Algorithmic Differentiation in Python with Application Examples

Why not Symbolic Differentiation
A Raytracing Example
1.0
0.5
0.0 8
6
−0.5
1
3
5
7
9
Cylindrical mirror described by 0 = g(x) = x 2 1 + x2 2 − 1,
laser beam enters at x = (0, −1) and direction v.
Recursive algorithm for next reflection point x + and
direction v + :
„ x +
v + «
0
B
= F(x, v) = @ x +
r !
“ ” x T 2
v ‖x‖
‖v‖ − 2 −1
2 ‖v‖ 2 − xT v
‖v‖ 2
P(x + )v
4
2
0
−1.0
−1.0 −0.5 0.0 0.5 1.0
where P(x + ) = I − 2 wwT
‖w‖ 2 , w = w(x + ) = ∇ xg(x + )
Goal: compute sensitivity of the 10’th reflection point x (10) w.r.t. initial
direction v (0) , i.e. dx(10)
dv (0) .
Sebastian F. Walter, Humboldt-Universität zu Berlin Algorithmic () Differentiation in Python with Application ExamplesWednesday, 10.07.2010 5 / 27