GAMM Annual Meeting 2013 Novi Sad, March 18–22, 2013
THE SYMPLECTIC LANCZOS PROCESS FOR HAMILTONIAN-POSITIVE MATRICES (Symplectic Lanczos for Hamiltonian-positive matrices encounters no serious breakdown!)
Peter Benner
Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Magdeburg, Germany TU Chemnitz Mathematics in Industry and Technology Chemnitz, Germany Joint work with Ahmed Salam (ULCO, Calais) Introduction Symplectic Lanczos Hamiltonian-positiv References Introduction Hamiltonian Eigenproblems
Definition 0 In Let J = , then H ∈ R2n×2n is called Hamiltonian if −In 0 (HJ)T = HJ.
Note: J−1 = JT = −J.
Explicit block form of Hamiltonian matrices
AB , where A, B, C ∈ n×n and B = BT , C = C T . C −AT R
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 2/13 Introduction Symplectic Lanczos Hamiltonian-positiv References Introduction Hamiltonian Eigenproblems
Definition 0 In Let J = , then H ∈ R2n×2n is called Hamiltonian if −In 0 (HJ)T = HJ.
Note: J−1 = JT = −J.
Explicit block form of Hamiltonian matrices
AB , where A, B, C ∈ n×n and B = BT , C = C T . C −AT R
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 2/13 Typical Hamiltonian spectrum
Introduction Symplectic Lanczos Hamiltonian-positiv References Introduction Spectral Properties
Hamiltonian Eigensymmetry Hamiltonian matrices exhibit the Hamiltonian eigensymmetry: if λ is a finite eigenvalue of H, then λ,¯ −λ, −λ¯ are eigenvalues of H, too.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 3/13 Introduction Symplectic Lanczos Hamiltonian-positiv References Introduction Spectral Properties
Hamiltonian Eigensymmetry Hamiltonian matrices exhibit the Hamiltonian eigensymmetry: if λ is a finite eigenvalue of H, then λ,¯ −λ, −λ¯ are eigenvalues of H, too.
Typical Hamiltonian spectrum
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 3/13 Introduction Symplectic Lanczos Hamiltonian-positiv References Hamiltonian Eigenproblems
Goal Structure-preserving algorithm, i.e., if λ˜ is a computed eigenvalue of H, then λ,˜ −λ,˜ −λ˜ should also be computed eigenvalues. Goal cannot be achieved by general methods for matrices or matrix pencils like the QR/QZ, Lanczos, Arnoldi algorithms! For an algorithm based on similarity transformations, the goal is achieved if the Hamiltonian structure is preserved.
Definition V ∈ R2n×2n is symplectic if V T JV = J, i.e., V −1 = JT V T J. 2n×2k T Vk ∈ R is symplectic or a J-isometry if Vk JnVk = Jk . Lemma If H is Hamiltonian and V is symplectic, then V −1HV is Hamiltonian, too.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 4/13 Introduction Symplectic Lanczos Hamiltonian-positiv References Hamiltonian Eigenproblems
Goal Structure-preserving algorithm, i.e., if λ˜ is a computed eigenvalue of H, then λ,˜ −λ,˜ −λ˜ should also be computed eigenvalues. Goal cannot be achieved by general methods for matrices or matrix pencils like the QR/QZ, Lanczos, Arnoldi algorithms! For an algorithm based on similarity transformations, the goal is achieved if the Hamiltonian structure is preserved.
Definition V ∈ R2n×2n is symplectic if V T JV = J, i.e., V −1 = JT V T J. 2n×2k T Vk ∈ R is symplectic or a J-isometry if Vk JnVk = Jk . Lemma If H is Hamiltonian and V is symplectic, then V −1HV is Hamiltonian, too.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 4/13 Introduction Symplectic Lanczos Hamiltonian-positiv References Hamiltonian Eigenproblems
Goal Structure-preserving algorithm, i.e., if λ˜ is a computed eigenvalue of H, then λ,˜ −λ,˜ −λ˜ should also be computed eigenvalues. Goal cannot be achieved by general methods for matrices or matrix pencils like the QR/QZ, Lanczos, Arnoldi algorithms! For an algorithm based on similarity transformations, the goal is achieved if the Hamiltonian structure is preserved.
