AD Tutorial at the TU Berlin

Higher-Order Mixed Partial Derivatives by UTP Arithmetic
Always possible to compute higher-order mixed partial derivatives by
evaluation/interpolation 2
Example: Hessian H = [[f xx , f xy ], [f yx , f yy ]]
〈s 1 |H|s 2 〉 = 1 2 [〈s 1|H|s 2 〉 + 〈s 2 |H|s 1 〉]
= 1 2 [〈s 1 + s 2 − s 2 |H|s 2 〉 + 〈s 2 + s 1 − s 1 |H|s 1 〉]
= 1 2 [〈s 1 + s 2 |H|s 1 + s 2 〉 − 〈s 1 |H|s 1 〉 − 〈s 2 |H|s 2 〉] .
s T 1 ∇2 f (x)s 2 = ∂2 f (x + t 1 s 1 + t 2 s 2 )
∂t 1 ∂t 2
˛˛˛˛t1 =t 2 =0
= 1 » ∂ 2 –
f (x + t(s 1 + s 2 ))
2 ∂t
˛˛˛˛t=0 2 − ∂2 f (x + s 1 t)
∂t 2 − ∂2 f (x + s 2 t)
∂t 2 .
to get H ij use s 1 = e i and s 2 = e j cartesian basis vectors
2 Griewank et al., Evaluating Higher Derivative Tensors by Forward Propagation of
Univariate Taylor Series,Mathematics of Computation, 2000
Sebastian F. Walter, HU Berlin () Not So Short Tutorial OnAlgorithmic Differentiation Wednesday, 04.06.2010 11 / 39