M. P. Calvo, A. Iserles, and A. Zanna. Numerical solution of isospectral flows. Maths Comput., 66:1461--1486, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....R factors. The differential equation satisfied by Q is a nonlinear polynomial equation. From the numerical point of view, the key issue is how to properly integrate the Q equation, in particular how to make sure that the computed solution stays orthogonal. This issue has been recently addressed in [CaIsZa], DiRuVa1] DiVa] GoSuOr] Hi] and [MeRa] among others. The basic techniques in these works rest upon integrating the differential equations defining Q, so that, either directly, or through post processing with a projection step, the solution stays orthogonal. In this paper we look at a new ....

....ij ; i j 0; i = j Gamma(Q T AQ) ji ; i j : 2:4) So called automatic unitary integrators are those methods which automatically maintain orthogonality of Q when integrating (2. 3) Amongst the popular schemes, Gauss Runge Kutta (GRK) schemes are automatic integrators ( Coop] DiRuVa1] [CaIsZa]) The main (and possibly only) drawback with these schemes is the cost. One needs to solve a nonlinear matrix equation at each step, and unfortunately Newton s method appears prohibitively expensive; an orthogonality preserving linear iteration can be used (see [DiRuVa1, 4.2) 4.3) but for ....

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Calvo, M.P., Iserles, A. and Zanna, A.: "Numerical Solution of Isospectral Flows", Math. Comp., 66 (1997), pp. 1461-1486.

....from one coordinate representation to another involves B, and we will here discuss optimization of this computation. We use the DiffMan [4] free Lie algebra module to compute B. The algorithm used is given in [17] For order 6, the DiffMan bch program yields format rat; z = bch(6) z = [1] [2] 1 2 [1,2] 1 12 [1, 1,2] 1 12 [2, 1,2] 1 24 [2, 1, 1,2] 1 720 [1, 1, 1, 1,2] 1 180 [2, 1, 1, 1,2] 1 180 [2, 2, 1, 1,2] 1 720 [2, 2, 2, 1,2] 1 120 [ 1,2] 1, 1,2] 1 360 [ 1,2] 2, 1,2] 1 1440 [2, 1, 1, 1, 1,2] 1 360 [2, 2, 1, 1, 1,2] ....

....coordinate representation to another involves B, and we will here discuss optimization of this computation. We use the DiffMan [4] free Lie algebra module to compute B. The algorithm used is given in [17] For order 6, the DiffMan bch program yields format rat; z = bch(6) z = 1] 2] 1 2 [1,2] 1 12 [1, 1,2] 1 12 [2, 1,2] 1 24 [2, 1, 1,2] 1 720 [1, 1, 1, 1,2] 1 180 [2, 1, 1, 1,2] 1 180 [2, 2, 1, 1,2] 1 720 [2, 2, 2, 1,2] 1 120 [ 1,2] 1, 1,2] 1 360 [ 1,2] 2, 1,2] 1 1440 [2, 1, 1, 1, 1,2] 1 360 [2, 2, 1, 1, 1,2] ....

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M. P. Calvo, A. Iserles, and A. Zanna. Numerical solution of isospectral flows. Math. Comp., 66(220):1461--1486, 1997.

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Calvo, M.P., Iserles, A. and Zanna, A. (1995). Numerical solution of isospectral flows. DAMTP Tech. Rep. 1995/NA3, University of Cambridge.

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M.P. Calvo, A. Iserles and A. Zanna, "Numerical solution of isospectral flows", DAMTP Tech. Rep. 1995/NA03 (1995).

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M.P. Calvo, A. Iserles, A. Zanna, Numerical solution of isospectral flows, to appear in Maths Comput. (1997).

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M. P. Calvo, A. Iserles, and A. Zanna. Numerical Solution of Isospectral Flows. Technical Report NA1995.

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M.P. Calvo, A. Iserles & A. Zanna, "Numerical solution of isospectral flows", DAMTP Tech. Rep. 1995.

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Calvo, M.P., Iserles, A., Zanna, A.: Numerical solution of isospectral flows. Cambridge University Tech. Rep. 1995/NA3 (1995).

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M.P. Calvo, A. Iserles & A. Zanna, "Numerical solution of isospectral flows", DAMTP Tech. Rep. 1995.

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M. P. Calvo, A. Iserles & A. Zanna, "Numerical solution of isospectral flows", DAMTP Tech. Rep. 1995/NA03 (1995), University of Cambridge.

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M. P. Calvo, A. Iserles, and A. Zanna, Numerical solution of isospectral flows, Maths Comput. 66 (1997), 1461--1486.

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M. P. Calvo, A. Iserles, and A. Zanna. Numerical Solution of Isospectral Flows. Math. of Comp., 66:1461--1486, 1997.

....[IMKNZ00] In other words, the eigenvalues of Y (t) in (1.2) and, by implication, in the DBF (1.1) are invariants of the flow and do not vary as t increases. We note in passing that the equation (1.3) is the key to practical computation of the solution of (1. 2) whilst respecting its invariants [CIZ97, Ise02]. The second feature of the DBF (1.1) is that it is a gradient system, with a global Lyapunov function, therefore it is assured of convergence to a fixed point of the flow as t [Bro91] More precisely, as shown in [BBR92] BFR90] it is a gradient The work of AMB was supported in part by the ....

