Lecture 6 Structure-Preserving Dynamical Systems

Transcription

1 Lecture 6 Structure-Preserving Dynamical Systems Algorithm Designs Moody T. Chu North Carolina State University June 14, MCM, CAS

2 Outline Staircase Structure QR Algorithm QZ Algorithm SVD Algorithm Lancaster Structure General Quadratic Model Structure Preserving Transformation! Flow Approach Optimal Control Hamiltonian Structure Basic Hamiltonian Matrix Canonical Form Building Flows Hamiltonian Pencils shh Flow Simply Hamiltonian Flows Group Structure

3 Outline Staircase Structure QR Algorithm QZ Algorithm SVD Algorithm Lancaster Structure General Quadratic Model Structure Preserving Transformation! Flow Approach Optimal Control Hamiltonian Structure Basic Hamiltonian Matrix Canonical Form Building Flows Hamiltonian Pencils shh Flow Simply Hamiltonian Flows Group Structure

4 Outline Staircase Structure QR Algorithm QZ Algorithm SVD Algorithm Lancaster Structure General Quadratic Model Structure Preserving Transformation! Flow Approach Optimal Control Hamiltonian Structure Basic Hamiltonian Matrix Canonical Form Building Flows Hamiltonian Pencils shh Flow Simply Hamiltonian Flows Group Structure

5 Outline Staircase Structure QR Algorithm QZ Algorithm SVD Algorithm Lancaster Structure General Quadratic Model Structure Preserving Transformation! Flow Approach Optimal Control Hamiltonian Structure Basic Hamiltonian Matrix Canonical Form Building Flows Hamiltonian Pencils shh Flow Simply Hamiltonian Flows Group Structure

6 Outline Staircase Structure QR Algorithm QZ Algorithm SVD Algorithm Lancaster Structure General Quadratic Model Structure Preserving Transformation! Flow Approach Optimal Control Hamiltonian Structure Basic Hamiltonian Matrix Canonical Form Building Flows Hamiltonian Pencils shh Flow Simply Hamiltonian Flows Group Structure

7 Why Search for Structure? Critical for retrieving latent information. Spectral decomposition for symmetric matrices. Singular value decomposition for rectangular matrices. Schur decomposition for general square matrices. Efficient for numerical computation. QR algorithm Upper Hessenberg structure. QZ algorithm Upper Hessenberg/triangular structure. SVD algorithm Bidiagonal structure. Improve physical feasibility and interpretability. Reduce information leakage or disturbance. Pejorative manifold (Kahan 72). The solution structure is lost when the problem leaves the manifold due to an arbitrary perturbation. The problem may not be sensitive at all if the problem stays on the manifold, unless it is near another pejorative manifold.

8 Two Questions Given a dynamical system, what are the structures invariant under the the flow? Given a set of structures related to a fixed matrix, can a dynamical system, discrete or continuous, be designed to preserve the specified structures?

9 Given A = [a ij ] R m n, Define Staircase Structure j ff t k (A) := max k, max {i a ik 0}, k = 1,..., n. k<i m A is in staircase form if and only if t k (A) t k+1 (A), k = 1,..., n 1. Examples with step indices {1, 3, 4, 4, 5}: , } {{ } full staircase

10 QR Algorithm (Arbenz & Golub 95) Assume that A 0 is symmetric and {A k } are the iterates generated by the QR algorithm. 1. If A 0 is reducible by some permutation matrix P, that is,» PA 0 P A01 A = 02, 0 A 03 then so is each A k by means of the same permutation P. 2. If A 0 is irreducible, then the zero pattern of A 0 is preserved throughout {A k } if and only if A 0 is a full staircase matrix.

11 Zero Pattern and Irreducibility Two nearly identical matrices: , Differ only at the (1, 6) and (6, 1) positions. No significant staircase form. Totally different dynamics: Zero pattern for the left matrix is preserved because it is reducible, Zero pattern for the right matrix is totally destroyed even after one iteration.

12 Preserving Staircase For non-symmetric matrices, Reducibility cannot be preserved. If A 0 is in the staircase form, then so is {A k } throughout the QR algorithm. If X 0 is in the staircase form, then so is X(t) throughout the Toda lattice. The staircase form preservation between the QR algorithm and the Toda lattice is not directly related. Even if X 0 is in the staircase form, the corresponding A 0 = exp(x 0 ) may not be.

