A KRYLOV-SCHUR-TYPE ALGORITHM FOR EIGENPROBLEMS WITH HAMILTONIAN SPECTRAL SYMMETRY Peter Benner Professur Mathematik in Industrie und Technik Fakult¨ at f¨ ur Mathematik Technische Universit¨ at Chemnitz RANMEP2008 National Tsinghua University, Taiwan, January 4–8, 2008 Joint work with Heike Faßbender and Martin Stoll Dedicated to Ralph Byers (1955–2007)
Overview Hamiltonian 1 Introduction Krylov-Schur Hamiltonian Eigenproblems Peter Benner Applications Introduction Symplectic 2 The Symplectic Lanczos Algorithm Lanczos 3 The SR Algorithm The SR Algorithm 4 A Hamiltonian Krylov-Schur-Type Algorithm HKS Quadratic Derivation Eigenvalue Problems Shift-and-invert Conclusions and Numerical Example Outlook References 5 Quadratic Eigenvalue Problems Shift-and-invert Corner singularities Gyroscopic systems 6 Conclusions and Outlook 7 References
Introduction Hamiltonian Eigenproblems Hamiltonian Definition Krylov-Schur � � 0 I n Peter Benner , then H ∈ R 2 n × 2 n is called Let J = − I n 0 Introduction Hamiltonian Hamiltonian, if ( HJ ) T = HJ , Eigenproblems Applications skew-Hamiltonian, if ( HJ ) T = − HJ . Symplectic Lanczos A matrix pencil λ N − H is called a Hamiltonian/skew-Hamiltonian The SR Algorithm pencil, if H is Hamiltonian and N is skew-Hamiltonian. HKS Quadratic Explicit block form Eigenvalue Problems Conclusions and of Hamiltonian matrices: Outlook � A � G References , where A , G , Q ∈ R n × n and G = G T , Q = Q T , − A T Q of skew-Hamiltonian Matrices: � A � G , where A , G , Q ∈ R n × n and G = − G T , Q = − Q T . A T Q
Introduction Hamiltonian Eigenproblems Hamiltonian Definition Krylov-Schur � � 0 I n Peter Benner , then H ∈ R 2 n × 2 n is called Let J = − I n 0 Introduction Hamiltonian Hamiltonian, if ( HJ ) T = HJ , Eigenproblems Applications skew-Hamiltonian, if ( HJ ) T = − HJ . Symplectic Lanczos A matrix pencil λ N − H is called a Hamiltonian/skew-Hamiltonian The SR Algorithm pencil, if H is Hamiltonian and N is skew-Hamiltonian. HKS Quadratic Explicit block form Eigenvalue Problems Conclusions and of Hamiltonian matrices: Outlook � A � G References , where A , G , Q ∈ R n × n and G = G T , Q = Q T , − A T Q of skew-Hamiltonian Matrices: � A � G , where A , G , Q ∈ R n × n and G = − G T , Q = − Q T . A T Q
Introduction Hamiltonian Eigenproblems Hamiltonian Definition Krylov-Schur � � 0 I n Peter Benner , then H ∈ R 2 n × 2 n is called Let J = − I n 0 Introduction Hamiltonian Hamiltonian, if ( HJ ) T = HJ , Eigenproblems Applications skew-Hamiltonian, if ( HJ ) T = − HJ . Symplectic Lanczos A matrix pencil λ N − H is called a Hamiltonian/skew-Hamiltonian The SR Algorithm pencil, if H is Hamiltonian and N is skew-Hamiltonian. HKS Quadratic Explicit block form Eigenvalue Problems Conclusions and of Hamiltonian matrices: Outlook � A � G References , where A , G , Q ∈ R n × n and G = G T , Q = Q T , − A T Q of skew-Hamiltonian Matrices: � A � G , where A , G , Q ∈ R n × n and G = − G T , Q = − Q T . A T Q
Introduction Spectral Properties Hamiltonian Krylov-Schur Hamiltonian Eigensymmetry Peter Benner Hamiltonian matrices and Hamiltonian/skew-Hamiltonian pencils Introduction exhibit the Hamiltonian eigensymmetry: Hamiltonian Eigenproblems if λ is a finite eigenvalue of H − λ N , then ¯ λ, − λ, − ¯ λ are eigenvalues Applications of H − λ N , too. Symplectic Lanczos The SR Algorithm HKS Quadratic Eigenvalue Problems Conclusions and Outlook References
Introduction Spectral Properties Hamiltonian Krylov-Schur Hamiltonian Eigensymmetry Peter Benner Hamiltonian matrices and Hamiltonian/skew-Hamiltonian pencils Introduction exhibit the Hamiltonian eigensymmetry: Hamiltonian Eigenproblems if λ is a finite eigenvalue of H − λ N , then ¯ λ, − λ, − ¯ λ are eigenvalues Applications of H − λ N , too. Symplectic Lanczos The SR Algorithm Typical Hamiltonian spectrum: HKS Quadratic Eigenvalue Problems Conclusions and Outlook References
Hamiltonian Eigenproblems Hamiltonian Goal Krylov-Schur Structure-preserving algorithm, i.e., if ˜ Peter Benner λ is a computed eigenvalue of H − λ N , then ˜ λ, − ˜ λ, − ˜ Introduction λ should also be computed eigenvalues. Hamiltonian Eigenproblems Applications Goal cannot be achieved by general methods for matrices or matrix Symplectic pencils like the QR/QZ, Lanczos, Arnoldi algorithms! Lanczos The SR For an algorithm based on similarity transformations, the goal is Algorithm HKS achieved if the Hamiltonian structure is preserved. Quadratic Eigenvalue Problems Definition Conclusions and S ∈ R 2 n × 2 n is symplectic iff i.e., S − 1 = J T S T J . S T JS = J , Outlook References Lemma If H is Hamiltonian (skew-Hamiltonian) and S is symplectic, then S − 1 HS is Hamiltonian (skew-Hamiltonian), too.
