On an integrable discretization of the Rayleigh quotient gradient system and the power method with a shift

Nakamura, Yoshimasa; Kajiwara, Kenji; Shiotani, Heinosuke

doi:10.1016/s0377-0427(98)00084-3

Cited by 9 publications

(8 citation statements)

References 28 publications

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“…Simultaneously, the first column of Q converges to a normalized eigenvector of the maximal eigenvalue µ 1 of the symmetric matrix N . It is shown in [32] that by discretizing the Rayleigh quotient gradient system a recurrence relation of the power method with shift, for computing largest eigenvalue and its eigenvector, is derived. The optimal shift is then found by using an explicit solution.…”

Section: Integrable Gradient Flows Of Brockett Formmentioning

confidence: 99%

“…Such discretization has explicit solutions, conserved quantities and some other integrable features and is sometimes called an "integrable discretization" [30]. Indeed in [32] we see that an integrable discretization of the Rayleigh quotient gradient system gives rise to a recurrence relation of the power method with shift. The basic idea is as follows.…”

Section: Discretizations Of Integrable Systems To Design Numerical Almentioning

confidence: 99%

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A new approach to numerical algorithms in terms of integrable systems

Nakamura¹

International Conference on Informatics Research for Development of Knowledge Society Infrastructure, 2004. ICKS 2004.

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Almost four decades passed after the discovery of solitons and infinite dimensional integrable systems. The theory of integrable systems has had great impact to wide area in physics and mathematics. In this paper an approach to numerical algorithms in terms of integrable systems is surveyed. Some integrable systems of Lax form describe continuous flows of efficient numerical algorithms, for example, the QR algorithm and the Jacobi algorithm. Discretizations of integrable systems in tau-function level enable us to formulate algorithms for computing continued fractions such as the qd algorithm and the discrete Schur flow. A new singular value decomposition (I-SVD) algorithm is designed by using a discrete integrable system defined by the Christoffel-Darboux identity for orthogonal polynomials.

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Section: Integrable Gradient Flows Of Brockett Formmentioning

confidence: 99%

Section: Discretizations Of Integrable Systems To Design Numerical Almentioning

confidence: 99%

A new approach to numerical algorithms in terms of integrable systems

Nakamura¹

International Conference on Informatics Research for Development of Knowledge Society Infrastructure, 2004. ICKS 2004.

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“…(For a more sophisticated discretization in the unpreconditioned Hermitian case, along with a cautionary note about use of large step-size in the forward Euler method, see [28]. )…”

Section: Discrete Dynamical Systemsmentioning

confidence: 99%

“…Simple discretizations, such as Euler's method (1.1), do not typically respect such invariants, giving approximate solutions that drift from the manifold. Invariant-preserving alternatives (see, e.g., [18,26]) generally require significantly more computation per step (though a tractable method for the unpreconditioned, Hermitian case has been proposed by Nakamura, Kajiwara, and Shiotani [28]). Our goal is to explain the relationship between convergence and stability of the continuous and discrete dynamical systems.…”

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confidence: 99%

Dynamical Systems and Non-Hermitian Iterative Eigensolvers

Embree¹,

Lehoucq²

2009

SIAM J. Numer. Anal.

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Abstract. Simple preconditioned iterations can provide an efficient alternative to more elaborate eigenvalue algorithms. We observe that these simple methods can be viewed as forward Euler discretizations of well-known autonomous differential equations that enjoy appealing geometric properties. This connection facilitates novel results describing convergence of a class of preconditioned eigensolvers to the leftmost eigenvalue, provides insight into the role of orthogonality and biorthogonality, and suggests the development of new methods and analyses based on more sophisticated discretizations. These results also highlight the effect of preconditioning on the convergence and stability of the continuous-time system and its discretization.

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“…The third example is the power method (cf [23]) for finding maximal eigenvalue of symmetric matrices. The power method with a shift of origin is shown to be an integrable discretization of the Rayleigh quotient gradient system [17], where no such tau function appears.…”

Section: Introductionmentioning

confidence: 99%