M. P. Calvo, A. Iserles, and A. Zanna, Numerical solution of isospectral flows, Maths Comput. 66 (1997), 1461--1486.

....0 = B(Y ) Y ] Y (0) Y 0 , we may thus solve X 0 = B(XY 0 X Gamma1 )X , X(0) I , and let Y = XY 0 X Gamma1 . This brings isospectral flows into the realm of geometric integration: it is possible to show that no classical numerical method can respect isospectral structure for m 3 [4], while methods that evolve in SO(m) are widely available, not least in this paper. An extensive survey of Lie group methods is available and it goes into considerable detail in the many aspects of this fast evolving subject [14] Our purpose in this paper is considerably more modest, to introduce ....

Calvo, M.P., Iserles, A. and Zanna, A. (1997). Numerical solution of isospectral flows, Maths Comput. 66, 1461--1486.

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M.P. Calvo, A. Iserles and A. Zanna, "Numerical solution of isospectral flows", DAMTP Techn. Rep. NA03/95, University of Cambridge, 1995.

.... Toda flow, associated with the Toda lattice equations governing the motion of the particles on a one dimensional lattice under exponential nearest neighbour interaction [24] When B(L) f(L) Gamma Gammaf (L) f being an analytic function of the spectrum of L, then we have the QR flow (see [6] or [27] for a list of references) When B(L) N; L] for a fixed diagonal matrix N , then we have the double bracket flow [2] and so on. It is well known that the solution of (2:1) is isospectral, namely the the spectrum of the integral curve L(t) of (2:1) does not change with time [24] The d ....

....d independent conditions, that can be chosen as tr(L r ) d X i=1 r i ; r = 1; d; 2. 2) 2 an alternative to the symmetric polynomials [22, 24] Such conditions are at the basis of the Calvo Iserles Zanna analysis of numerical methods and their retention of isospectrality [6]. They proved that classical numerical methods, such as multistep and Runge Kutta schemes cannot be isospectral for d 3, in the sense that, given a classical numerical method, it is always possible to construct an isospectral flow for which it fails to retain isospectrality. In the 2 Theta 2 ....

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M.P. Calvo, A. Iserles and A. Zanna, "Numerical solution of isospectral flows", DAMTP Techn. Rep. NA03, University of Cambridge, 1995, to appear in Math. of Comp.

.... obtained by means of the formulae ff j = 1 2 e Gamma(q j 1 Gammaq j ) 2 ; j = 1; 2; 3; fi j = 1 2 p j ; with the periodicity conditions ff j 3 = ff j ; fi j 3 = fi j ; j = 1; 2; 3; 7, 21] It is well known that classical numerical methods cannot retain isospectrality (see for instance [2, 25]) However, isospectrality can be recovered by solving numerically the orthogonal flow y 0 = fl(y)y; y(t n ) I ; t 2 [t n ; t n h] 4.5) 2 i.e. the eigenvalues of the solution L(t) do not change with time. 19 where fl(y) B(yLn y T ) with either an orthogonal method or with a ....

M. P. Calvo, A. Iserles, and A. Zanna. Numerical Solution of Isospectral Flows. Math. of Comp., 66:1461--1486, 1997.

.... Theta 0 = dexp Gamma1 Theta F (t; e Theta Y 0 ; y 0 ) Y 0 = F (t; Y; y 0 ) Y y 0 = y (F (t; y) Technical details were given by Munthe Kaas (1999) but the reader might easily work out explicitly the case of isospectral equations, whereby (X; Z) XZX Gamma1 , X;Z 2 GL(n; R) (Calvo, Iserles Zanna 1997, Zanna 1999) 3 Lie group solvers Inasmuch as we wish to focus in this paper on methods that adhere to the general scheme (1.2) it is worthwhile to mention methods that solve Lie group equations by different means. Perhaps the two most interesting examples are provided by the methods of rigid ....

....b n 3 7 7 7 7 7 7 7 5 P = 2 6 6 6 6 6 6 6 4 0 a 1 0 : 0 Gammaa 1 . 0 . 0 . an Gamma1 0 : 0 Gammaa n Gamma1 0 3 7 7 7 7 7 7 7 5 : Other sources of isospectral flows are numerical linear algebra and control theory (Calvo et al. 1997). The manifold of all matrices similar to L 0 2 Sym(n; R) is a homogeneous space, subject to the action of the orthogonal group O(n; R) namely (X; L) XLX T , X 2 O(n; R) L 2 Sym(n; R) We solve (5.9) with n = 30 using a fourth order RK MK method and compare expm with an approximant based on ....

Calvo, M. P., Iserles, A. & Zanna, A. (1997), `Numerical solution of isospectral flows', Maths Comp. 66, 1461--1486.