Hamiltonian Eigenproblems Hamiltonian Goal Krylov-Schur Structure-preserving algorithm, i.e., if ˜ Peter Benner λ is a computed eigenvalue of H − λ N , then ˜ λ, − ˜ λ, − ˜ Introduction λ should also be computed eigenvalues. Hamiltonian Eigenproblems Applications Goal cannot be achieved by general methods for matrices or matrix Symplectic pencils like the QR/QZ, Lanczos, Arnoldi algorithms! Lanczos The SR For an algorithm based on similarity transformations, the goal is Algorithm HKS achieved if the Hamiltonian structure is preserved. Quadratic Eigenvalue Problems Definition Conclusions and S ∈ R 2 n × 2 n is symplectic iff i.e., S − 1 = J T S T J . S T JS = J , Outlook References Lemma If H is Hamiltonian (skew-Hamiltonian) and S is symplectic, then S − 1 HS is Hamiltonian (skew-Hamiltonian), too.
Hamiltonian Eigenproblems Hamiltonian Goal Krylov-Schur Structure-preserving algorithm, i.e., if ˜ Peter Benner λ is a computed eigenvalue of H − λ N , then ˜ λ, − ˜ λ, − ˜ Introduction λ should also be computed eigenvalues. Hamiltonian Eigenproblems Applications Goal cannot be achieved by general methods for matrices or matrix Symplectic pencils like the QR/QZ, Lanczos, Arnoldi algorithms! Lanczos The SR For an algorithm based on similarity transformations, the goal is Algorithm HKS achieved if the Hamiltonian structure is preserved. Quadratic Eigenvalue Problems Definition Conclusions and S ∈ R 2 n × 2 n is symplectic iff i.e., S − 1 = J T S T J . S T JS = J , Outlook References Lemma If H is Hamiltonian (skew-Hamiltonian) and S is symplectic, then S − 1 HS is Hamiltonian (skew-Hamiltonian), too.
Introduction Applications Hamiltonian eigenproblems arise in many different applications, e.g.: Hamiltonian Krylov-Schur Systems and control: Peter Benner Solution methods for algebraic and differential Riccati equations. Introduction Design of LQR/LQG/ H 2 / H ∞ controllers and filters for Hamiltonian Eigenproblems continuous-time linear control systems. Applications Stability radii and system norm computations; optimization of Symplectic Lanczos system norms. The SR Passivity-preserving model reduction based on balancing. Algorithm Reduced-order control for infinite-dim. systems based on inertial HKS manifolds. Quadratic Eigenvalue Computational physics: Problems exponential integrators for Hamiltonian dynamics. Conclusions and Outlook [ Eirola ’03, Lopez/Simoncini ’06 ] References Quantum chemistry: computing excitation energies in many-particle systems using random phase approximation (RPA). Quadratic eigenvalue problems...
Introduction Applications Hamiltonian Krylov-Schur Quadratic Eigenproblems with Hamiltonian Symmetry Peter Benner Introduction Q ( λ ) x := ( λ 2 M + λ G + K ) x = 0 , Hamiltonian Eigenproblems Applications where M = M T , K = K T , G = − G T , Symplectic Lanczos The SR These QEPs arise in Algorithm linear elasticity HKS Quadratic gyroscopic systems Eigenvalue Problems vibro-acoustics Conclusions and Outlook opto-electronics References
Introduction Applications Hamiltonian Krylov-Schur Quadratic Eigenproblems with Hamiltonian Symmetry Peter Benner Introduction Q ( λ ) x := ( λ 2 M + λ G + K ) x = 0 , Hamiltonian Eigenproblems Applications where M = M T , K = K T , G = − G T , Symplectic Lanczos The SR These QEPs arise in Algorithm linear elasticity HKS computation of corner singularities in 3D anisotropic Quadratic Eigenvalue elastic structures [ Apel/Mehrmann/Watkins ’01 ] ; Problems Conclusions and gyroscopic systems Outlook vibro-acoustics References opto-electronics
Introduction Applications Hamiltonian Krylov-Schur Quadratic Eigenproblems with Hamiltonian Symmetry Peter Benner Introduction Q ( λ ) x := ( λ 2 M + λ G + K ) x = 0 , Hamiltonian Eigenproblems Applications where M = M T , K = K T , G = − G T , Symplectic Lanczos The SR These QEPs arise in Algorithm linear elasticity HKS Quadratic gyroscopic systems Eigenvalue Problems used for modeling vibrations of spinning structures such as Conclusions and the simulation of tire noise, helicopter rotor blades, inertial Outlook navigation systems and components, or spin-stabilized References satellites with appended solar panels or antennas [ Lancaster ’99, Nackenhorst ’04, Elssel/Voss ’06, . . . ] ; vibro-acoustics opto-electronics
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