Algorithmic Differentiation in Python with Application Examples
Algorithmic Differentiation in Python with Application Examples
Algorithmic Differentiation in Python with Application Examples
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Univariate Taylor Polynomial Arithmetic (UTP)<br />
Basic Observation 1: Let f : R N → R, then<br />
d<br />
dt f (x + e it)<br />
∣ = (∇ x f (x)) T · e i = ∂f<br />
t=0<br />
∂x i<br />
Basic Observation 2: Hessian<br />
d 2<br />
f (x + e i t 1 + e j t 2 ) ∣<br />
dt 1 dt 2<br />
∣<br />
t1 =t 2 =0<br />
= e T i ∇ 2 xf (x)e j = ∂2 f<br />
∂x i ∂x j<br />
e i = (0, . . . , 1, . . . , 0) is the i’th cartesian basis vector.<br />
Sebastian F. Walter, Humboldt-Universität zu Berl<strong>in</strong> <strong>Algorithmic</strong> () <strong>Differentiation</strong> <strong>in</strong> <strong>Python</strong> <strong>with</strong> <strong>Application</strong> <strong>Examples</strong> Wednesday, 10.07.2010 13 / 27
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