A new approach to numerical algorithms in terms of integrable systems

Nakamura, Y.

doi:10.1109/icks.2004.1313425

Cited by 7 publications

(11 citation statements)

References 44 publications

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“…The right-hand-side of ( 19) defines orispherical coordinate system (q n , z nm ), n > m, on the space of symmetric positive-definite matrices G. Explicit time dependence of G(t) given by (19) provides that…”

Section: [H A]]mentioning

confidence: 99%

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Toda chain flow in Krylov space

Dymarsky,

Gorsky

2019

Preprint

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Section: [H A]]mentioning

confidence: 99%

“…The connection between the recursion method and Toda chain is likely to lead to new practical improvements in numerical applications, as suggested by many other uses of Toda chain in the context of computational algorithms [19].…”

Section: [H A]]mentioning

confidence: 99%

Toda chain flow in Krylov space

Dymarsky,

Gorsky

2019

Preprint

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“…For such a connection, see also [203,219]. Other equations are related to acceleration methods [205]. For example in [204], the authors wrote abstract.…”

Section: Integrable Systemsmentioning

confidence: 99%

The genesis and early developments of Aitken’s process, Shanks’ transformation, the ε–algorithm, and related fixed point methods

Brezinski

Redivo–Zaglia

2018

Numer Algor

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In this paper, we trace back the genesis of Aitken's ∆ 2 process, and Shanks' sequence transformation. These methods, which are extrapolation methods, are used for accelerating the convergence of sequences of scalars, vectors, matrices, and tensors. They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions, and Padé approximants. We go back to the roots of these methods, and analyze the original contributions. New and detailed explanations on the building and properties of Shanks' transformation and its kernel are provided. We then review their historical algebraic and algorithmic developments. We also analyze how they were involved in the solution of systems of linear and nonlinear equations, in particular in the methods of Steffensen, Pulay, and Anderson. Testimonies by various actors of the domain are given. The paper can also serve as an introduction to this domain of numerical analysis.

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“…With appropriate discretization, for example, it can be shown that the above formula leads to the Rutishauser qd algorithm (Nakamura 2004, Rutishauser 1954. Instead of detailing here how this can be done for the eigenvalue computation which is a well studied subject, we shall demonstrate a similar application to the much more sophisticated singular value decomposition in the next section.…”

Section: Tau Functions and Determinantal Solutionmentioning

confidence: 99%

Linear algebra algorithms as dynamical systems

Chu

2008

Acta Numerica

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Any logical procedure that is used to reason or infer either deductively or inductively so as to draw conclusions or make decisions can be called, in a broad sense, a realization process. A realization process usually assumes the recursive form that one state gets developed into another state by following a certain specific rule. Such an action is qualified as what is generally known as a dynamical system. In mathematics, especially for existence questions, a realization process often appears in the form of an iterative procedure or a differential equation. For years researchers have taken great effort to describe, analyze, and modify realization processes for various applications.The thrust in this exposition is to exploit the notion of dynamical systems as a special realization process for problems arising from the field of linear algebra. Several differential equations whose solutions evolve in some submanifolds of matrices are cast in fairly general frameworks of which special cases have been found to afford unified and fundamental insights into the structure and behavior of existing discrete methods and, now and then, suggest new and improved numerical methods. In some cases, there are remarkable connections between smooth flows and discrete numerical algorithms. In other cases, the flow approach seems advantageous in tackling very difficult open problems. Various aspects of the recent development and application in this direction are discussed in this paper.

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A new approach to numerical algorithms in terms of integrable systems

Cited by 7 publications

References 44 publications

Toda chain flow in Krylov space

Toda chain flow in Krylov space

The genesis and early developments of Aitken’s process, Shanks’ transformation, the ε–algorithm, and related fixed point methods

Linear algebra algorithms as dynamical systems

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