AD Tutorial at the TU Berlin

Algorithm: Forward UTPM of Symmetric Eigenvalue Decomposition
input : [A] D = [A 0 , . . . , A D− 1], where A d ∈ R N×N symmetric positive definite, d = 0, . . . , D − 1
output: [˜Λ] D = [˜Λ 0 , . . . , ˜Λ D− 1], where Λ 0 ∈ R N×N diagonal and Λ d ∈ R N×N block diagonal
d = 1, . . . , D − 1.
output: b ∈ N N b+1 , array of integers defining the blocks. The integer N B is the number of blocks. Each
block has the size of the multiplicity of an eigenvalue λ nb of Λ 0 s.t. for sl = b[n b ] : b[n b + 1] one
has (Q 0 [:, sl ]) T A 0 Q 0 [:, sl ] = λ nb I.
Λ 0 , Q 0 = eigh (A 0 )
E ij = (Λ 0 ) jj − (Λ 0 ) ii
H = P B ◦ (1/E)
for d = 1 to D − 1 do
S = − 1 2
end
P d−1
k=1 QT d−k Q k
K = ∆F + ˜Q T 0 A d ˜Q 0 + SΛ 0 + Λ 0 S
˜Q d = Q 0 (S + H ◦ K)
˜Λ d = ¯P B ◦ K
for the special case of distinct eigenvalues, this algorithm suffices
for repeated eigenvalues this algorithm is one step in a little more involved algorithm
Sebastian F. Walter, HU Berlin () Not So Short Tutorial OnAlgorithmic Differentiation Wednesday, 04.06.2010 30 / 39