13 QZ Algorithm Generalized eigenvalue problem, A 0 x = λb 0 x. First reduce A 0 to an upper Hessenberg form and B 0 to an upper triangular form. Orthogonal equivalence transformations are used. Critical components: Simulate the effect of the QR algorithm on the matrix B 1 0 A 0 without explicitly forming the inverse or the product. Throughout the iteration, preserve the upper Hessenberg/triangular structure.

14 QZ Flow Consider a smooth orthogonal equivalence transformation on the pencil B 0 λ A 0, L (t) = Q(t)(B 0 λ A 0 )Z (t), Q(t), Z (t) O(n). Dynamical system for the isospectral flow L (t) dl dt = L R LL, L (0) = B 0 λ A 0, Dynamical system for the coordinate transformations ( dq = LQ, dt dz dt = ZR, L, R o(n). The choice of skew-symmetric matrix parameters L(t) and R(t) determines the dynamics.

15 Preserving Upper Hesserberg/Triangularity Write { X(t) = Q(t)A0 Z (t), Y (t) = Q(t)B 0 Z (t). Mimic the QZ algorithm. Choose L(t) and R(t) so that dx / dy remain upper dt dt Hessenberg/triangular whenever X(t)/Y (t) are. Many choices. Out of naïveté but with proper symmetry, { L := Π0 (XY 1 ), The QZ flow: dl dt R := Π 0 (Y 1 X). = L Π 0 (Y 1 X) Π 0 (XY 1 )L, L (0) = B 0 λ A 0.

16 Related to the Toda Lattice If X(t)/Y (t) are upper Hessenberg/triangular, then both L(t) and R(t) are tridiagonal. Define { E(t) := X(t)Y 1 (t), then F (t) := Y 1 (t)x(t), { de dt = [E, Π 0 (E)], df dt = [F, Π 0 (F)]. The QZ flow is related to the QZ algorithm in the same way as the Toda flow is related to the QR algorithm.

17 Conjecture 1 The QZ flow was designed solely for the purpose of maintaining the upper Hessenberg/triangular form. If both A 0 and B 0 are staircase matrices, not necessarily of the same pattern, then the structures of A 0 and B 0 are preserved by X(t) and Y (t), respectively, under the QZ flow. Observed numerically, but no formal proof.

18 Algebraic Manipulation? Direct manipulation is hard. A 0 = , B 0 = , Y 1 is usually full and dense. The QZ flow is somehow able to mix and then separate the different staircase forms.

19 SVD Algorithm First reduce A 0 to a bidiagonal matrix via orthogonal equivalence transformations. Critical components: Performing the QR algorithm on the product A 0 A 0 without explicitly forming the product. The bidiagonal structure is preserved throughout the iteration.

20 Assume SVD Flow X(t) = U(t)B 0 V (t), U(t) O(m), V (t) O(n). Necessary format: dx dt = XR LX, X(0) = B 0. Coordinate transformation: ( du dt = LU, dv dt = VR, L, R o(n). How to choose skew-symmetric matrix parameters L(t) and R(t)?

21 Want Maintain the Bidiagonal Structure X(t) remains bidiagonal for all t. L(t), R(t) are tridiagonal and skew-symmetric. Good convergence. Among many other choices, L = Π 0 (XX ), R = Π 0 (X X). The gradient flow will reduce the off-diagonal magnitude but will not keep the bidiagonal structure. L = 1 ( X diag(x) diag(x) X ), 2 R = 1 ( Xdiag(X) diag(x)x ). 2

22 Related to the Toda Lattice Define Y (t) = X (t)x(t). Then dy dt = [Y, Π 0 (Y )]. Convergence follows from the Toda dynamics.

23 Conjecture 2 The Lokta-Volterra system was discovered with the preservation of the bidiagonal form in mind. Suppose B 0 is a staircase matrix. Then the SVD flow B(t) defined by the Lokta-Volterra equation and the corresponding SVD algorithm maintains the same staircase structure. For small size matrices, the validity can be proved by an ad hoc calculation.

24 Second-Order Vibration System Dynamical system with n-degree-of-freedom: Mẍ + (C + G)ẋ + (K + N)x = F. Some interpretations: M := Mass matrix M = M 0. C := Damping matrix C = C. K := Stiffness matrix K = K 0. G := Gyroscopic matrix G = G. N := Dissipation matrix N = N. F := External force.

