Solving large Hamiltonian eigenvalue problems
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1 Solving large Hamiltonian eigenvalue problems David S. Watkins Department of Mathematics Washington State University Adaptivity and Beyond, Vancouver, August 2005 p.1
2 Some Collaborators Adaptivity and Beyond, Vancouver, August 2005 p.2
3 Some Collaborators Volker Mehrmann Adaptivity and Beyond, Vancouver, August 2005 p.2
4 Some Collaborators Volker Mehrmann Thomas Apel Adaptivity and Beyond, Vancouver, August 2005 p.2
5 Some Collaborators Volker Mehrmann Thomas Apel Peter Benner Adaptivity and Beyond, Vancouver, August 2005 p.2
6 Some Collaborators Volker Mehrmann Thomas Apel Peter Benner Heike Faßbender Adaptivity and Beyond, Vancouver, August 2005 p.2
7 Some Collaborators Volker Mehrmann Thomas Apel Peter Benner Heike Faßbender... Adaptivity and Beyond, Vancouver, August 2005 p.2
8 Problem: Linear Elasticity Elastic Deformation (3D, anisotropic, composite materials) Adaptivity and Beyond, Vancouver, August 2005 p.3
9 Problem: Linear Elasticity Elastic Deformation (3D, anisotropic, composite materials) Singularities at cracks, interfaces Adaptivity and Beyond, Vancouver, August 2005 p.3
10 Problem: Linear Elasticity Elastic Deformation (3D, anisotropic, composite materials) Singularities at cracks, interfaces Lamé Equations (PDE, spherical coordinates) Adaptivity and Beyond, Vancouver, August 2005 p.3
11 Problem: Linear Elasticity Elastic Deformation (3D, anisotropic, composite materials) Singularities at cracks, interfaces Lamé Equations (PDE, spherical coordinates) Separate radial variable. Adaptivity and Beyond, Vancouver, August 2005 p.3
12 Problem: Linear Elasticity Elastic Deformation (3D, anisotropic, composite materials) Singularities at cracks, interfaces Lamé Equations (PDE, spherical coordinates) Separate radial variable. Get quadratic eigenvalue problem. (λ 2 M + λg + K)v = 0 M = M > 0 G = G K = K < 0 Adaptivity and Beyond, Vancouver, August 2005 p.3
13 Problem: Linear Elasticity Elastic Deformation (3D, anisotropic, composite materials) Singularities at cracks, interfaces Lamé Equations (PDE, spherical coordinates) Separate radial variable. Get quadratic eigenvalue problem. (λ 2 M + λg + K)v = 0 M = M > 0 G = G K = K < 0 solve numerically Adaptivity and Beyond, Vancouver, August 2005 p.3
14 Numerical Solution Adaptivity and Beyond, Vancouver, August 2005 p.4
15 Numerical Solution Discretize using finite elements Adaptivity and Beyond, Vancouver, August 2005 p.4
16 Numerical Solution Discretize using finite elements (λ 2 M + λg + K)v = 0 M T = M > 0 G T = G K T = K < 0 Adaptivity and Beyond, Vancouver, August 2005 p.4
17 Numerical Solution Discretize using finite elements (λ 2 M + λg + K)v = 0 M T = M > 0 G T = G K T = K < 0 matrix quadratic eigenvalue problem (large, sparse) Adaptivity and Beyond, Vancouver, August 2005 p.4
18 Numerical Solution Discretize using finite elements (λ 2 M + λg + K)v = 0 M T = M > 0 G T = G K T = K < 0 matrix quadratic eigenvalue problem (large, sparse) Find few smallest eigenvalues (and corresponding eigenvectors). Adaptivity and Beyond, Vancouver, August 2005 p.4
19 Other Applications Adaptivity and Beyond, Vancouver, August 2005 p.5
20 Other Applications Gyroscopic systems λ 2 M + λg + K Adaptivity and Beyond, Vancouver, August 2005 p.5
21 Other Applications Gyroscopic systems λ 2 M + λg + K Quadratic regulator (optimal control) [ ] [ ] C T C A T 0 E T A BB T λ E 0 symmetric/skew-symmetric Adaptivity and Beyond, Vancouver, August 2005 p.5
