Eigenvalues of Symmetrix Hierarchical Matrices

20 February 2012 Eigenvalue Algorithms for Symmetric Hierarchical Matrices Thomas Mach Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURGMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Eigenvalue Problem Deﬁnition The pair (λ, v ) ∈ R × Rn is called an eigenpair of the symmetric matrix M = M T ∈ Rn×n , if Mv = v λ. The set Λ(M) = {λ|∃v : (λ, v ) eigenpair of M} is the spectrum of M. Similarity Transformation Λ(M) = Λ(P −1 MP) ∀P invertibleMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Classiﬁcation of Eigenvalue Problems [Golub, Van der Vorst ’00] Is M real or complex?Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Classiﬁcation of Eigenvalue Problems [Golub, Van der Vorst ’00] Is M real or complex? M ∈ Rn×nMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Classiﬁcation of Eigenvalue Problems [Golub, Van der Vorst ’00] Is M real or complex? M ∈ Rn×n Special properties (symmetric, Hermitian, skew-symmetric or unitary)?Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Classiﬁcation of Eigenvalue Problems [Golub, Van der Vorst ’00] Is M real or complex? M ∈ Rn×n Special properties (symmetric, Hermitian, skew-symmetric or unitary)? symmetric: M = M TMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Classiﬁcation of Eigenvalue Problems [Golub, Van der Vorst ’00] Is M real or complex? M ∈ Rn×n Special properties (symmetric, Hermitian, skew-symmetric or unitary)? symmetric: M = M T Further structure?Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Classiﬁcation of Eigenvalue Problems [Golub, Van der Vorst ’00] Is M real or complex? M ∈ Rn×n Special properties (symmetric, Hermitian, skew-symmetric or unitary)? symmetric: M = M T Further structure? Yeah.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Classiﬁcation of Eigenvalue Problems [Golub, Van der Vorst ’00] Is M real or complex? M ∈ Rn×n Special properties (symmetric, Hermitian, skew-symmetric or unitary)? symmetric: M = M T Further structure? Yeah. M ∈ H(TI × I , k) ⇒ see next slideMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Classiﬁcation of Eigenvalue Problems [Golub, Van der Vorst ’00] Is M real or complex? M ∈ Rn×n Special properties (symmetric, Hermitian, skew-symmetric or unitary)? symmetric: M = M T Further structure? Yeah. M ∈ H(TI × I , k) ⇒ see next slide Which eigenvalues required?Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Classiﬁcation of Eigenvalue Problems [Golub, Van der Vorst ’00] Is M real or complex? M ∈ Rn×n Special properties (symmetric, Hermitian, skew-symmetric or unitary)? symmetric: M = M T Further structure? Yeah. M ∈ H(TI × I , k) ⇒ see next slide Which eigenvalues required? some (inner) or all eigenvaluesMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions H-Matrices [Hackbusch 1998] Some dense matrices, e.g. BEM or FEM, can be approximated by H-matrices in a data-sparse manner. hierarchical tree TI block H-tree TI × I I = {1, 2, 3, 4, 5, 6, 7, 8} 12345678 12345678 12345678 12345678 1 1 1 1 2 2 2 2 {1, 2, 3, 4} {5, 6, 7, 8} 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 {1, 2} {3, 4} {5, 6} {7, 8} 7 7 7 7 8 8 8 8 {1}{2}{3}{4}{5}{6}{7}{8} dense matrices, rank-k-matrices rank-k-matrix: Ma×b = AB T , A ∈ Rn×k , B ∈ Rm×k (k n, m),Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions H-Matrices [Hackbusch 1998] Hierarchical Matrices H(TI × I , k) = M ∈ RI × I rank (Ma×b ) ≤ k ∀a × b admissible 3 3 22 7 10 14 8 11 9 8 5 3 7 19 10 11 8 3 10 10 31 11 11 9 16 12 8 15 8 12 11 8 adaptive rank k(ε) 19 10 11 8 14 8 11 11 10 11 