25 Quadratic Eigenvalue Problem Assume the homogeneous solution x(t): x = e λt u. Look for nontrivial solution to the QEP: Q(λ)u := (λ 2 M + λc + K )u = 0. If M is nonsingular, then there are 2n eigenpairs. Many applications. Many numerical techniques.

26 Model Reduction Can the original n-degree-of-freedom system be reduced to n totally independent single-degree-of-freedom subsystems? Must maintain isospectrality. Must be done via real-valued transformation.

27 Common Knowledge Reduction means simultaneous diagonalization. In general, it is impossible to diagonalize three matrices M, C, and K simultaneously. Those can be done are called proportionally or classically clamped very limited. Is simultaneous diagonalization the wrong question to ask? Any other way to achieve the reduction?

28 Symmetric Linearization Lancaster pair: L(λ) := L(λ; M, C, K ) = [ C M M 0 ] [ K 0 λ 0 M ]. Equivalence between Q(λ) and L(λ). ([ ] [ ]) [ C M K 0 u λ M 0 0 M v { (λc + K )u + λmv = 0, λmu Mv = 0. ] = 0 If M is nonsingular, then v = λu.

29 Structure Preserving Transformation Look for nonsingular matrices Π l, Π r R 2n 2n such that Lancaster structure is preserved:» CD M Π l L(λ)Π r = L(λ; M D, C D, K D ) = D M D 0 M D, C D and K D are all diagonal matrices, Isospectral equivalence: ( λ 2 M D + λc D + K D ) z = 0 [ z λz ]» KD 0 λ 0 M D = Π r [ u λu ],.

30 Not a Conventional Transformation Write Π l = [ π [l] 11 π [l] 12 π [l] 21 π [l] 22 ], Π r = [ π [r] 11 π [r] 12 π [r] 21 π [r] 22 ]. π [l] ij, π [r] ij R n n. Do the structure preserving transformations Π l and Π r exist? Can the transformations Π l and Π r be real-valued? Is there any relationship between Π l and Π r?, say, Π l = Π r? How to find the real-valued transformations Π l and Π r numerically?

31 Nonlinear Algebraic System To maintain the Lancaster structure: π [l] 11 K π[r] 12 + π[l] 12 Mπ[r] 22 = 0, π [l] 21 K π[r] 11 + π[l] 22 Mπ[r] 21 = 0, π [l] 21 Cπ[r] 12 + π[l] 22 Mπ[r] 12 + π[l] 21 Mπ[r] 22 = 0, π [l] 11 Cπ[r] 12 + π[l] 12 Mπ[r] 12 + π[l] 11 Mπ[r] 22 = π [l] 21 Cπ[r] 11 + π[l] 22 Mπ[r] 11 + π[l] 21 Mπ[r] 21 = π [l] 21 K π[r] 12 + π[l] 22 Mπ[r] 22. To attain the diagonal form: π [l] 21 K π[r] 12 + π[l] 22 Mπ[r] 22 = M D, π [l] 11 Cπ[r] 11 + π[l] 12 Mπ[r] 11 + π[l] 11 Mπ[r] 21 = C D, π [l] 11 K π[r] 11 π[l] 12 Mπ[r] 21 = K D, A nonlinear algebraic system of 8n 2 3n equations in 8n 2 unknowns.

32 Existence For almost all regular quadratic pencils, Real-valued equivalence transformations Π l and Π r do exist. (Garvey, Friswell, & Prells, 02), has flaws and is incomplete. (Chu & Del Buono, 05), simpler and complete proof. For self-adjoint quadratic pencils, Π l = Π r. This is congruence transformation. Proof is based on the availability of complete spectral information. Not numerically feasible. Any constructive way to establish Π l and Π r?

33 Moving Frame Denote A 0 := [ ] [ K0 0 C0 M, B 0 M 0 := 0 0 M 0 0 ]. Assume the transformation changes as a one-parameter family: { A(t) = T l (t)a 0 T r (t), B(t) = T l (t)b 0T r (t). subject to the rule: [ l11 (t) l Ṫ l (t) = T l (t)l(t) = T l (t) 12 (t) l 21 (t) l 22 (t) [ r11 (t) r Ṫ r (t) = T r (t)r(t) = T r (t) 12 (t) r 21 (t) r 22 (t) ], ].

34 Equivalence Flow The transformation is governed by { da dt = AR + L A, db dt = BR + L B. L(t) and R(t) effectuate the dynamical behavior. This is an isospectral flow. Need to preserve the Lancaster structure.