Definition V ∈ R2n×2n is symplectic if V T JV = J, i.e., V −1 = JT V T J. 2n×2k T Vk ∈ R is symplectic or a J-isometry if Vk JnVk = Jk . Lemma If H is Hamiltonian and V is symplectic, then V −1HV is Hamiltonian, too.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 4/13 Introduction Symplectic Lanczos Hamiltonian-positiv References Introduction Applications Hamiltonian eigenproblems arise in many different applications, e.g.: Systems and control: Solution methods for algebraic and differential Riccati equations. Design of LQR/LQG/H2/H∞ controllers and filters for continuous-time linear control systems. Stability radii and system norm computations; optimization of system norms. Passivity-preserving model reduction based on balancing. Reduced-order control for infinite-dimensional systems based on inertial manifolds. Computational physics: exponential integrators for Hamiltonian dynamics. [Eirola ’03, Lopez/Simoncini ’06, B./Meister ’13] Quantum chemistry: computing excitation energies in many-particle systems using random phase approximation (RPA). Quadratic eigenvalue problems: in particular, gyroscopic systems.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 5/13 Introduction Symplectic Lanczos Hamiltonian-positiv References The Symplectic Lanczos Algorithm
Symplectic Lanczos Algorithm for Hamiltonian operators H is based on transpose-free unsymmetric Lanczos process [Freund ’94]; computes partial J-tridiagonalization; 2n×2k provides a symplectic (J-orthogonal) Lanczos basis Vk ∈ R , T i.e., Vk JnVk = Jk ; was derived in several variants: [Freund/Mehrmann ’94, Ferng/Lin/Wang ’97, B./Faßbender ’97, Watkins ’04]; requires re-J-orthogonalization using, e.g., modified symplectic Gram-Schmidt; can be restarted implicitly using implicit SR steps [B./Faßbender ’97] or Krylov-Schur restarting [B./Faßbender/Stoll ’11] which allows easy locking & purging procedure; does not provide an orthogonal Lanczos basis and is prone to serious breakdown.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 6/13 Introduction Symplectic Lanczos Hamiltonian-positiv References The Hamiltonian J-Tridiagonal Form or Hamiltonian J-Hessenberg Form
δ1 β1 ζ2 δ2 ζ2 β2 ζ3 .. .. δ ζ . . 3 3 . . . . . . . . . ζn 2n×2n Tn = δn ζn βn ∈ R , ν1 −δ1 ν2 −δ2 ν3 −δ3 . . .. .. νn −δn
−1 can be computed by symplectic similarity Tn = V HV for almost (serious breakdown!) any V1 = V (:, 1), is computed partially by symplectic Lanczos process, based on symplectic Lanczos recursion T HVk = Vk Tk + ζk+1vk+1e2k , Vk = [V (:, 1 : k), V (:, n + 1 : n + k)].
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 7/13 Introduction Symplectic Lanczos Hamiltonian-positiv References The Symplectic Lanczos Algorithm T Algorithm based on symplectic Lanczos recursion HVk = Vk Tk + ζk+1vk+1e2k 2n×2n 2n INPUT: H ∈ R , k ∈ N, and start vectorv ˜1 6= 0 ∈ R . 2k×2k 2n×2k OUTPUT: Tk ∈ R , Vk ∈ R , ζk+1, and vk+1.
