Algorithmic Differentiation in Python with Application Examples

Algorithm: Forward UTPM of the Rectangular QR Decomposition
input : [A] D = [A 0 , . . . , A D− 1], where A d ∈ R M×N , d = 0, . . . , D − 1, M ≥ N.
output: [Q] D = [Q 0 , . . . , Q D− 1] matrix with orthonormal column vectors, where Q d ∈ R M×N ,
d = 0, . . . , D − 1
output: [R] D = [R 0 , . . . , R D− 1] upper triangular, where R d ∈ R N×N , d = 0, . . . , D − 1
Q 0 , R 0 = qr (A 0 )
for d = 1 to D − 1 do
∆F = A d − P d−1
k=1 Q d−kR k
S = − 1 P d−1
2 k=1 QT d−k Q k
P L ◦ X = P L ◦ (Q T 0 ∆FR−1 0 − S)
X = P L ◦ X − (P L ◦ X) T
R d = Q T 0 ∆F − (S + X)R 0
Q d = (∆F − Q 0 R d )R −1
0
end
Sebastian F. Walter, Humboldt-Universität zu Berlin Algorithmic () Differentiation in Python with Application Examples Wednesday, 10.07.2010 27 / 27