35 Determining the Vector Field To maintain the Lancaster structure for A(t) and B(t): l 21M Kr 12 = 0, l 12K + Mr 21 = 0, l 12M + Mr 12 = 0, l 11M + Cr 12 + Mr 22 = l 12C + l 22M + Mr 11 = l 22M + Mr 22. There are 5n 2 equations in 8n 2 unknowns Can be solved in terms of three matrix parameters.

36 Forming L(t) and R(t) r 12 = DM, l 21 = D K, l 12 = D M, r 21 = DK, r 11 r 22 = DC, l 11 l 22 = D C. One possible formation: [ ] [ ] [ r11 (t) r 12 (t) D 0 C = 2 M r 21 (t) r 22 (t) 0 D C K 2 [ ] [ ] [ l11 (t) l 12 (t) D C 0 = l 21 (t) l 22 (t) 0 D 2 M K ] [ Nr 0 + C 2 Determined up to three free parameters D, N l and N r. ] 0 N r ], + [ ] Nl 0. 0 N l

37 Isospectral Flow The corresponding flow (Garvey et al,04): Ṁ = 1 2 (MDC CDM) + MN r + N l M, Ċ = (MDK KDM) + CN r + N l C, K = 1 2 (CDK KDC) + KN r + N l K. How to choose D, N l and N r so as to attain convergence?

38 Maintaining Symmetry Assume (M(0), C(0), K (0)) has some symmetry to begin with. Take N r (t) = N l (t). Then symmetry is preserved: D(t) M(t) C(t) K (t) skew-symmetric symmetric symmetric symmetric symmetric symmetric skew-symmetric symmetric symmetric skew-symmetric skew-symmetric skew-symmetric skew-symmetric skew-symmetric symmetric skew-symmetric. Still..., need to control the convergence.

39 A Control Problem An open-loop control: minimize f (x), subject to ẋ = g(x)u, x(0) = x 0, u = control. A possible control: A closed-loop control: u = g(x) f (x). ẋ = g(x)g(x) f (x), x(0) = x 0. This is a gradient flow!

40 Objective Function Minimize Subject to F(M, C, K ) := M 2 F (1 + δ) diag(m) 2 F + C 2 F (1 + δ) diag(c) 2 F + K 2 F (1 + δ) diag(k ) 2 F. Ṁ = 1 2 (MDC CDM) + MN + N M, Ċ = (MDK KDM) + CN + N C, K = 1 2 (CDK KDC) + KN + N K. (D, N) = control.

41 Basic Ideas While minimizing off-diagonal entries of (M, C, K ), also penalize growth of diagonal entries by a factor of δ. Assume (M 0, C 0, K 0 ) are all symmetric and, hence, N l = N r and D = D. Tangent vectors in the orbit of equivalence at (M, C, K ) are linear in the control parameters (D, N). Need to rewrite the vector field in terms of an outer product form.

42 Structure Preserving Isospectral Flow The resulting (M(t), C(t), K (t)) has the following properties: It is isospectral to (M 0, C 0, K 0 ). It preserves the Lancaster structure implicitly. It moves in the direction to minimize the off-diagonal entries while keeping the diagonal entries at bay. Ideally, (M(t), C(t), K (t)) converges to (M D, C D, K D ).

43 Hamiltonian Structure Define J := [ 0 In I n 0 ]. J 2 = I. H R 2n 2n is Hamiltonian (HJ) = HJ. H has the structure:» M P H = Q M, P and Q are symmetric. W R 2n 2n is skew-hamiltonian (WJ) = WJ. W has the structure:» M F W = G M, F and G are skew-symmetric.

44 Importance of Hamiltonian Structure Many applications: Systems and controls. Algebraic Riccati equations. Quadratic eigenvalue problems. Structures carry underlying physical settings. Many inherent properties: Eigenvalues of H are symmetric with respect to the imaginary axis. Eigenvalues of W have even algebraic and geometric multiplicities.

45 Preserving Hamiltonian Structure Conventional algorithms usually fail to preserve the Hamiltonian structure. Considerable research effort in deriving special methods for matrices with Hamiltonian structure. Iterative procedures are carefully carved, but usually complicated. Most Hamiltonian structure-preserving dynamical systems can be characterized as a single line equation. Strong numerical evidence for convergence. Lack complete asymptotic analysis.