31 11 31 11 9 16 11 7 6 15 5 6 13 8 8 15 9 15 12 13 11 8 3 3 8 7 5 19 13 11 61 11 10 7 8 9 16 11 11 8 11 8 13 13 storage NSt,H (T , k) = O(n log n k(ε)) 9 3 25 10 13 8 7 11 10 19 11 6 5 8 11 9 11 16 3 8 11 11 31 11 15 6 9 15 11 11 8 8 8 15 10 10 15 9 5 8 11 61 3 6 14 9 11 10 10 7 13 6 15 10 3 6 25 10 6 13 8 12 complexity of approximate arithmetic 10 19 10 11 8 12 5 6 10 14 6 10 31 10 16 9 8 15 11 20 13 8 7 8 11 15 9 10 10 5 8 11 8 11 10 51 7 9 3 7 3 9 7 6 15 10 25 11 8 13 13 7 13 11 5 6 10 16 10 9 7 3 3 11 25 10 10 19 9 13 8 8 11 11 MH v O(n log n k(ε)) 11 8 3 7 8 15 8 39 10 10 10 6 5 10 3 9 15 3 10 3 25 7 10 3 6 3 15 10 13 6 11 13 7 10 7 22 7 10 11 9 8 5 7 12 10 3 7 19 10 11 8 12 O(n log n k(ε)2 ) 10 11 16 12 +H , −H 6 3 10 10 31 9 8 15 8 20 13 8 8 11 8 34 10 13 10 6 5 10 9 15 15 11 10 25 7 11 13 6 11 13 13 11 8 13 7 −1 ∗H , HLU(·), (·)H O(n (log n)2 k(ε)2 ) 7 11 10 9 16 10 11 61 11 23 13 9 6 13 20 9 9 7 11 8 3 7 9 8 8 15 5 6 12 9 39 10 3 10 15 10 12 13 11 8 8 8 15 9 10 15 9 7 11 11 5 8 7 3 7 3 10 61 10 9 11 19 13 9 6 13 20 9 9 7 9 8 13 8 8 11 5 6 10 15 10 9 34 10 13 12 7 11 12 11 23 10 15 8 9 11 9 7 10 13 51Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions H-Matrices [Hackbusch 1998] Hierarchical Matrices H(TI × I , k) = M ∈ RI × I rank (Ma×b ) ≤ k ∀a × b admissible 3 3 22 7 10 14 8 11 9 8 5 3 7 19 10 11 8 3 10 10 31 11 11 9 16 12 8 15 8 12 11 8 adaptive rank k(ε) 19 10 11 8 14 8 11 11 10 11 31 11 31 11 9 16 11 7 6 15 5 6 13 8 8 15 9 15 12 13 11 8 3 3 8 7 5 19 13 11 61 11 10 7 8 9 16 11 11 8 11 8 13 13 storage NSt,H (T , k) = O(n log n k(ε)) 9 3 25 10 13 8 7 11 10 19 11 6 5 8 11 9 11 16 3 8 11 11 31 11 15 6 9 15 11 11 8 8 8 15 10 10 15 9 5 8 11 61 3 6 14 9 11 10 10 7 13 6 15 10 3 6 25 10 6 13 8 12 complexity of approximate arithmetic 10 19 10 11 8 12 5 6 10 14 6 10 31 10 16 9 8 15 11 20 13 8 7 8 11 15 9 10 10 5 8 11 8 11 10 51 7 9 3 7 3 9 7 6 15 10 25 11 8 13 13 7 13 11 5 6 10 16 10 9 7 3 3 11 25 10 10 19 9 13 8 8 11 11 MH v O(n log n k(ε)) 11 8 3 7 8 15 8 39 10 10 10 6 5 10 3 9 15 3 10 3 25 7 10 3 6 3 15 10 13 6 11 13 7 10 7 22 7 10 11 9 8 5 7 12 10 3 7 19 10 11 8 12 O(n log n k(ε)2 ) 10 11 16 12 +H , −H 6 3 10 10 31 9 8 15 8 20 13 8 8 11 8 34 10 13 10 6 5 10 9 15 15 11 10 25 7 11 13 6 11 13 13 11 8 13 7 −1 ∗H , HLU(·), (·)H O(n (log n)2 k(ε)2 ) 7 11 10 9 16 10 11 61 11 23 13 9 6 13 20 9 9 7 11 8 3 7 9 8 8 15 5 6 12 9 39 10 3 10 15 10 12 13 11 8 8 8 15 9 10 15 9 7 11 11 5 8 7 3 7 3 10 61 10 9 11 19 13 9 6 13 20 9 9 7 9 8 13 8 8 11 5 6 10 15 10 9 34 10 13 12 7 11 12 11 23 10 15 8 9 11 9 7 10 13 51 T B1 T B2 T B1 A1 + A2 = A1 A2 T B2Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Special Case: H -Matrices [Hackbusch 1998] F1 B2 AT 2 B4 AT 4 T F3 A2 B2 B8 AT 8 F5 B6 AT 6 T A4 B4 T F7 A6 B 6 F9 B10 AT 10 B12 AT 12 T A10 B10 F11 T A8 B8 F13 B14 AT 14 T A12 B12 T A14 B14 F15 Structure of a symmetric H3 (k)-matrix.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Hlib Hlib ¨ [Borm, Grasedyck, et al.] We use the Hlib (www.hlib.org) for the H-arithmetic operations and some examples out of the library for testing the eigenvalue algorithm.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Eigenvalues of Symmetric H-Matrices M = M T ∈ H(T , k) ⇓ ΛH (M) = {λ1 , . . . , λn } in O(n2 (log n)α k β ) {λi } ∈ ΛH (M) in O(n (log n)α k β )?Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Eigenvalues of Symmetric H-Matrices M = M T ∈ H(T , k) ⇓ ΛH (M) = {λ1 , . . . , λn } in O(n2 (log n)α k β ) {λi } ∈ ΛH (M) in O(n (log n)α k β )? dense: M + N, Mv in O(n2 ) and Λ(M) in O(n3 )Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions LR Cholesky Algorithm QR-like AlgorithmMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions LR Cholesky Algorithm LR Cholesky AlgorithmMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions LR Cholesky Algorithm [Rutishauser 1958] LR-Cholesky Transformation for i = 1, . . . do Li LT = Mi i Mi+1 = LT Lii endMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions LR Cholesky Algorithm [Rutishauser 1958] LR-Cholesky Transformation for i = 1, . . . do Li LT = Mi ⇒ Li = Mi L−T i i Mi+1 = LT Li = LT Mi L−T i i i end lim Mi = diag (λ1 , λ2 , . . . , λn ) ∈ H(T , 0) i→∞Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions LR Cholesky Algorithm [Rutishauser 1958] LR-Cholesky Transformation for i = 1, . . . do Li LT = Mi − µi I i Mi+1 = LT Li + µi I i end lim Mi = diag (λ1 , λ2 , . . . , λn ) ∈ H(T , 0) i→∞ ∀i: Mi − µi I symmetric positive deﬁniteMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions LR Cholesky Algorithm [Rutishauser 1958] LR-Cholesky Transformation for i = 1, . . . do Li LT = Mi − µi I i Mi+1 = LT Li + µi I i end lim Mi = diag (λ1 , λ2 , . . . , λn ) ∈ H(T , 0) i→∞ ∀i: Mi − µi I symmetric positive deﬁnite H-LR-Cholesky Transformation for i = 1, . . . do ˜ i = H-Cholesky factorization(Mi − µi I) L ˜ M˜ i+1 = ˜ T ∗H ˜ i + µi I Li L endMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions LR Cholesky Algorithm [Rutishauser 1958] LR-Cholesky Transformation for i = 1, . . . do Li LT = Mi − µi I i Mi+1 = LT Li + µi I i end lim Mi = diag (λ1 , λ2 , . . . , λn ) ∈ H(T , 0) i→∞ ∀i: Mi − µi I symmetric positive deﬁnite H-LR-Cholesky Transformation for i = 1, . . . do ˜ i = H-Cholesky factorization(Mi − µi I) L ˜ M˜ i+1 = ˜ T ∗H ˜ i + µi I Li L shift strategy end deﬂationMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Example - H-Fill-In 32 8 4 8 32 4 8 4 32 8 4 8 32 8 8 32 8 4 8 32 4 8 4 32 8 4 8 32 Matrix FEM16 (∆2,h , 16 inner discr. points).Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Example - H-Fill-In 32 10 4 10 32 5 6 8 4 5 32 9 6 9 32 6 8 8 32 10 6 6 10 32 5 4 8 6 5 32 10 4 10 32 Matrix FEM16 (∆2,h , 16 inner discr. points), after 1 step.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Example - H-Fill-In 32 10 4 10 32 7 7 8 4 7 32 10 7 10 32 7 8 8 32 11 7 7 11 32 7 4 8 7 7 32 10 4 10 32 Matrix FEM16 (∆2,h , 16 inner discr. points), after 2 steps.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Example - H-Fill-In 32 8 4 8 32 4 8 4 32 8 4 4 8 32 8 4 8 32 8 4 8 32 4 8 8 4 32 8 4 4 8 32 4 8 4 32 8 4 4 8 32 4 8 4 32 8 4 8 32 8 4 8 32 8 4 4 8 32 4 8 8 4 32 8 4 8 32 8 8 32 8 4 8 32 4 8 8 4 32 8 4 4 8 32 8 4 8 32 8 4 8 32 4 8 4 32 8 4 4 8 32 4 8 4 32 8 4 4 8 32 4 8 8 4 32 8 4 8 32 8 4 8 32 8 4 4 8 32 4 8 4 32 8 4 8 32 Matrix FEM32 (∆2,h , 32 inner discr. points).Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Example - H-Fill-In 32 11 4 11 32 10 11 8 4 10 32 14 4 4 4 11 14 32 18 15 8 8 4 8 32 14 11 5 18 14 32 17 14 8 7 8 4 4 15 11 17 32 20 8 5 14 20 32 18 9 5 15 13 10 16 8 4 8 32 14 7 4 4 4 4 14 32 10 12 14 7 5 18 7 4 10 32 11 12 11 32 14 3 11 4 3 26 9 15 4 14 32 20 14 6 8 5 13 4 7 14 20 32 18 13 8 3 11 14 18 32 15 7 10 5 3 4 6 13 15 32 16 8 8 32 15 11 4 4 3 16 15 32 18 14 11 3 4 7 5 8 11 18 32 20 8 7 12 4 4 14 20 32 18 17 9 14 8 7 26 4 11 32 15 11 6 3 3 18 15 32 11 9 4 8 7 17 11 11 32 20 9 6 9 20 32 22 4 8 4 8 12 14 32 19 14 4 6 4 16 7 4 8 19 32 20 13 15 4 8 5 7 22 14 20 32 15 4 13 15 32 21 8 4 8 6 15 32 18 11 4 4 4 21 18 32 14 4 8 11 14 32 11 4 11 32 Matrix FEM32 (∆2,h , 32 inner discr. points), after 1 step.