22 Other Applications Gyroscopic systems λ 2 M + λg + K Quadratic regulator (optimal control) [ ] [ ] C T C A T 0 E T A BB T λ E 0 symmetric/skew-symmetric Higher-order systems λ n A n + λ n 1 A n λa 1 + A 0 Adaptivity and Beyond, Vancouver, August 2005 p.5
23 Hamiltonian Structure Adaptivity and Beyond, Vancouver, August 2005 p.6
24 Hamiltonian Structure Our matrices are real. Adaptivity and Beyond, Vancouver, August 2005 p.6
25 Hamiltonian Structure Our matrices are real. λ, λ, λ, λ occur together. Adaptivity and Beyond, Vancouver, August 2005 p.6
26 Hamiltonian Structure Our matrices are real. λ, λ, λ, λ occur together. spectra of Hamiltonian matrices Adaptivity and Beyond, Vancouver, August 2005 p.6
27 Hamiltonian Structure Our matrices are real. λ, λ, λ, λ occur together. spectra of Hamiltonian matrices Adaptivity and Beyond, Vancouver, August 2005 p.6
28 Hamiltonian Structure Our matrices are real. λ, λ, λ, λ occur together. spectra of Hamiltonian matrices Adaptivity and Beyond, Vancouver, August 2005 p.6
29 Hamiltonian Matrices Adaptivity and Beyond, Vancouver, August 2005 p.7
30 Hamiltonian Matrices H R 2n 2n Adaptivity and Beyond, Vancouver, August 2005 p.7
31 Hamiltonian Matrices H R 2n 2n [ 0 I J = I 0 ] R 2n 2n Adaptivity and Beyond, Vancouver, August 2005 p.7
32 Hamiltonian Matrices H R 2n 2n [ 0 I J = I 0 ] R 2n 2n H is Hamiltonian iff JH is symmetric. Adaptivity and Beyond, Vancouver, August 2005 p.7
33 Hamiltonian Matrices H R 2n 2n [ 0 I J = I 0 ] R 2n 2n H is Hamiltonian iff JH is symmetric. [ ] A K H = N A T, where K = K T and N = N T Adaptivity and Beyond, Vancouver, August 2005 p.7
34 Reduction of Order λ 2 Mv + λgv + Kv = 0 Adaptivity and Beyond, Vancouver, August 2005 p.8
35 Reduction of Order λ 2 Mv + λgv + Kv = 0 w = λv, Adaptivity and Beyond, Vancouver, August 2005 p.8
36 Reduction of Order λ 2 Mv + λgv + Kv = 0 w = λv, Mw = λmv Adaptivity and Beyond, Vancouver, August 2005 p.8
37 Reduction of Order λ 2 Mv + λgv + Kv = 0 w = λv, Mw = λmv [ ] [ ] [ K 0 v λ 0 M w G M M 0 ] [ v w ] = 0 Adaptivity and Beyond, Vancouver, August 2005 p.8
38 Reduction of Order λ 2 Mv + λgv + Kv = 0 w = λv, Mw = λmv [ ] [ ] [ K 0 v λ 0 M w G M M 0 ] [ v w ] = 0 Ax λnx = 0 Adaptivity and Beyond, Vancouver, August 2005 p.8
39 Reduction of Order λ 2 Mv + λgv + Kv = 0 w = λv, Mw = λmv [ ] [ ] [ K 0 v λ 0 M w G M M 0 ] [ v w ] = 0 Ax λnx = 0 symmetric/skew-symmetric Adaptivity and Beyond, Vancouver, August 2005 p.8
40 Reduction of Order λ 2 Mv + λgv + Kv = 0 w = λv, Mw = λmv [ ] [ ] [ K 0 v λ 0 M w G M M 0 ] [ v w ] = 0 Ax λnx = 0 symmetric/skew-symmetric Structure has been preserved. Adaptivity and Beyond, Vancouver, August 2005 p.8
41 Reduction to Hamiltonian Matrix A λn (symmetric/skew-symmetric) Adaptivity and Beyond, Vancouver, August 2005 p.9
42 Reduction to Hamiltonian Matrix A λn (symmetric/skew-symmetric) ( [ ]) N = R T 0 I JR J = I 0 Adaptivity and Beyond, Vancouver, August 2005 p.9
43 Reduction to Hamiltonian Matrix A λn (symmetric/skew-symmetric) ( [ ]) N = R T 0 I JR J = I 0 always possible, sometimes easy Adaptivity and Beyond, Vancouver, August 2005 p.9
44 Reduction to Hamiltonian Matrix A λn (symmetric/skew-symmetric) ( [ ]) N = R T 0 I JR J = I 0 always possible, sometimes easy Transform: Adaptivity and Beyond, Vancouver, August 2005 p.9
