Algorithmic Differentiation in Python with Application Examples

Why not Symbolic Differentiation (cont)
Compute recursively x (k) , v (k) = F(x (k−1) , v (k−1) ) as symbolic
expression
Use sum,product and chainrule to compute the wanted derivative
Problem: Expression swell (show live example)
import p y l a b ; import numpy ; from numpy import s q r t , dot , cos , s i n , pi , l i n
import sympy ; from sympy import s q r t
def F ( x , v ) :
””” computes next r e f l e c t i o n p o i n t x and d i r e c t i o n v ”””
c = d o t ( v , v )
x2 = [ x [ 0 ] + v [ 0 ] ∗ ( s q r t ( ( d o t ( x , v ) / c )∗∗2 −( d o t ( x , x ) − 1 . ) / c)− d o t ( x , v ) / c ) ,
x [ 1 ] + v [ 1 ] ∗ ( s q r t ( ( d o t ( x , v ) / c )∗∗2 −( d o t ( x , x ) − 1 . ) / c)− d o t ( x , v ) / c ) ]
w = x2
v2 = [ ( v [ 0 ] − 2∗ w[ 0 ] ∗ d o t (w, v ) / d o t (w,w) ) ,
( v [ 1 ] − 2∗ w[ 1 ] ∗ d o t (w, v ) / d o t (w,w ) ) ]
return x2 , v2
x1 , x2 , v1 , v2 = sympy . symbols ( ’ x1 ’ , ’ x2 ’ , ’ v1 ’ , ’ v2 ’ )
x = [ x1 , x2 ] ; v = [ v1 , v2 ]
x , v = F ( x , v )
p r i n t ’x , v=\n ’ , x , v
#x , v = F ( x , v )
Sebastian # p rF. i Walter, n t ’x Humboldt-Universität , v=\n ’ , x , v zu Berlin Algorithmic () Differentiation in Python with Application ExamplesWednesday, 10.07.2010 6 / 27