## On the Discretization of Double-Bracket Flows (2001)

Venue: | Found. Comput. Math |

Citations: | 13 - 4 self |

### BibTeX

@ARTICLE{Iserles01onthe,

author = {Arieh Iserles},

title = {On the Discretization of Double-Bracket Flows},

journal = {Found. Comput. Math},

year = {2001},

volume = {2},

pages = {305--329}

}

### OpenURL

### Abstract

This paper extends the method of Magnus series to Lie-algebraic equations originating in double-bracket flows. We show that the solution of the isospectral flow Y = [[Y; N ]; Y ], Y (0) = Y0 2 Sym(n), can be represented in the form Y (t) = e \Omega\Gamma Y0e , where the Taylor expansion of\Omega can be constructed explicitly, term-by-term, identifying individual expansion terms with certain rooted trees with bicolour leaves. This approach is extended to other Lie-algebraic equations that can be appropriately expressed in terms of a finite `alphabet'. 1

### Citations

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99 |
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(Show Context)
Citation Context ...diate consequence is that, provided that Y 0 itself is not in the commutative Banach algebra generated by N (in which case Y (t) j Y 0 ), a fixed pointsY = lim t!1 Y (t) is a (local) minimiser of OE (=-=Brockett 1991-=-). This minimality of (1.1) makes it relevant to a wide range of applications. The most obvious is computing eigenvalues of symmetric matrices: If N is a diagonal matrix with distinct diagonal element... |

80 |
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(Show Context)
Citation Context ...g grades in the usual way, ([X 1 ; X 2 ]) = (X 1 ) + (X 2 ), where (X i ) is the grade of X i , the Witt--Birkhoff formula gives the dimension of the linear space spanned by all terms of given grade (=-=Bourbaki 1975-=-). This, however, although immensely useful in reducing the computational cost for standard Magnus expansions (Iserles et al. 2000), falls short of being a sufficiently powerful tool to help in reduci... |

71 | Rungeâ€“Kutta methods on Lie groups - Munthe-Kaas - 1998 |

51 | Computations in a free Lie algebra - Munthe-Kaas, Owren - 1999 |

49 | Numerical solution of isospectral flows - Calvo, Iserles, et al. - 1997 |

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(Show Context)
Citation Context ...inear programming problems (Brockett 1991) and, perhaps with greater relevance to practical linearalgebra computations, calculating inverse eigenvalue and least squares problems (Chu & Driessel 1990, =-=Chu 1998-=-). A special choice of N corresponds to a variant of the familiar Toda lattice equations (Bloch 1990). An obvious means of discretising (1.2) is by standard numerical methods, e.g. Runge--Kutta or mul... |

27 | Magnus and Fer expansions for matrix differential equations: the convergence problem - Blanes, Casas, et al. - 1998 |

22 |
Steepest descent, linear programming and Hamiltonian flows
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(Show Context)
Citation Context ...algebra computations, calculating inverse eigenvalue and least squares problems (Chu & Driessel 1990, Chu 1998). A special choice of N corresponds to a variant of the familiar Toda lattice equations (=-=Bloch 1990-=-). An obvious means of discretising (1.2) is by standard numerical methods, e.g. Runge--Kutta or multistep. Unfortunately, for ns3 this is bound to lead to the loss of the most important structural fe... |

18 | E.: Double Bracket Equations and Geodesic Flows on Symmetric Spaces - Bloch, Brockett, et al. - 1997 |

15 |
On the numerical solution of isospectral flows
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(Show Context)
Citation Context ...ential in applications, it is vital to discretise (1.2) with a method that respects this feature. Such methods, which have been recently introduced in (Calvo et al. 1997, Calvo, Iserles & Zanna 1999, =-=Zanna 1998-=-), share a common denominator: they all regard the isospectral orbit I Y0 ae Sym(n) of all symmetric matrices similar to Y 0 as a homogeneous space, subjected to the transitive SO(n) action (Q)X = QXQ... |

12 |
Lie-group methods, Acta Numerica 9
- Iserles, Munthe-Kaas, et al.
- 2000
(Show Context)
Citation Context ...at YN+1 cannot be any longer guaranteed to evolve in Sym(n) -- unless, there is, QN+1 is itself orthogonal.) No classical numerical methods can respect the Lie-group structure of (1.6) for general G (=-=Iserles et al. 2000-=-). In the case of orthogonal flows (1.4), exceptionally, symplectic Runge--Kutta methods preserve orthogonality (Dieci, Russell & van Vleck 1994). However, they are implicit, thereby expensive. An alt... |

11 |
On the solution of linear differential equations in Lie groups
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(Show Context)
Citation Context ...; A] =) h R t 0 [ R x 0 A; A]; [ R t 0 A; A] i =) R t 0 \ThetaR x 1 0 [ R x 2 0 A; A]; [ R x 1 0 A; A] : The coefficients in (1.8) can be also obtained recursively. We note in passing that (Iserles & =-=Nrsett 1999-=-) and more recent publications contain a wealth of further material, e.g. on efficient numerical approximation of multivariate integrals in the Magnus expansion. The reader is referred to (Iserles et ... |

3 | Optimal control and geodesic flows - Bloch, Crouch - 1996 |

2 | Conservative methods for the Toda lattice equations - Calvo, Iserles, et al. - 1996 |