45 Reduction to Hamiltonian Matrix A λn (symmetric/skew-symmetric) ( [ ]) N = R T 0 I JR J = I 0 always possible, sometimes easy Transform: A λr T JR Adaptivity and Beyond, Vancouver, August 2005 p.9
46 Reduction to Hamiltonian Matrix A λn (symmetric/skew-symmetric) ( [ ]) N = R T 0 I JR J = I 0 always possible, sometimes easy Transform: A λr T JR R T AR 1 λj Adaptivity and Beyond, Vancouver, August 2005 p.9
47 Reduction to Hamiltonian Matrix A λn (symmetric/skew-symmetric) ( [ ]) N = R T 0 I JR J = I 0 always possible, sometimes easy Transform: A λr T JR R T AR 1 λj J 1 R T AR 1 λi Adaptivity and Beyond, Vancouver, August 2005 p.9
48 Reduction to Hamiltonian Matrix A λn (symmetric/skew-symmetric) ( [ ]) N = R T 0 I JR J = I 0 always possible, sometimes easy Transform: A λr T JR R T AR 1 λj J 1 R T AR 1 λi H = J 1 R T AR 1 is Hamiltonian. Adaptivity and Beyond, Vancouver, August 2005 p.9
49 Example (our application) [ K 0 0 M ] [ v w ] λ [ G M M 0 ] [ v w ] = 0 Adaptivity and Beyond, Vancouver, August 2005 p.10
50 Example (our application) [ ] [ K 0 0 M [ G M N = M 0 v w ] ] λ [ G M M 0 ] [ v w ] = 0 Adaptivity and Beyond, Vancouver, August 2005 p.10
51 Example (our application) [ ] [ K 0 0 M [ G M N = M 0 v w ] ] λ [ G M M 0 ] [ v w ] = 0 N = R T JR = Adaptivity and Beyond, Vancouver, August 2005 p.10
52 Example (our application) [ ] [ ] [ ] [ K 0 v G M λ 0 M w M 0 [ ] G M N = M 0 [ ] [ ] [ N = R T I 1 JR = 2 G 0 I 0 M I 0 v w ] = 0 I G M ] Adaptivity and Beyond, Vancouver, August 2005 p.10
53 Example (our application) [ ] [ ] [ ] [ K 0 v G M λ 0 M w M 0 [ ] G M N = M 0 [ ] [ ] [ N = R T I 1 JR = 2 G 0 I 0 M I 0 cost: zero flops v w ] = 0 I G M ] Adaptivity and Beyond, Vancouver, August 2005 p.10
54 H = J 1 R T AR 1 Adaptivity and Beyond, Vancouver, August 2005 p.11
55 H = J 1 R T AR 1 After some algebra... Adaptivity and Beyond, Vancouver, August 2005 p.11
56 H = J 1 R T AR 1 After some algebra... [ ] [ I 0 0 M 1 H = 1 2 G I K 0 ] [ I G I ] Adaptivity and Beyond, Vancouver, August 2005 p.11
57 H = J 1 R T AR 1 After some algebra... [ ] [ I 0 0 M 1 H = 1 2 G I K 0 ] [ I G I ] Do not form the product explicitly Adaptivity and Beyond, Vancouver, August 2005 p.11
58 H = J 1 R T AR 1 After some algebra... [ ] [ I 0 0 M 1 H = 1 2 G I K 0 ] [ I G I ] Do not form the product explicitly Krylov subspace methods Adaptivity and Beyond, Vancouver, August 2005 p.11
59 H = J 1 R T AR 1 After some algebra... [ ] [ I 0 0 M 1 H = 1 2 G I K 0 ] [ I G I ] Do not form the product explicitly Krylov subspace methods We just need to apply the operator. Adaptivity and Beyond, Vancouver, August 2005 p.11
60 H = J 1 R T AR 1 After some algebra... [ ] [ I 0 0 M 1 H = 1 2 G I K 0 ] [ I G I ] Do not form the product explicitly Krylov subspace methods We just need to apply the operator. M = LL T (done once) Adaptivity and Beyond, Vancouver, August 2005 p.11
61 H = J 1 R T AR 1 After some algebra... [ ] [ I 0 0 M 1 H = 1 2 G I K 0 ] [ I G I ] Do not form the product explicitly Krylov subspace methods We just need to apply the operator. M = LL T (done once) backsolve Adaptivity and Beyond, Vancouver, August 2005 p.11
62 However,... Adaptivity and Beyond, Vancouver, August 2005 p.12
63 However,... Adaptivity and Beyond, Vancouver, August 2005 p.12
64 However, we really want H 1. Adaptivity and Beyond, Vancouver, August 2005 p.12
65 However, we really want H 1. H = [ I G I ] [ 0 M 1 K 0 ] [ I G I ] Adaptivity and Beyond, Vancouver, August 2005 p.12