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Example - H-Fill-In 32 11 4 11 32 22 13 12 4 22 32 22 4 4 4 13 22 32 43 25 17 16 4 12 32 28 24 32 43 28 32 30 25 10 9 8 4 4 25 24 30 32 32 17 32 25 32 32 73 30 32 32 32 38 38 8 4 16 32 32 23 30 31 30 4 32 32 31 32 32 32 46 73 23 31 32 25 30 32 25 32 57 32 31 29 32 104 30 32 31 32 32 32 32 32 10 32 32 30 32 57 32 32 32 32 8 32 29 32 32 32 32 9 38 46 31 32 32 32 32 32 40 8 8 32 32 32 32 30 31 38 32 32 32 32 32 30 42 33 10 8 32 32 32 32 32 32 32 30 32 32 32 32 63 32 32 32 30 13 104 30 32 32 31 32 31 31 30 63 31 32 31 28 42 32 32 32 31 32 32 32 32 31 28 32 32 75 4 18 4 8 32 32 32 32 32 23 19 6 40 33 30 30 32 32 32 30 25 7 8 10 13 75 32 32 32 31 23 30 31 32 48 8 4 18 19 25 32 28 14 4 6 7 48 28 32 20 4 8 14 20 32 11 4 11 17 Matrix FEM32 (∆2,h , 32 inner discr. points), after 50 steps.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Computation Time 107 H-LR algorithm 106 LAPACK dsyev 105 O(n2 (log2 n)2 ) 104 O(n3 ) CPU time in s 103 102 101 100 10−1 10−2 1 10 102 103 104 105 106 DimensionMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Computation Time 107 H-LR algorithm 106 LAPACK dsyev 105 O(n2 (log2 n)2 ) 104 O(n3 ) CPU time in s 103 102 101 Name n # iterations 100 FEM8 64 101 10−1 FEM16 256 556 10−2 1 FEM32 1 024 2 333 10 102 103 4 FEM64 10 4 096 105 1010 6 320 DimensionMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Theorem Adaption of [Fasino ’05/Plestenjak, Van Barel, Van Camp ’08] r M = diag (d) + i=1 tril ui viT + triu vi uiTMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Theorem Adaption of [Fasino ’05/Plestenjak, Van Barel, Van Camp ’08] r M = diag (d) + i=1 tril ui viT + triu vi uiT Structure Preservation of dpss Matrices Let M be a symmetric positive definite diagonal plus semiseparable matrix, with a decomposition as in the definition. The Cholesky factor L of M = LLT can be written in the form ˜ r L = diag d + i=1 tril ui ˜iT . v Multiplying the Cholesky factors in reverse order gives the next iterate N = LT L of the LR Cholesky algorithm. The matrix N has the same form as M, ˆ r N = diag d + i=1 tril î ˜iT + triu ˜i îT uv vu .Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Proof Idea = L1:p−1,1:p−1 LT p,1:p−1 = M1:p−1,p = vi uiT i ⇒ L1:p−1,1:p−1 ˜i |1:p−1 = vi |1:p−1 and Lp,1:p−1 = v ui |p ˜iT v 1:p−1 i ˜ dp + ui |p ˜i |p =Lpp = v Mpp − Lp,1:p−1 LT p,1:p−1 i L is a dpss matrix.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Proof Idea T N=L L= T ˜ diag d + tril ui ˜iT v i ˜ diag d + tril ui ˜iT v i ˆ ˜ u i = Z + diag d ui , with Zp,: = ˜j |p 0 · · · 0 v uj |p uj |p+1 · · · uj |n j tril (N, −1) = tril ˜ diag(d)ui + Zui ˜iT , −1 v i = tril ˆi ˜iT , −1 uv i N is a dpss matrix.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Structure of ˆ and ˜ u v r M = diag (d) + tril ui viT + . . . i=1 r N = diag (d) + tril ˆi ˜iT + . . . uv i=1         0 0 0 * . . . . . . . . . . . . 0 0 0 *         ∗ ∗ ∗ ∗ .         vi =  .  ˜i =  .  ui =  .  ˆi =  .  . . . v u . . . . ∗ ∗ ∗ ∗         0 * 0 0         . . . . . . . . . . . . 0 * 0 0Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Hierarchical Matrices rank (Ma:b,c:d ) = k Ma:b,c:d = AB T uiT = 0 · · · 0 AT r j,r 0 ··· 0 viT r = 0 ··· 0 T Bj,r 0 ··· 0 , r = 1, . . . , k T ˆir u = ∗ ··· ∗ ∗ 0 ··· 0 ˜iT vr = 0 ··· 0 ∗ ∗ ··· ∗ 0 0     0 0  0 ∗  0 0 0 0 ∗ ∗       T T tril ui vi = 0 0 0 0 tril ˆi ˜i uv = 0 ∗ ∗ ∗       0 ∗ ∗ 0 0 0 ∗ ∗ ∗ ∗       0 ∗ ∗ 0 0 0  0 ∗ ∗ ∗ ∗ ∗  0 0 0 0 0 0 0 0 0 0 0 0 0 0Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Hierarchical Matrices rank (Ma:b,c:d ) = k Ma:b,c:d = AB T uiT = 0 · · · 0 AT r j,r 0 ··· 0 viT r = 0 ··· 0 T Bj,r 0 ··· 0 , r = 1, . . . , k T ˆir u = ∗ ··· ∗ ∗ 0 ··· 0 ˜iT vr = 0 ··· 0 ∗ ∗ ··· ∗ 0 0     0 0  0 ∗  0 0 0 0 ∗ ∗       T T tril ui vi = 0 0 0 0 tril ˆi ˜i uv = 0 ∗ ∗ ∗       0 ∗ ∗ 0 0 0 ∗ ∗ ∗ ∗       0 ∗ ∗ 0 0 0  0 ∗ ∗ ∗ ∗ ∗  0 0 0 0 0 0 0 0 0 0 0 0 0 0 The structure of hierarchical matrices is not preserved under LR Cholesky transformations.