Algorithmic Differentiation in Python with Application Examples

Algorithms for Univariate Taylor Polynomials over Scalars (UTPS)
binary operations
unary operations
z = φ(x, y) d = 0, . . . , D OPS MOVES
x + cy z d = x d + cy d 2D 3D
x × y z d = P d
k=0 h
x ky d−k D 2 3D
x/y z d = 1 x y d − P i
d−1
0 k=0 z ky d−k D 2 3D
y = φ(x) d = 0, . . . , D OPS MOVES
h
ln(x) ỹ d = 1 ˜x x d − P i
d−1
0 k=1 x d−kỹ k D 2 2D
exp(x) ỹ d = P d
k=1 y d−k˜x k D 2 2D
√ h
x yd = 1 x 2y d − P i
d−1
0 k=1 y 1
ky d−k 2 D2 3D
h
x r ỹ d = 1 r P d
x 0 k=1 y d−k˜x k − P i
d−1
k=1 x d−kỹ k 2D 2 2D
sin(v) ˜s d = P d
j=1 ṽjc d−j 2D 2 3D
cos(v) ˜c d = P d
j=1 −ṽ js d−j
tan(v) ˜φd = P d
j=1 w d−jṽ j
˜w d = 2 P d
j=1 φ d−j “
˜φ j
arcsin(v) ˜φd = w −1
0 ṽ d − P d−1
j=1 w d−j ˜φ
”
j
˜w d = − P d
j=1 v d−j “
˜φ j
arctan(v) ˜φd = w −1
0 ṽ d − P d−1
j=1 w d−j ˜φ
”
j
˜w d = 2 P d
j=1 v d−jṽ j
Sebastian F. Walter, Humboldt-Universität zu Berlin Algorithmic () Differentiation in Python with Application Examples Wednesday, 10.07.2010 16 / 27