br />
}<br />
≡y d<br />
where f : R → R, y = f (x)<br />
Required: algorithms to compute y d for <strong>the</strong> elem. funcs. +, −, ∗, /, . . .<br />
Sebastian F. Walter, HU <strong>Berlin</strong> () Not So Short <strong>Tutorial</strong> OnAlgorithmic Differenti<strong>at</strong>ion Wednesday, 04.06.2010 7 / 39