1 ζ1 = kv˜1k2 1 2 v1 = v˜1 ζ1 3 FOR m = 1, 2,..., k
(a) t = Hvm (b) δm = ht, vmi % B./Faßbender ’97: δm = 1, Watkins ’04: δm = 0. (c)˜wm = t − δmvm (d) νm = ht, vmiJ 1 (e) wm = w˜m νm (f) u = Hwm (g) βm = −hu, wmiJ (h)˜vm+1 = u − ζmvm−1 − βmvm + δmwm (i) ζm+1 = kv˜m+1k2 1 (j) vm+1 = v˜m+1 ζm+1 ENDFOR
Note: 3(b) yields orthogonality of vk , wk [Ferng/Lin/Wang ’97] and optimal conditioning of Lanczos basis [B. ’03] if kvk2 = 1 is forced.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 8/13 Introduction Symplectic Lanczos Hamiltonian-positiv References The Symplectic Lanczos Algorithm T Algorithm based on symplectic Lanczos recursion HVk = Vk Tk + ζk+1vk+1e2k 2n×2n 2n INPUT: H ∈ R , k ∈ N, and start vectorv ˜1 6= 0 ∈ R . 2k×2k 2n×2k OUTPUT: Tk ∈ R , Vk ∈ R , ζk+1, and vk+1.
1 ζ1 = kv˜1k2 1 2 v1 = v˜1 ζ1 3 FOR m = 1, 2,..., k
(a) t = Hvm (b) δm = ht, vmi % B./Faßbender ’97: δm = 1, Watkins ’04: δm = 0. (c)˜wm = t − δmvm (d) νm = ht, vmiJ 1 (e) wm = w˜m νm (f) u = Hwm (g) βm = −hu, wmiJ (h)˜vm+1 = u − ζmvm−1 − βmvm + δmwm (i) ζm+1 = kv˜m+1k2 1 (j) vm+1 = v˜m+1 ζm+1 ENDFOR
Note: 3(b) yields orthogonality of vk , wk [Ferng/Lin/Wang ’97] and optimal conditioning of Lanczos basis [B. ’03] if kvk2 = 1 is forced.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 8/13 Introduction Symplectic Lanczos Hamiltonian-positiv References The Symplectic Lanczos Algorithm Numerical example: rolling tire Results from [B./Faßbender/Stoll ’08/’11]
Modeling the noise of rolling tires requires to determine the transient vibrations, [Nackenhorst/von Estorff ’01].
FEM model of a deformable wheel rolling on a rigid plane surface results in a gyroscopic system of order n = 124, 992 [Nackenhorst ’04].
Here, use reduced-order model of size n = 2, 635 computed by AMLS [Elssel/Voß ’06].
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 9/13 Introduction Symplectic Lanczos Hamiltonian-positiv References The Symplectic Lanczos Algorithm Numerical example: rolling tire Results from [B./Faßbender/Stoll ’08/’11]
Compare eigs and symplectic Lanczos with Krylov-Schur restarting (HKS) applied to H−1 to compute the 12 smallest eigenvalues. eigs needs 8, HKS 6 iterations. max(cond (SR)) = 331. Eigenvalues scaled by 1,000.
eigs HKS Eigenvalue Residual Eigenvalue Residual 4 · 10−12 +1 .73705142673ı 2 · 10−14 1.73705142671ı 5 · 10−17 −3 · 10−12 +1 .66795405953ı 8 · 10−15 1.66795405955ı 2 · 10−15 2 · 10−13 +1 .66552788164ı 2 · 10−15 1.66552788164ı 1 · 10−16 4 · 10−14 +1 .58209209804ı 1 · 10−16 1.58209209804ı 5 · 10−17 −1 · 10−14 +1 .13657108578ı 8 · 10−17 1.13657108578ı 7 · 10−18 1 · 10−14 +0 .80560062107ı 1 · 10−16 0.80560062107ı 6 · 10−18
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 10/13 Introduction Symplectic Lanczos Hamiltonian-positiv References The Symplectic Lanczos Algorithm Numerical example: rolling tire Results from [B./Faßbender/Stoll ’08/’11]
Compare eigs and HKS applied to H−1 to compute the 180 smallest eigenvalues.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 10/13 Introduction Symplectic Lanczos Hamiltonian-positiv References
Symplectic Lanczos for Hamiltonian-positive Matrices Hamiltonian-positive Matrices
Definition A Hamiltonian matrix H ∈ R2n×2n is called Hamiltonian-positive if its symmetric generator S = JT H is positive definite.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 11/13 Introduction Symplectic Lanczos Hamiltonian-positiv References
Symplectic Lanczos for Hamiltonian-positive Matrices Hamiltonian-positive Matrices
Definition A Hamiltonian matrix H ∈ R2n×2n is called Hamiltonian-positive if its symmetric generator S = JT H is positive definite.