....subject to M = O, this crucial structural feature of orthogonal flows survives under discretization by a Runge Kutta method. This has been independently proved by Dieci, Russell and Van Vleck [11] while Calvo, Iserles and Zanna showed that the condition M = O is necessary (as well as sufficient) [4]. These results can be also extended with ease to flows on the Stiefel manifold. The behaviour of multistep methods in the present context is radically different. Theorem 2 (Calvo, Iserles Zanna, 7] For every multistep method (2:2) there exists a system (2:1) with a quadratic invariant which ....

....is assured. Insofar as M 2 is concerned, this is a quadratic conservation law, therefore Theorem 1 proves that M = O is sufficient for its retention. Moreover, it is possible to construct an isospectral problem (5:1) for which M = O is necessary as well for the retention of the quadratic invariant [4]. In the case of the third integral of motion, M 3 , it has been proved in [4] that the departure of an arbitrary Runge Kutta method from the manifold can be written in the form ffh 2 fih 3 O(h 3 ) where ff = 0 for all problems of the form (5:1) if and only if M = O, while fi = 0 for ....

[Article contains additional citation context not shown here]

M.P. Calvo, A. Iserles & A. Zanna (1995). Numerical solution of isospectral flows, DAMTP Tech. Rep. 1995/NA03, University of Cambridge.

.... obtained by means of the formulae ff j = 1 2 e Gamma(q j 1 Gammaq j ) 2 ; j = 1; 2; 3; fi j = 1 2 p j ; with the periodicity conditions ff j 3 = ff j ; fi j 3 = fi j ; j = 1; 2; 3; 7, 22] It is well known that classical numerical methods cannot retain isospectrality (see for instance [2, 26]) However, isospectrality can be recovered by solving numerically the orthogonal flow y 0 = fl(y)y; y(t n ) I ; t 2 [t n ; t n h] 4.5) where fl(y) B(yLn y T ) with either an orthogonal method or with a Lie group scheme. Thus, if Ln L(t n ) is given and yn 1 y(t n 1 ) is an ....

M. P. Calvo, A. Iserles, and A. Zanna. Numerical Solution of Isospectral Flows. Math. of Comp., 66:1461--1486, 1997.

....is well known, since it characterizes isospectral flows. The solution L(t) of (10) is similar to the initial condition L 0 , as it can be evinced by the group action, hence both L(t) and L 0 have the same spectrum. The numerical solution of these flows has been discussed with greater generality in [2], to which we refer the reader for further references. The knowledge of the fundamental solution U (t) that solves the Lie group equation (8) U 0 = B(L)U; U(0) I; 11) allows us to obtain the solution to (10) We have L(t) U (t) Delta L 0 = U (t)L 0 U (t) T : In this light, the ....

....refer the reader for further references. The knowledge of the fundamental solution U (t) that solves the Lie group equation (8) U 0 = B(L)U; U(0) I; 11) allows us to obtain the solution to (10) We have L(t) U (t) Delta L 0 = U (t)L 0 U (t) T : In this light, the approach proposed in [2] to solve numerically (10) whilst retaining isospectrality, can be depicted as finding the fundamental solution to (11) by means of an SO(d) invariant numerical method. This allowed the authors to introduce isospectral numerical methods, whereas classical methods for ODEs fail to retain the ....

Calvo, M.P., Iserles, A., Zanna, A.: Numerical solution of isospectral flows, DAMTP Techn. Rep. NA03/95, University of Cambridge, 1995.

....there has been an increasing interest in the design of numerical integration techniques which preserve important qualitative properties of differential equations. This includes the recent work on preserving the symplectic structure of Hamiltonian systems [15] orthogonality [5] isospectrality [2] and on designing numerical methods which stay on a prescribed manifold [3, 4, 6, 7, 11, 12, 13] In this paper we are concerned with the latter problem, in particular numerical integration of differential equations evolving on homogeneous spaces. The structure of a homogeneous space is both ....

....is the homogeneous space related to the action given by the matrix vector product (A; y) Ay. Thus, y 0 (V ) V y, and the general initial value problem on the sphere is y 0 t = f(y t )y t ; y 0 = p; where f : S n so(n 1) 4. 3 Isospectral Flows This topic is covered thoroughly in [2]. Let m 0 be a symmetric n Theta n matrix 3 , m 0 2 Sym(n) The matrix differential equation y 0 = f(y) y] f(y)y Gamma yf(y) y 0 = m 0 ; 6) where f(y) 2 so(n) is an example of an isospectral flow evolving in Sym(n) The adjective isospectral reflects the fact that the eigenvalues ....

Calvo, M.P., Iserles, A., Zanna, A.: Numerical solution of isospectral flows, DAMTP Techn. Rep. NA03/95, University of Cambridge, 1995.

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M. P. Calvo, A. Iserles, and A. Zanna. Numerical solution of isospectral flows. Maths Comput., 66:1461--1486, 1997.

No context found.

M.P. Calvo, A. Iserles and A. Zanna, Numerical solution of isospectral flows, Math. Comp. 66 (1997), 1461--1486.

No context found.

M. P. Calvo, A. Iserles, A. Zanna, Numerical solution of isospectral flows, Math. of Comp. 66 (1997) 1461--1486.

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