66 However, we really want H 1. [ ] [ I 0 H = 1 2 G I [ ] [ H 1 I 0 = 1 2 G I 0 M 1 K 0 0 ( K) 1 M 0 ] [ I G I ] [ I G I ] ] Adaptivity and Beyond, Vancouver, August 2005 p.12
67 However, we really want H 1. [ ] [ I 0 H = 1 2 G I [ ] [ H 1 I 0 = 1 2 G I K = LL T 0 M 1 K 0 0 ( K) 1 M 0 ] [ I G I ] [ I G I ] ] Adaptivity and Beyond, Vancouver, August 2005 p.12
68 Shift and Invert? Adaptivity and Beyond, Vancouver, August 2005 p.13
69 Shift and Invert? (H τi) 1 Adaptivity and Beyond, Vancouver, August 2005 p.13
70 Shift and Invert? (H τi) 1 is not Hamiltonian Adaptivity and Beyond, Vancouver, August 2005 p.13
71 Shift and Invert? (H τi) 1 is not Hamiltonian Structure is lost. Adaptivity and Beyond, Vancouver, August 2005 p.13
72 Shift and Invert? (H τi) 1 is not Hamiltonian Structure is lost. How can we recover it? Adaptivity and Beyond, Vancouver, August 2005 p.13
73 Exploitable Structures Adaptivity and Beyond, Vancouver, August 2005 p.14
74 Exploitable Structures symplectic (first idea) (H τi) 1 (H + τi) Adaptivity and Beyond, Vancouver, August 2005 p.14
75 Exploitable Structures symplectic (first idea) (H τi) 1 (H + τi) skew-hamiltonian (easiest?) H 2 Adaptivity and Beyond, Vancouver, August 2005 p.14
76 Exploitable Structures symplectic (first idea) (H τi) 1 (H + τi) skew-hamiltonian H 2 (easiest?) (H τi) 1 (H + τi) 1 Adaptivity and Beyond, Vancouver, August 2005 p.14
77 Exploitable Structures symplectic (first idea) (H τi) 1 (H + τi) skew-hamiltonian H 2 (easiest?) (H τi) 1 (H + τi) 1 Hamiltonian H 1 Adaptivity and Beyond, Vancouver, August 2005 p.14
78 Exploitable Structures symplectic (first idea) (H τi) 1 (H + τi) skew-hamiltonian H 2 (easiest?) (H τi) 1 (H + τi) 1 Hamiltonian H 1 H 1 (H τi) 1 (H + τi) 1 Adaptivity and Beyond, Vancouver, August 2005 p.14
79 Structured Lanczos Processes Adaptivity and Beyond, Vancouver, August 2005 p.15
80 Structured Lanczos Processes Unsymmetric Lanczos Process u k+1 b k d k = Bu k u k a k d k u k 1 b k 1 d k w k+1 d k+1 b k = B T w k w k d k a k w k 1 d k 1 b k 1 Adaptivity and Beyond, Vancouver, August 2005 p.15
81 Hamiltonian Lanczos Process u k+1 b k+1 = Hv k u k a k u k 1 b k 1 v k+1 d k+1 = Hu k+1 Adaptivity and Beyond, Vancouver, August 2005 p.16
82 Symplectic Lanczos Process v k+1 d k+1 b k = Sv k v k d k a k v k 1 d k 1 b k 1 + u k d k u k+1 d k+1 = S 1 v k+1 Adaptivity and Beyond, Vancouver, August 2005 p.17
83 Symplectic Lanczos Process v k+1 d k+1 b k = Sv k v k d k a k v k 1 d k 1 b k 1 + u k d k u k+1 d k+1 = S 1 v k+1 many interesting relationships Adaptivity and Beyond, Vancouver, August 2005 p.17
84 Implicit Restarts Adaptivity and Beyond, Vancouver, August 2005 p.18
85 Implicit Restarts short Lanczos runs (breakdowns!!, no look-ahead) Adaptivity and Beyond, Vancouver, August 2005 p.18
86 Implicit Restarts short Lanczos runs (breakdowns!!, no look-ahead) Restart implicitly as in IRA (Sorensen 1991), ARPACK Adaptivity and Beyond, Vancouver, August 2005 p.18
87 Implicit Restarts short Lanczos runs (breakdowns!!, no look-ahead) Restart implicitly as in IRA (Sorensen 1991), ARPACK Restart with HR, not QR Adaptivity and Beyond, Vancouver, August 2005 p.18
88 Remarks on Stability Adaptivity and Beyond, Vancouver, August 2005 p.19
89 Remarks on Stability Both Hamiltonian and symplectic Lanczos processes are potentially unstable. Adaptivity and Beyond, Vancouver, August 2005 p.19