Lemma [Amodio 2003] All eigenvalues of Hamiltonian-positive matrices are purely imaginary, i.e., Λ(H) ⊂ R.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 11/13 Introduction Symplectic Lanczos Hamiltonian-positiv References
Symplectic Lanczos for Hamiltonian-positive Matrices Hamiltonian-positive Matrices
Definition A Hamiltonian matrix H ∈ R2n×2n is called Hamiltonian-positive if its symmetric generator S = JT H is positive definite.
Comparison of spectra of spd generator and its Hamiltonian
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 11/13 Introduction Symplectic Lanczos Hamiltonian-positiv References
Symplectic Lanczos for Hamiltonian-positive Matrices Hamiltonian-positive Matrices
Definition A Hamiltonian matrix H ∈ R2n×2n is called Hamiltonian-positive if its symmetric generator S = JT H is positive definite.
Example: weakly coupled Hamiltonian systems
p˙ = H , 1 q for Hamiltonian H(p, q) = pT p + qT Kq + 2pT Wq q˙ = −Hp, 2
with kinetic energy determined by K = K T > 0 and weak coupling, i.e., kW k ”small” " # In W S = JT H = > 0 if kW k ”small” enough. W T K
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 11/13 Introduction Symplectic Lanczos Hamiltonian-positiv References
Symplectic Lanczos for Hamiltonian-positive Matrices Hamiltonian-positive Matrices
Definition A Hamiltonian matrix H ∈ R2n×2n is called Hamiltonian-positive if its symmetric generator S = JT H is positive definite.
Theorem There is no serious breakdown in symplectic Lanczos for Hamiltonian-positive matrices.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 11/13 Introduction Symplectic Lanczos Hamiltonian-positiv References
Symplectic Lanczos for Hamiltonian-positive Matrices Hamiltonian-positive Matrices
2n×2n 2n INPUT: H ∈ R , k ∈ N, and start vectorv ˜1 6= 0 ∈ R . 2k×2k 2n×2k OUTPUT: Tk ∈ R , Vk ∈ R , ζk+1, and vk+1.
1 ζ1 = kv˜1k2 1 2 v1 = v˜1 ζ1 3 FOR m = 1, 2,..., k
(a) t = Hvm (b) δm = ht, vmi % B./Faßbender ’97: δm = 1, Watkins ’04: δm = 0. (c)˜wm = t − δmvm (d) νm = ht, vmiJ 1 (e) wm = w˜m νm (f) u = Hwm (g) βm = −hu, wmiJ (h)˜vm+1 = u − ζmvm−1 − βmvm + δmwm (i) ζm+1 = kv˜m+1k2 1 (j) vm+1 = v˜m+1 ζm+1 ENDFOR
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 11/13 Introduction Symplectic Lanczos Hamiltonian-positiv References
Symplectic Lanczos for Hamiltonian-positive Matrices Hamiltonian-positive Matrices
Definition A Hamiltonian matrix H ∈ R2n×2n is called Hamiltonian-positive if its symmetric generator S = JT H is positive definite.
Theorem There is no serious breakdown in symplectic Lanczos for Hamiltonian-positive matrices.
Proof. If vm = 0 =⇒ benign breakdown, else
νm = hHvm, vmiJ
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 11/13 Introduction Symplectic Lanczos Hamiltonian-positiv References
Symplectic Lanczos for Hamiltonian-positive Matrices Hamiltonian-positive Matrices
Definition A Hamiltonian matrix H ∈ R2n×2n is called Hamiltonian-positive if its symmetric generator S = JT H is positive definite.
Theorem There is no serious breakdown in symplectic Lanczos for Hamiltonian-positive matrices.
