Convergence of LR algorithm for a one-point spectrum tridiagonal matrix

Ferreira, Carla; Parlett, Beresford Ν.

doi:10.1007/s00211-009-0238-2

Cited by 3 publications

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“…Our code 3dqds computes this eigenvalue exactly (and also the generalized eigenvectors) using the following method which is part of the prologue. See [12].…”

Section: Liu Matrixmentioning

confidence: 99%

A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

Ferreira

Parlett

2022

Numer. Math.

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The paper discusses the following topics: attractions of the real tridiagonal case, relative eigenvalue condition number for matrices in factored form, dqds, triple dqds, error analysis, new criteria for splitting and deflation, eigenvectors of the balanced form, twisted factorizations and generalized Rayleigh quotient iteration. We present our fast real arithmetic algorithm and compare it with alternative published approaches.

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“…Our code 3dqds computes this eigenvalue exactly (and also the generalized eigenvectors) using the following method which is part of the prologue. See [12].…”

Section: Liu Matrixmentioning

confidence: 99%

A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

Ferreira

Parlett

2022

Numer. Math.

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Sensitivity of eigenvalues of an unsymmetric tridiagonal matrix

2012

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Calcium Ion Fluctuations Alter Channel Gating in a Stochastic Luminal Calcium Release Site Model

Weinberg

2017

IEEE/ACM Trans. Comput. Biol. and Bioinf.

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Stochasticity and small system size effects in complex biochemical reaction networks can greatly alter transient and steady-state system properties. A common approach to modeling reaction networks, which accounts for system size, is the chemical master equation that governs the dynamics of the joint probability distribution for molecular copy number. However, calculation of the stationary distribution is often prohibitive, due to the large state-space associated with most biochemical reaction networks. Here, we analyze a network representing a luminal calcium release site model and investigate to what extent small system size effects and calcium fluctuations, driven by ion channel gating, influx and diffusion, alter steady-state ion channel properties including open probability. For a physiological ion channel gating model and number of channels, the state-space may be between approximately 10-10 elements, and a novel modified block power method is used to solve the associated dominant eigenvector problem required to calculate the stationary distribution. We demonstrate that both small local cytosolic domain volume and a small number of ion channels drive calcium fluctuations that result in deviation from the corresponding model that neglects small system size effects.

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Convergence of LR algorithm for a one-point spectrum tridiagonal matrix

Abstract: We prove convergence for the basic LR algorithm on a real unreduced tridiagonal matrix with a one-point spectrum-the Jordan form is one big Jordan block. First we develop properties of eigenvector matrices. We also show how to deal with the singular case.

Cited by 3 publications

References 7 publications

A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

Sensitivity of eigenvalues of an unsymmetric tridiagonal matrix

Calcium Ion Fluctuations Alter Channel Gating in a Stochastic Luminal Calcium Release Site Model

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