90 Remarks on Stability Both Hamiltonian and symplectic Lanczos processes are potentially unstable. Breakdowns can occur. Adaptivity and Beyond, Vancouver, August 2005 p.19
91 Remarks on Stability Both Hamiltonian and symplectic Lanczos processes are potentially unstable. Breakdowns can occur. Are the answers worth anything? Adaptivity and Beyond, Vancouver, August 2005 p.19
92 Remarks on Stability Both Hamiltonian and symplectic Lanczos processes are potentially unstable. Breakdowns can occur. Are the answers worth anything? right and left eigenvectors Adaptivity and Beyond, Vancouver, August 2005 p.19
93 Remarks on Stability Both Hamiltonian and symplectic Lanczos processes are potentially unstable. Breakdowns can occur. Are the answers worth anything? right and left eigenvectors residuals Adaptivity and Beyond, Vancouver, August 2005 p.19
94 Remarks on Stability Both Hamiltonian and symplectic Lanczos processes are potentially unstable. Breakdowns can occur. Are the answers worth anything? right and left eigenvectors residuals condition numbers for eigenvalues Adaptivity and Beyond, Vancouver, August 2005 p.19
95 Remarks on Stability Both Hamiltonian and symplectic Lanczos processes are potentially unstable. Breakdowns can occur. Are the answers worth anything? right and left eigenvectors residuals condition numbers for eigenvalues Don t skip these tests. Adaptivity and Beyond, Vancouver, August 2005 p.19
96 Example Adaptivity and Beyond, Vancouver, August 2005 p.20
97 Example λ 2 Mv + λgv + Kv = 0 Adaptivity and Beyond, Vancouver, August 2005 p.20
98 Example λ 2 Mv + λgv + Kv = 0 n = 3423 Adaptivity and Beyond, Vancouver, August 2005 p.20
99 Example λ 2 Mv + λgv + Kv = 0 n = 3423 H = [ I G I ] [ 0 M 1 K 0 ] [ I G I ] Adaptivity and Beyond, Vancouver, August 2005 p.20
100 Example λ 2 Mv + λgv + Kv = 0 n = 3423 H = [ I G I ] [ 0 M 1 K 0 ] [ I G I ] H 1 = [ I G I ] [ 0 ( K) 1 M 0 ] [ I G I ] Adaptivity and Beyond, Vancouver, August 2005 p.20
101 Compare various approaches: Adaptivity and Beyond, Vancouver, August 2005 p.21
102 Compare various approaches: Hamiltonian(1) H 1 Adaptivity and Beyond, Vancouver, August 2005 p.21
103 Compare various approaches: Hamiltonian(1) Hamiltonian(3) H 1 H 1 (H τi) 1 (H + τi) 1 Adaptivity and Beyond, Vancouver, August 2005 p.21
104 Compare various approaches: Hamiltonian(1) Hamiltonian(3) H 1 H 1 (H τi) 1 (H + τi) 1 symplectic (H τi) 1 (H + τi) Adaptivity and Beyond, Vancouver, August 2005 p.21
105 Compare various approaches: Hamiltonian(1) Hamiltonian(3) H 1 H 1 (H τi) 1 (H + τi) 1 symplectic (H τi) 1 (H + τi) unstructured (H τi) 1 + ordinary Lanczos with implicit restarts Adaptivity and Beyond, Vancouver, August 2005 p.21
106 Compare various approaches: Hamiltonian(1) Hamiltonian(3) H 1 H 1 (H τi) 1 (H + τi) 1 symplectic (H τi) 1 (H + τi) unstructured (H τi) 1 + ordinary Lanczos with implicit restarts Get 6 smallest eigenvalues in right half-plane. Adaptivity and Beyond, Vancouver, August 2005 p.21
107 Compare various approaches: Hamiltonian(1) Hamiltonian(3) symplectic H 1 H 1 (H τi) 1 (H + τi) 1 (H τi) 1 (H + τi) unstructured (H τi) 1 + ordinary Lanczos with implicit restarts Get 6 smallest eigenvalues in right half-plane. Tolerance = 10 8 Adaptivity and Beyond, Vancouver, August 2005 p.21
108 Compare various approaches: Hamiltonian(1) Hamiltonian(3) symplectic H 1 H 1 (H τi) 1 (H + τi) 1 (H τi) 1 (H + τi) unstructured (H τi) 1 + ordinary Lanczos with implicit restarts Get 6 smallest eigenvalues in right half-plane. Tolerance = 10 8 Take 20 steps and restart with 10. Adaptivity and Beyond, Vancouver, August 2005 p.21
