AD Tutorial at the TU Berlin

More documents

Recommendations

Info

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
Test Example for the Symmetric Eigenvalue Decomposition 4 Orthonormal Matrix: 0 1 cos(x(t)) 1 sin(x(t)) −1 1 Q(t) = √ B− sin(x(t)) −1 cos(x(t)) −1 C 3 @ 1 − sin(x(t)) 1 cos(x(t)) A −1 cos(x(t)) 1 sin(x(t)) Λ(t) = diag(x 2 − x + 1 2 , 4x2 − 3x, δ(− 1 2 x3 + 2x 2 − 3 2 x + 1) + (x3 + x 2 − 1), 3x − 1) , where x ≡ x(t) := 1 + t. constant δ = 0 means repeated eigenvalues, δ > 0 distinct but close In Taylor arithmetic one obtains Λ 0 = diag(1/2, 1, 1 + δ, 2) Λ 1 = diag(1, 5, 5 + δ, 3) Λ 2 = diag(2, 8, 8 + δ, 0) Λ 3 = diag(0, 0, 6 − 3δ, 0) Λ d = diag(0, 0, 0, 0), ∀d ≥ 4 . Define A(t) = Q(t)Λ(t)Q(t) and try to reconstruct Λ(t) and Q(t). 4 Example adapted from Andrew and Tan, Computation of Derivatives of Repeated Eigenvalues and the Corresponding Eigenvectors of Symmetric Matrix Pencils, SIAM Journal on Matrix Analysis and Applications Sebastian F. Walter, HU Berlin () Not So Short Tutorial OnAlgorithmic Differentiation Wednesday, 04.06.2010 31 / 39
Page 1 and 2: Not So Short Tutorial On Algorithmi
Page 3 and 4: Computational Model computer progra
Page 5 and 6: PART I: The Forward Mode of AD: Tay
Page 7 and 8: Univariate Taylor Polynomial Arithm
Page 9 and 10: Algorithms for Univariate Taylor Po
Page 11 and 12: Higher-Order Mixed Partial Derivati
Page 13 and 14: PART II: The Reverse Mode of AD Seb
Page 15 and 16: Live Example: Semi-Automatic Revers
Page 17 and 18: PART III: Advanced AD and Applicati
Page 19 and 20: Optimum Experimental Design in Chem
Page 21 and 22: Differentiation of ODE Solvers Expl
Page 23 and 24: Application Example: DAESOL-II for
Page 25 and 26: Univariate Taylor Propagation on Ma
Page 27 and 28: Algorithm: Reverse UTPM of the Rect
Page 29: ALGOPY Live Example: Moore-Penrose
Page 33 and 34: Test Example for the Symmetric Eige
Page 35 and 36: Test Example for the Symmetric Eige
Page 37 and 38: Sebastian F. Walter, HU Berlin () N
Page 39: Summary: Have shown many aspects of

AD Tutorial at the TU Berlin

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?