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Example - H-Fill-In 32 8 4 32 11 4 8 32 4 8 11 32 10 11 8 4 32 8 4 4 10 32 14 4 4 4 4 8 32 8 4 11 14 32 18 15 8 8 4 8 32 8 4 8 32 14 11 5 8 32 4 8 18 14 32 17 14 8 7 8 8 4 4 32 8 8 32 4 4 8 4 4 15 11 17 32 20 8 5 14 20 32 18 9 5 15 13 10 16 8 4 32 8 4 4 8 32 14 7 4 4 4 4 8 32 4 8 4 14 32 10 12 14 7 5 4 4 32 8 8 32 8 18 7 4 10 32 11 12 11 32 14 3 11 4 3 26 4 8 32 8 4 9 15 4 14 32 20 14 6 4 8 32 4 8 8 5 13 4 7 14 20 32 18 13 8 8 4 32 8 3 11 14 18 32 15 4 8 32 8 7 10 5 3 4 6 13 15 32 16 8 8 32 8 4 8 32 15 11 4 4 3 8 32 4 8 16 15 32 18 14 11 3 4 7 5 8 4 32 8 4 8 11 18 32 20 8 7 12 4 4 8 32 8 4 4 14 20 32 18 17 9 14 8 7 26 8 32 8 4 4 11 32 15 11 6 8 32 4 3 3 18 15 32 11 9 8 4 4 32 8 8 32 4 4 4 8 7 17 11 11 32 20 9 6 9 20 32 22 4 8 4 8 4 32 8 4 8 12 14 32 19 14 4 6 4 4 8 32 4 8 16 7 4 8 19 32 20 13 15 4 8 4 4 32 8 8 32 8 8 5 7 22 14 20 32 15 4 13 15 32 21 8 4 8 32 8 4 4 8 6 15 32 18 11 4 8 32 4 4 4 4 21 18 32 14 4 8 4 32 8 8 11 14 32 11 4 8 32 4 11 32Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions H -Matrices F1 B2 AT 2 B4 AT 4 T F3 A2 B2 B8 AT 8 F5 B 6 AT 6 T A4 B4 T F7 A 6 B6 F9 B10 AT 10 B12 AT 12 T A10 B10 F11 T A8 B8 F13 B14 AT 14 T A12 B12 T A14 B14 F15Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions H -Matrices F1 F3 F5 F7 F9 F11 I F13 F15Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions H -Matrices F1 I F3 F5 I I F7 F9 I F11 I F13 I I F15Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions H -Matrices F1 I F3 F5 II I F7 F9 I F11 I F13 II I F15Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions H -Matrices F1 II F3 F5 II II F7 F9 II F11 I F13 II II F15Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions H -Matrices F1 III F3 F5 II III F7 F9 III F11 I F13 II III F15Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions H -Matrices F1 III F3 F5 II III F7 F9 III F11 I F13 II III F15 ⇒ rank bounded by k instead of k ⇒ total storage required by the low-rank parts of M is increased −1) only from 2nk to 2nk ( 2Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Computation Time H -Matrices 107 H-LR H (1) H-LR H (2) 105 dsyev CPU time in s O(n2 log2 n) 103 O(n3 ) O(n2 ) 101 10−1 105# Iterations # It. H (1) 104 # It. H (2) 103 O(n) 102 102 103 104 Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Slicing the Spectrum Slicing the SpectrumMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Bisectioning [Parlett ’80] -6 -4 -2 -1 0 1 2 4 6Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Bisectioning [Parlett ’80] λ3 = ? -6 -4 -2 -1 0 1 2 4 6 -6 -4 -2 -1 0 1 2 4 6Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Bisectioning [Parlett ’80] λ3 = ? -6 -4 -2 -1 0 1 2 4 6 -6 -4 -2 -1 0 1 2 4 6 a0 = −6.5 b0 = 5.5Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Bisectioning [Parlett ’80] λ3 = ? -6 -4 -2 -1 0 1 2 4 6 a0 = −6.5 b0 = 5.5 -6 -4 -2 -1 0 1 2 4 6 a0 = −6.5 b0 = 5.5 µ1 = −0.5Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Bisectioning [Parlett ’80] λ3 = ? -6 -4 -2 -1 0 1 2 4 6 a0 = −6.5 b0 = 5.5 µ1 = −0.5 ν(−0.5) = 4 -6 -4 -2 -1 0 1 2 4 6 a0 = −6.5 b0 = 5.5 µ1 = −0.5Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Bisectioning [Parlett ’80] λ3 = ? ν(−0.5) = 4 -6 -4 -2 -1 0 1 2 4 6 a0 = −6.5 b0 = 5.5 µ1 = −0.5 ν(−3.5) = 2 -6 -4 -2 -1 0 1 2 4 6 a1 = −6.5 b1 = −0.5 µ2 = −3.5Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Bisectioning [Parlett ’80] λ3 = ? ν(−3.5) = 2 -6 -4 -2 -1 0 1 2 4 6 a1 = −6.5 b1 = −0.5 µ2 = −3.5 ν(−2) = 4 -6 -4 -2 -1 0 1 2 4 6 a2 = −3.5 b2 = −0.5 µ3 = −2Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Bisectioning [Parlett ’80] ˆ λ3 ∈ [−3.5, −2.75], λ3 = −3.125 ν(−2) = 4 -6 -4 -2 -1 0 1 2 4 6 a2 = −3.5 b2 = −0.5 µ3 = −2 ν(−2.75) = 3 -6 -4 -2 -1 0 1 2 4 6 a3 = −3.5 b3 = −2 µ4 = −2.75 b 0 − a0 ≈ 2 M 2 ˆ λi − λi < ⇔ bn − an < 2 bi+1 − ai+1 = 1 (bi − ai ) 2 ⇒ O (log2 ( M 2 / ))Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Evaluation of ν(µ) Sylvester’s Law of Inertia Each matrix M is congruent to a matrix diag −Iν , Irank(M)−ν , 0n−rank(M) , where ν is the number of negative eigenvalues. The triple (ν, rank (M) − ν, n − rank (M)) is called the inertia of M. M = LDLT ⇒ ν(M) = ν(D) M − µI = Lµ Dµ LT µ ⇒ ν(µ) = ν(M − µI ) = ν(Dµ )Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Complexity Flops per factorization (for H -matrices): O nk 2 (log n)4 . Factorization per eigenvalue: O (log( M 2/ )) . Flops per eigenvalue: O nk 2 (log n)4 log M 2/ . Slicing the whole spectrum: O n2 k 2 (log n)4 log( M 2/ ) .Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Absolute Error for H5 (1)-Matrix 10−7 abs. error 1 10−8 2 ev Absolute error 10−9 10−10 10−11 0 100 200 300 400 500 600 700 800 900 1 000 Eigenvalue ˆ Absolute error |λi − λi | for the 1 024 × 1 024 H5 (1)-matrix, ev = 10−8 .Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions CPU Time for 10 Eigenvalues O(n (log n)4 ) O(n (log n)2 ) O(n log n) O(n) Computation Time 103 Time in s 102 101 100 104 105 106 Dimension Computation times for 10 eigenvalues of H (1)-matrices ( = 8, . . . , 15).Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Parallelization ν(−0.5) = 4 -6 -4 -2 -1 0 1 2 4 6 a0 = −6.5 b0 = 5.5 µ1 = −0.5 ν(−3.5) = 2 ν(2.5) = 9 -6 -4 -2 -1 0 1 2 4 6 1 1 2 2 a1 = −6.5 b1 = −0.5 = a1 b1 = 5.5 µ1 = −3.5 2 µ2 = 2.5 2Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Parallelization Speedup OpenMP: Name n t1 core in s t1c /t2c t1c /t4c t8 core t1c /t8c H2 r1 128 0.33 1.83 3.30 0.06 5.50 H4 r1 512 9.44 1.94 3.67 1.43 6.60 H6 r1 2 048 219.28 1.91 3.64 33.88 6.47 H8 r1 8 192 4 022.80 1.87 3.44 676.57 5.95 H10 r1 32 768 49 012.24 1.93 3.18 10 006.60 4.90Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Parallelization Speedup Open MPI: (H9 (1) ∈ R16 384×16 384 ) No. of Processes t in s Speedup Eﬃciency 1 16 564.58 1.00 1.00 2+1 8 340.22 1.99 0.66 4+1 4 044.13 4.10 0.82 6+1 2 678.95 6.18 0.88 11+1 1 494.33 11.08 0.92 23+1 713.80 23.21 0.96 35+1 476.44 34.77 0.96 47+1 364.30 45.47 0.95 95+1 188.92 87.68 0.91 191+1 100.61 164.64 0.86 287+1 71.86 230.50 0.80 383+1 61.91 267.56 0.70Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions H-Matrices Shifting aﬀects the structure. The LDLT factorization is of almost linear complexity only for the original H-matrix and not necessarily for the shifted ones. For H -matrices we use the exact LDLT factorization. For general H-matrices: The truncation introduces errors in the LDLT factorization — the computed D may have another inertia ⇒ some eigenvalues may lie outside the computed interval ⇒ larger errorsMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Preconditioned Inverse Iteration Preconditioned Inverse Iteration for Hierarchical MatricesMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Preconditioned Inverse Iteration [Knyazev, Neymeyr, et al.] Deﬁnition The function x T Mx µ(x) = µ(x, M) = xT x is called the Rayleigh quotient.