46 Symplectic Group S R 2n 2n is symplectic S JS = J. Natural symmetry SJS = J. From a matrix group Sp(2n). S 1 = JS J. g = T I2n Sp(2n) = {all Hamiltonian matrices}. Hamiltonian matrices as tangent vectors to Sp(2n) is analogous to skew-symmetric matrices to O(n).

47 Schur-Hamiltonian Form Given Hamiltonian H with no purely imaginary eigenvalues, There exists an orthogonal symplectic matrix U R 2n 2n such that eh = U HU is Hamiltonian, and is of the form eh =» R P 0 R P is symmetric and R is upper quasitriangular. Given skew-hamiltonian W, There exists an orthogonal symplectic matrix U R 2n 2n such that fw = U WU is skew-hamiltonian, and is of the form fw =» R F 0 R F is skew-symmetric and R is upper quasitriangular.,,

48 URV Form Given Hamiltonian H, There exist orthogonal symplectic matrices U, V R 2n 2n such that H b = U HV is of the form» T N bh = 0 R, N has no particular structure, T is upper triangular and R is upper quasitriangular.

49 Hamiltonian Eigenvalue Computation Critical components: Reduce a matrix of Hamiltonian structure to its Schur-Hamiltonian form. Employ classical iterative schemes to the reduced eigenproblem. Stable eigenvalue computation procedures for skew-hamiltonian matrices are well developed (Benner et al. 05, Van Loan, 84). Much harder task for For Hamiltonian matricesr. H 2 is skew-hamiltonian. By URV,» U H 2 TR TN NT U = 0 R T Eigenvalues of H are the square roots of those from TR. A QZ -type algorithm can be applied to find the eigenvalues of the product TR without explicitly forming the product..

50 Symplectic Flow A smooth curve S(t) on the manifold of symplectic group Sp(2n) is necessarily governed by ds dt = SK, (or KS), K is Hamiltonian. If the symplectic S(t) is also orthogonal, then [ ] M Q K =, Q M M is skew-symmetric and Q is symmetric.

51 Hamiltonian Flow Given H 0 R 2n 2n, consider the Lax dynamics, dx dt = [X, P 0 (X)], X(0) = H 0, P 0 acting on X is defined by» 0 X P 0 (X) := 21 X 21 0» X11 X, if X = 12 X 21 X 22. Corresponding parameter dynamical system, dg dt = gp 0 (X), g(0) = I 2n. P 0 (X) is Hamiltonian g(t) is orthogonal symplectic. H 0 is Hamiltonian X(t) = g (t)h 0 g(t) remains Hamiltonian. X 21 (t) 0 as t (Chu & Norris 88). The limit point is not exactly of the Schur-Hamiltonian form yet. The flow approach is remarkably simple.

52 Skew-Hamiltonian Flow X is skew-hamiltonian P 0 (X) is not Hamiltonian. Skew-Hamiltonian eigenproblem is supposed to be relatively easier than the Hamiltonian eigenproblem by iterative methods. Every real skew-hamiltonian matrix has a real Hamiltonian square root (Faßbender et al. 99). Given a skew-hamiltonian matrix W 0, define H 0 := W 1/2 0. Apply the Hamiltonian flow to obtain X(t) W(t) := X 2 (t) is skew-hamiltonian and converges to an upper block triangular form. The very same parameter g(t) serves as the continuous coordinate transformation for W(t) = g (t)w 0 g(t) and leads to convergence. Symbolic dynamical system, dw dt = [W, P 0 (W 1/2 )], W(0) = W 0, A skew-hamiltonian matrix W has infinitely many Hamiltonian square roots.

53 Using Π 0 (X) only Conjecture 3 Convergence to the real Schur form. Cannot preserve Hamiltonian structure. Define P 1 (X) := [ Π0 (X 11 ) X 21 X 21 Π 0 (X 11 ) ] Appears to be a compromise. P 1 for a Hamiltonian matrix X differ from Π 0 only in the (2, 2)-block. Toda-Hamiltonian flow: dh dt = [H, P 1 (H)], H(0) = H 0. Suppose H 0 is Hamiltonian with no purely imaginary eigenvalues. Then the Toda-Hamiltonian flow H(t) remains Hamiltonian and converges to the real Schur-Hamiltonian form.

