Ordinary Differential Equations and the Symmetric Eigenvalue Problem

Deift, Percy; Nanda, T.; Tomei, Carlos

doi:10.1137/0720001

Cited by 160 publications

(82 citation statements)

References 7 publications

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“…The seminal initial paper on the nonperiodic Toda flow is [21], where its gradient nature is demonstrated. Other early key papers include [8,22,23].…”

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confidence: 99%

On the optimality of double‐bracket flows

Bloch

Iserles

2004

International Journal of Mathematics and Mathematical Sciences

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“…The seminal initial paper on the nonperiodic Toda flow is [21], where its gradient nature is demonstrated. Other early key papers include [8,22,23].…”

mentioning

confidence: 99%

On the optimality of double‐bracket flows

Bloch

Iserles

2004

International Journal of Mathematics and Mathematical Sciences

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“…where L± denotes the strictly upper (respectively lower) triangular part of L. The corresponding equations (1) are known as the Toda Lattice first considered by Flaschka [2] and Moser [3] for a real symmetric tri di agona 1 matri x L. In [1] the authors used the Toda equati ons (1) to compute the eigenvalues of a tridiagonal symmetric matrix L .…”

Section: Matri X Consi Sting Of the Ei Genvalues Of La As T -+ Too mentioning

confidence: 99%

“…the flow (1) is isospectral. For certain very speci a 1 choices of the matri x B thi s system has another i nteresti ng feature: L(t) converges to a di agona 1 …”

Section: Introbuctionmentioning

confidence: 99%

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Differential Equations and the $QR$ Algorithm

Nanda¹

1985

SIAM J. Numer. Anal.

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In this paper we consider a variety of isos~ectral flows on the set of nx n matrices. These flows arise from Lax pairs and can all be interpreted in terms of the QR decomposition for nonsingular matrices. The asymptotics of these differential equations are considered in detail and for symmetric matrices these asymptotics provide a new method of solving the eigenvalue problem.-2- INTRObuCTION:In this article we consider the system of differential equations (1) (which will be called isospectral flows) for an nx n rea 1 matri x L(1)where La is an arbitrary nxn matrix and B(t) is an nxn skew symmetric matri x. The flow (1) has the property that the ei genva 1 ues of L(t) are independent of t, i.e. the flow (1)

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CMV: The unitary analogue of Jacobi matrices

Killip

Nenciu

2006

Comm Pure Appl Math

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We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices studied recently by Cantero, Moral, and Velázquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogues of well-known properties of Jacobi matrices: foliation by co-adjoint orbits, a natural symplectic structure, algorithmic reduction to this shape, Lax representation for an integrable lattice system (Ablowitz-Ladik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite Ablowitz-Ladik hierarchy and describe the long-time behavior of this system.

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Ordinary Differential Equations and the Symmetric Eigenvalue Problem

Cited by 160 publications

References 7 publications

On the optimality of double‐bracket flows

On the optimality of double‐bracket flows

Differential Equations and the $QR$ Algorithm

CMV: The unitary analogue of Jacobi matrices

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