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Preconditioned Inverse Iteration [Knyazev, Neymeyr, et al.] Deﬁnition The function x T Mx µ(x) = µ(x, M) = xT x is called the Rayleigh quotient. Minimize the Rayleigh quotient by a gradient method: 2 xi+1 := xi − α µ(xi ), µ(x) = (Mx − xµ(x)) , xT xMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Preconditioned Inverse Iteration [Knyazev, Neymeyr, et al.] Deﬁnition The function x T Mx µ(x) = µ(x, M) = xT x is called the Rayleigh quotient. Minimize the Rayleigh quotient by a gradient method: 2 xi+1 := xi − α µ(xi ), µ(x) = (Mx − xµ(x)) , xT x + preconditioning ⇒ update equation: xi+1 := xi − B −1 (Mxi − xi µ(xi )) .Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Preconditioned Inverse Iteration [Knyazev, Neymeyr 2009] xi+1 := xi − B −1 (Mxi − xi µ(xi )) If M ∈ Rn×n symmetric positive deﬁnite and B −1 approximates the inverse of M, such that I − B −1 M M ≤ c < 1, then Preconditioned INVerse ITeration (PINVIT) converges and the number of iterations is independent of n.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Algorithm and Complexity The number of iterations is independent of matrix size n. H-PINVIT Input: M ∈ Rn×n , X0 ∈ Rn×d (X0 X0 = I , e.g. randomly chosen) T Output: Xp ∈ R n×d , µ ∈ Rd×d , with MX − X µ ≤ p p B −1 = (M)−1 or B −1 = L−T L−1 H H H T R := MX0 − X0 µ, µ = X0 MX0 for (i := 1; R F > ; i + +) do Xi := Orthogonalize Xi−1 − B −1 R R := MXi − Xi µ, µ = XiT MXi endMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Algorithm and Complexity The number of iterations is independent of matrix size n. H-PINVIT Input: M ∈ Rn×n , X0 ∈ Rn×d (X0 X0 = I , e.g. randomly chosen) T Output: Xp ∈ R n×d , µ ∈ Rd×d , with MX − X µ ≤ p p B −1 = (M)−1 or B −1 = L−T L−1 H H H O(n (log n)2 k (c)2 ) T R := MX0 − X0 µ, µ = X0 MX0 for (i := 1; R F > ; i + +) do Xi := Orthogonalize Xi−1 − B −1 R O(n (log n) k (c)2 ) R := MXi − Xi µ, µ = XiT MXi O(n (log n) k (c)) end The complexity of the algorithm is determined by the H-matrix inversion/Cholesky decomposition: ⇒ O(n (log n)2 k (c)2 ).Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions CPU Time 104 H-Cholesky PINVIT MATLAB eigs 103 CPU time in s 102 101 100 10−1 10−2 FEM3D4 64 FEM3D8 512 FEM3D16 4 096 FEM3D32 32 768 FEM3D64 262 144 FEM3D128 2 079 152Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions CPU Time 104 H-Cholesky PINVIT MATLAB eigs 103 CPU time in s out of memory 102 101 100 10−1 10−2 FEM3D4 64 FEM3D8 512 FEM3D16 4 096 FEM3D32 32 768 FEM3D64 262 144 FEM3D128 2 079 152Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions CPU Time 104 H-Cholesky PINVIT MATLAB eigs 103 H-Cholesky decomposition CPU time in s out of memory 102 101 100 10−1 10−2 FEM3D4 64 FEM3D8 512 FEM3D16 4 096 FEM3D32 32 768 FEM3D64 262 144 FEM3D128 2 079 152Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions CPU Time 104 H-Cholesky PINVIT MATLAB eigs 103 H-Cholesky decomposition Slicing the Spectrum CPU time in s out of memory 102 101 100 10−1 10−2 FEM3D4 64 FEM3D8 512 FEM3D16 4 096 FEM3D32 32 768 FEM3D64 262 144 FEM3D128 2 079 152Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Folded Spectrum Method Folded Spectrum Method [Wang, Zunger 1994] Mσ = (M − σI)2 Mσ is s.p.d., if M is s.p.d. and σ = λi . (x − σ)2Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Folded Spectrum Method Folded Spectrum Method [Wang, Zunger 1994] Mσ = (M − σI)2 Mσ is s.p.d., if M is s.p.d. and σ = λi . The condition number of (M − σI )2 is large. −1 ⇒ The computation of Mσ is more expensive. −1 ⇒ Mσ has larger local ranks. −1 ⇒ Mσ v is more expensive.