54 URV Flow A flow X(t) = U (t)x 0 V (t) is necessarily governed by the system dx dt = XR LX, X(0) = X 0, (1) L and R to be determined. Similar to SVD and QZ flows. Same U transformation in the real Schur-Hamiltonian form for H 0 L = P 1 (U H 0 U). Same V transformation in the lower quasitriangular Schur-Hamiltonian for H0 R = P 2(V H0 V ). Define P 2 (X) :=» Π0 (X11) X 12 X 12 Π 0 (X11) Rewrite as the autonomous dynamical system, dx dt = XP 2 ((X JXJ) 1/2 ) P 1 ((XJX J) 1/2 )X, X(0) = H 0..

55 Three Types of Hamiltonian Pencils A linear pencil Bλ A is simply Hamiltonian BJA = AJB Equivalent to B 1 A being Hamiltonian, if B 1 exists. Has {λ, λ, λ, λ} as eigenvalues. A linear pencil Bλ A is shh B is skew-hamiltonian and A is Hamiltonian. Arise in gyroscopic systems, structural mechanics, linear response theory, and quadratic optimal control (Benner et al. 02). A linear pencil Bλ A is HH Both A and B are Hamiltonian. Rare in applications.

56 Preserving Isospectrality Assume Necessary format: H(t) = Q(t) (B 0 λ A 0 ) Z (t). dh dt = HR LH, H(0) = B 0 λ A 0. Coordinate transformation: ( dq dt = LQ, dz dt = ZR. So far, this is similar to the QZ flow.

57 Choices of R and L Mimicking the QZ flow, Choose R to be as much like Π 0 (B 1 A) as possible. Choose L to be as much like Π 0 (AB 1 ) as possible. Must subject to the structure preserving limitation. Further restrictions on R and L.

58 Isospectral shh Flow Write L (t) = W(t)λ H(t). WR LW remains skew-hamiltonian. HR LH remain Hamiltonian. Suffice to consider L = JR J. Q(t) and Z (t) are interchangeable. j Z (t) = JQ (t)j, Q(t) = JZ (t)j. Only one coordinate transformation is needed.

59 Mimicking QZ but Keeping shh Given 2n 2n matrix X, define [ Π0 (X P 4 (X) := 11 ) X21 X 21 Π 0 (X22 ) ]. Almost identical to Π 0 (X) except for a twist" at the (2, 2) block P 4 (X) = [...]

60 Modified shh Flow Define dl dt = L P 4 (W 1 H) P 4 (HW 1 ) L, L (0) = B 0 λ A 0, }{{}}{{} R L Inherent relationship: shh pencil HW 1 = J(W 1 H) J shh structure preserving. Only need to work with R. L (t) = JZ (t)j(w 0 λ H 0 )Z (t).

61 Conjecture 4 Suppose L (0) is an shh pencil. The flow L(t) with R := P 4 (W 1 H) maintains the shh structure and converges to the canonical form [ ] [ ] W11 W12 H11 H12 L = λ 0 W 11 0 H 11. f W11 and e H 11 are upper quasitriangular. f W12 is skew-symmetric. e H12 is symmetric. The canonical form is the same as that desirable in the literature (Benner et al. 02). Extremely complicated iterative procedure. If the convergence can be proved, then we have a very simple way to realize the canonical form.

62 Maintaining Simply Hamiltonian Bλ A is Hamiltonian if and only if Q(Bλ A)Z is Hamiltonian for arbitrary nonsingular Q and symplectic Z. To maintain the Hamiltonian structure, No restriction on L. R must be Hamiltonian.

63 Simply Hamiltonian Flow Both B 1 A and A 1 B are Hamiltonian, but AB 1 and BA 1 are not. Take { R = P1 (B 1 A) Simply Hamiltonian flow: L = Π 0 (AB 1 ). dl dt = L P 1 (B 1 A) Π 0 (AB 1 )L. Differ from the QZ flow only a P 1. Maintain the simply Hamiltonian structure.

64 Conjecture 5 B 0 λ A 0 has no purely imaginary eigenvalues L (t) converges to the canonical form [ ] [ ] B11 B12 Â11 Â L = λ 12, 0 B22 0 Â 22 b A11 and b B 11 are upper quasitriangular matrices with corresponding 1 1 or 2 2 blocks. b A22 and b B 22 are upper-left quasitriangular matrices with corresponding 1 1 or 2 2 blocks. B 0 λ A 0 has one pair of purely imaginary eigenvalues L (t) converges to the same canonical form as above, with the exception of a non-zero entry at the (n + 1, n) position which is periodic in t.