Proof. If vm = 0 =⇒ benign breakdown, else
νm = hHvm, vmiJ T T T T = vm H Jvm = vm J Hvm
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 11/13 Introduction Symplectic Lanczos Hamiltonian-positiv References
Symplectic Lanczos for Hamiltonian-positive Matrices Hamiltonian-positive Matrices
Definition A Hamiltonian matrix H ∈ R2n×2n is called Hamiltonian-positive if its symmetric generator S = JT H is positive definite.
Theorem There is no serious breakdown in symplectic Lanczos for Hamiltonian-positive matrices.
Proof. If vm = 0 =⇒ benign breakdown, else
νm = hHvm, vmiJ T T T T = vm H Jvm = vm J Hvm T √ = vm Svm > 0.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 11/13 Introduction Symplectic Lanczos Hamiltonian-positiv References
Symplectic Lanczos for Hamiltonian-positive Matrices Hamiltonian-positive Matrices
Definition A Hamiltonian matrix H ∈ R2n×2n is called Hamiltonian-positive if its symmetric generator S = JT H is positive definite.
Theorem There is no serious breakdown in symplectic Lanczos for Hamiltonian-positive matrices.
Proof. If vm = 0 =⇒ benign breakdown, else
νm = hHvm, vmiJ T T T T = vm H Jvm = vm J Hvm T √ = vm Svm > 0.
Remark: Hamiltonian-positiveness of Tk can be enforced; T βm = −hu, wmiJ = −u Su < 0.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 11/13 Introduction Symplectic Lanczos Hamiltonian-positiv References
How About Symplectic Arnoldi? Hamiltonian-positive Matrices
Symplectic Arnoldi recursion
HVk = Vk Hk + ζk vk+1,
where Vk is symplectic and orthogonal, Hk is in Hamiltonian Hessenberg form:
" H H # 11 12 T T Hk = with H11 = @ , H12 = H , H21 = hk+1,k ek e . T @ 12 k H21 −H11 @@
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 12/13 Introduction Symplectic Lanczos Hamiltonian-positiv References
How About Symplectic Arnoldi? Hamiltonian-positive Matrices
Symplectic Arnoldi recursion
HVk = Vk Hk + ζk vk+1,
where Vk is symplectic and orthogonal, Hk is in Hamiltonian Hessenberg form:
" H H # 11 12 T T Hk = with H11 = @ , H12 = H , H21 = hk+1,k ek e . T @ 12 k H21 −H11 @@
Theorem [Ammar/Mehrmann ’91]
There exists a symplectic Arnoldi reduction with Hn unreduced, i.e., hj+1,j 6= 0, if and only if ∃ x with
x T JH2k−1x = 0 (k = 1, 2,..., n − 1), x T x = 1,
that is not contained in an H-invariant subspace of dimension ≤ n. For H Hamiltonian-positive, no such vector exists! (Already observed in [Ammar/Mehrmann ’91, page 65].)
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 12/13 Introduction Symplectic Lanczos Hamiltonian-positiv References References
1 G. Ammar and V. Mehrmann. On Hamiltonian and symplectic Hessenberg forms. Linear Algebra Appl., 149:55–72, 1991. 2 P. Amodio. On the computation of few eigenvalues of positive definite Hamiltonian matrices. Future Generation Computer Systems, 22:403–411, 2006. 3 P. Benner. Structured Krylov subspace methods for eigenproblems with spectral symmetries. Workshop Theoretical and Computational Aspects of Matrix Algorithms, Dagstuhl, October 2003. 4 P. Benner and H. Faßbender. An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem. Linear Algebra Appl., 263:75–111, 1997. 5 P. Benner, H. Faßbender, and M. Stoll. A Krylov-Schur-type algorithm for Hamiltonian eigenproblems based on the symplectic Lanczos process. Linear Algebra Appl., 435(3):578–600, 2011. 6 W. R. Ferng, W. W. Lin, and C. S. Wang. The shift-inverted J-Lanczos algorithm for the numerical solutions of large sparse algebraic Riccati equations. Comp. Math. Appl., 33(10):23?40, 1997. 7 D. Watkins. On Hamiltonian and symplectic Lanczos processes. Linear Algebra Appl., 385:23–45, 2004.
Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 13/13