109 No-Clue Case (τ = 0) Adaptivity and Beyond, Vancouver, August 2005 p.22
110 No-Clue Case (τ = 0) Method Solves Eigvals Max. Found Resid. Hamiltonian(1) Adaptivity and Beyond, Vancouver, August 2005 p.22
111 No-Clue Case (τ = 0) Method Solves Eigvals Max. Found Resid. Hamiltonian(1) Unstructured Adaptivity and Beyond, Vancouver, August 2005 p.22
112 No-Clue Case (τ = 0) Method Solves Eigvals Max. Found Resid. Hamiltonian(1) Unstructured Unstructured code must find everything twice. 5 x Adaptivity and Beyond, Vancouver, August 2005 p.22
113 Conservative Shift (τ = 0.5) Adaptivity and Beyond, Vancouver, August 2005 p.23
114 Conservative Shift (τ = 0.5) Method Solves Eigvals Max. Found Resid. Hamiltonian(1) x Adaptivity and Beyond, Vancouver, August 2005 p.23
115 Conservative Shift (τ = 0.5) Method Solves Eigvals Max. Found Resid. Hamiltonian(1) Unstructured x Adaptivity and Beyond, Vancouver, August 2005 p.23
116 Conservative Shift (τ = 0.5) Method Solves Eigvals Max. Found Resid. Hamiltonian(1) Unstructured Hamiltonian(3) x Adaptivity and Beyond, Vancouver, August 2005 p.23
117 Conservative Shift (τ = 0.5) Method Solves Eigvals Max. Found Resid. Hamiltonian(1) Unstructured Hamiltonian(3) Symplectic x Adaptivity and Beyond, Vancouver, August 2005 p.23
118 Aggressive Shift (τ = 1.5) Adaptivity and Beyond, Vancouver, August 2005 p.24
119 Aggressive Shift (τ = 1.5) Method Solves Eigvals Max. Found Resid. Hamiltonian(1) Adaptivity and Beyond, Vancouver, August 2005 p.24
120 Aggressive Shift (τ = 1.5) Method Solves Eigvals Max. Found Resid. Hamiltonian(1) Unstructured Adaptivity and Beyond, Vancouver, August 2005 p.24
121 Aggressive Shift (τ = 1.5) Method Solves Eigvals Max. Found Resid. Hamiltonian(1) Unstructured Hamiltonian(3) Adaptivity and Beyond, Vancouver, August 2005 p.24
122 Aggressive Shift (τ = 1.5) Method Solves Eigvals Max. Found Resid. Hamiltonian(1) Unstructured Hamiltonian(3) Symplectic Adaptivity and Beyond, Vancouver, August 2005 p.24
123 The Last Slide Adaptivity and Beyond, Vancouver, August 2005 p.25
124 The Last Slide We have developed structure-preserving implicitly-restarted Lanczos methods for Hamiltonian and symplectic eigenvalue problems. Adaptivity and Beyond, Vancouver, August 2005 p.25
125 The Last Slide We have developed structure-preserving implicitly-restarted Lanczos methods for Hamiltonian and symplectic eigenvalue problems. The structure-preserving methods are more accurate than a comparable non-structured method. Adaptivity and Beyond, Vancouver, August 2005 p.25
126 The Last Slide We have developed structure-preserving implicitly-restarted Lanczos methods for Hamiltonian and symplectic eigenvalue problems. The structure-preserving methods are more accurate than a comparable non-structured method. By exploiting structure we can solve our problems more economically. Adaptivity and Beyond, Vancouver, August 2005 p.25
127 The Last Slide We have developed structure-preserving implicitly-restarted Lanczos methods for Hamiltonian and symplectic eigenvalue problems. The structure-preserving methods are more accurate than a comparable non-structured method. By exploiting structure we can solve our problems more economically. Thank you for your attention. Adaptivity and Beyond, Vancouver, August 2005 p.25
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