65 Initial Structure Dynamical System Limiting Behavior Operator X 0 =staircase Ẋ = [X, Π 0 (X)] Ashlock et al. 97 Π 0 (X) := X (X ) B 0 λ A 0 = staircase L = L Π 0 (Y 1 X) Π 0 (XY 1 )L Conjecture 1 B 0 = staircase Ḃ = BΠ 0 (B B) Π 0 (BB )B Conjecture 2 B 0 λ A 0 = Lancaster K = 1 2 (CDK KDC) + N L K + KN R D, N R, N L :=controls Ċ = (MDK KDM) + N L C + CN R Ṁ = 1 2 (MDC CDM) + N L M + MN R H 0 = Hamiltonian Ḣ = [H, P 0 (H)] Chu et al. 88 P 0 (X) := " 0 X 21 X 21 0 # W 0 = skew-hamiltonian Ẇ = [W, P 0 (W 1/2 )] H 0 = Hamiltonian Ḣ = [H, P 1 (H)] Conjecture 3 P 1 (X) :=» Π0 (X 11 ) X 21 X 21 Π 0 (X 11 ) W 0 = skew-hamiltonian Ẇ = [W, P 1 (W 1/2 )] X 0 = general Ẋ = XP 3 (X X) P 3 (XX )X Chu el al. 88 P 3 :=generalized P 0 H 0 = Hamiltonian Ẋ = XP 2 ((X JXJ) 1/2 ) P 1 ((XJX J) 1/2 )X URV flow " Π0 (X P 2 (X) := 11 ) X # 12 X 12 Π 0 (X 11 ) W 0 λ H 0 = shh " L = L P 4 (W 1 H) P 4 (HW 1 Π0 (X )L Conjecture 4 P 4 (X) := 11 ) X 21 # X 21 Π 0 (X 22 ) B 0 λ A 0 = Hamiltonian L = L P 1 (B 1 A) Π 0 (AB 1 )L Conjecture 5 B 0 λ A 0 = general Ȧ = AP 2 ((A B JB 1 AJ) 1/2 ) P 4 (AB 1 )A Not tested Ḃ = BP ((B 1 AJA B J) 1/2 ) P (AB 1 )A

66 HH Pencil Inherent relationships! AB 1 = J(B 1 A) J. A choice similar to that for the shh pencil will not work It misses a negative sign. Nor sure what the Hamiltonian Schur form is for the HH pencil. Not all Hamiltonian matrices have a Hamiltonian Schur form. Would it work if we choose R = P 1 (B 1 A), L = P 1 (AB 1 )?

67 One Final Question For all the Hamiltonian flows, is the staircase structure still preserved?

68 Generalizing into Manifolds Far too many applications where it is desirable that a specific structure is maintained throughout an evolving process. The notion of structure" should be interpreted quite liberally. Preserving volume, momentum, energy, symplecticity, or other kinds of physical quantities is an extremely important task with significant consequences. Lie theory is now a ubiquitous framework in many disciplines of sciences and engineering applications. Dynamical systems and numerical algorithms originally developed over Euclidean space need to be redeveloped over manifolds. Newton and the conjugate gradient methods have been generalized to the Grassmann and the Stiefel manifolds (Edelman et al. 99).

69 Newton Dynamics on a Lie Group (Owren & Welfert 00) The problem: Given a Lie group G and its corresponding Lie algebra g, want to find zeros(s)" of the map A typical Newton scheme: f : G g. Solve for a tangent vector u n T yn G via the linear equation Update y n to y n+1 via u n. df yn (u n) + f (y n) = 0.

70 Interpretation Bring back to local coordinates: All local charts of a Lie group can be obtained by translation. T y G = yg. Consider a representation of f restricted to a local chart at y n. f := f Lyn exp, L y (z) = yz. A classical Newton iteration over the Euclidean space. d f vn (u n ) + f (v n ) = 0. v n = ln y n. v n+1 = v n + u n. Lift to the new iterate on the manifold G. y n+1 = y n exp(u n).

71 Generalizations Under classical assumptions the proposed methods converge quadratically. This framework can be repeatedly applied to generalize other types of algorithms originally designed for Euclidean space to Lie groups. How far this generalization should go, and how practical such extensions might be, are yet to be seen.