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Folded Spectrum Method Folded Spectrum Method [Wang, Zunger 1994] Mσ = (M − σI)2 Mσ is s.p.d., if M is s.p.d. and σ = λi . The condition number of (M − σI )2 is large. −1 ⇒ The computation of Mσ is more expensive. −1 ⇒ Mσ has larger local ranks. −1 ⇒ Mσ v is more expensive. Multiple eigenvalues of Mσ may lead to incomplete subspace information. ⇒ v T Mv /v T v does not approximate λ.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Conclusions ConclusionsMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Slicing the Spectrum vs. LR Cholesky 102 103 104 105 106 Slicing the spectrum 105 LR Cholesky algorithm 104 LAPACK dsyev Time in s 103 102 10 1 10−1 10−2 102 Accuracy 10−2 10−6 H (1)-matrices ( = 1, . . . , 10) 10−10 all eigenvalues 102 103 104 105Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Slicing the Spectrum vs. LR Cholesky 102 103 104 105 106 Slicing the spectrum 105 LR Cholesky algorithm 104 LAPACK dsyev Time in s 103 102 10 1 10−1 10−2 102 Accuracy 10−2 10−6 10−10 102 103 104 105Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Slicing the Spectrum vs. H-PINVIT 102 103 104 105 106 104 Slicing the spectrum 103 HPINVIT MATLAB eigs Time in s 102 HPINVIT iteration 10 HPINVIT preconditioner 1 10−1 10−2 102 Accuracy 10−2 10−6 H (1)-matrices ( = 1, . . . , 15) 10−10 three smallest eigenvalues 102 103 104 105 106Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Slicing the Spectrum vs. H-PINVIT 102 103 104 105 106 104 Slicing the spectrum 103 HPINVIT MATLAB eigs Time in s 102 HPINVIT iteration 10 HPINVIT preconditioner 1 10−1 10−2 102 Accuracy 10−2 10−6 10−10 102 103 104 105 106Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Conclusions Three algorithms: H-LR Cholesky algorithm: eﬃcient only for H -matrices, but expensive otherwiseMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Conclusions Three algorithms: H-LR Cholesky algorithm: eﬃcient only for H -matrices, but expensive otherwise Slicing the spectrum: eﬃcient only for H -matrices, good parallelizableMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Conclusions Three algorithms: H-LR Cholesky algorithm: efficient only for H -matrices, but expensive otherwise Slicing the spectrum: efficient only for H -matrices, good parallelizable H-PINVIT: efficient for the smallest eigenvalue of positive definite H-matrices, computation of inner eigenvalues is difficultMax Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

Hierarchical (H-)Matrices LR Algorithm Slicing Algorithm PINVIT Conclusions Conclusions Three algorithms: H-LR Cholesky algorithm: efficient only for H -matrices, but expensive otherwise Slicing the spectrum: efficient only for H -matrices, good parallelizable H-PINVIT: efficient for the smallest eigenvalue of positive definite H-matrices, computation of inner eigenvalues is difficult Thank you for your attention.Max Planck Institute Magdeburg Thomas Mach, Eigenvalue Algorithms for Symmetric H-Matrices

http://people.cs.kuleuven.be	477
http://www.mpi-magdeburg.mpg.de	184
http://www.en.mpi-magdeburg.mpg.de	9
http://193.175.52.47	4
http://translate.googleusercontent.com	2

Eigenvalues of Symmetrix Hierarchical Matrices

by Thomas Mach on Feb 28, 2012

Accessibility

Categories

Upload Details

Usage Rights

5 Embeds 676

Statistics

Eigenvalues of Symmetrix Hierarchical Matrices Presentation Transcript