An algebraic method for quaternion and complex Least Squares coneigen-problem in quantum mechanics

Jiang, Tongsong; Jiang, Ziwu; Ling, Sitao

doi:10.1016/j.amc.2014.10.075

Cited by 23 publications

(15 citation statements)

References 15 publications

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“…b � (u 1 , θ 1 , u 2 , θ 2 , u 3 , θ 3 , u 4 , θ 4 , u 5 , θ 5 , u 6 , θ 6 ) Τ is the nonlinear parameter vector to be estimated. Equation (32) is written in the form of a matrix as…”

Section: Example 1: Gaussian Function Fitting Modelmentioning

confidence: 99%

“…e error equations were listed based on the model in (32) and 400 pairs of observed values (t i , y i ), and the parameters are solved using the LM unSep , LM VP + LS, and LM VP + CCVL methods, respectively. e parameter estimates are shown in Table 1, which are obtained by taking the average of 10 times.…”

Section: Example 1: Gaussian Function Fitting Modelmentioning

confidence: 99%

“…erefore, the LM algorithm overcomes this shortcoming and adjusts the damping parameters according to the idea of a trust region, to effectively control the direction of iterative descent. Once the nonlinear parameters were determined, the least squares (LS) method [32] is used to estimate the linear parameters.…”

Section: Introductionmentioning

confidence: 99%

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Efficient Parameters Estimation Method for the Separable Nonlinear Least Squares Problem

et al. 2020

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In this work, we combine the special structure of the separable nonlinear least squares problem with a variable projection algorithm based on singular value decomposition to separate linear and nonlinear parameters. en, we propose finding the nonlinear parameters using the Levenberg-Marquart (LM) algorithm and either solve the linear parameters using the least squares method directly or by using an iteration method that corrects the characteristic values based on the L-curve, according to whether or not the nonlinear function coefficient matrix is ill posed. To prove the feasibility of the proposed method, we compared its performance on three examples with that of the LM method without parameter separation. e results show that (1) the parameter separation method reduces the number of iterations and improves computational efficiency by reducing the parameter dimensions and (2) when the coefficient matrix of the linear parameters is well-posed, using the least squares method to solve the fitting problem provides the highest fitting accuracy. When the coefficient matrix is ill posed, the method of correcting characteristic values based on the L-curve provides the most accurate solution to the fitting problem.

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Section: Example 1: Gaussian Function Fitting Modelmentioning

confidence: 99%

Section: Example 1: Gaussian Function Fitting Modelmentioning

confidence: 99%

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Efficient Parameters Estimation Method for the Separable Nonlinear Least Squares Problem

et al. 2020

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“…Different filtering factors will result in different regularisation methods, and the TR method is a widely applied method [35,36]. e filtering factor obtained by the TR method can be expressed as…”

Section: Determination Of the Linear Parametersmentioning

confidence: 99%

Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems

Guo

2019

Complexity

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This paper considers the classical separable nonlinear least squares problem. Such problems can be expressed as a linear combination of nonlinear functions, and both linear and nonlinear parameters are to be estimated. Among the existing results, ill-conditioned problems are less often considered. Hence, this paper focuses on an algorithm for ill-conditioned problems. In the proposed linear parameter estimation process, the sensitivity of the model to disturbance is reduced using Tikhonov regularisation. The Levenberg–Marquardt algorithm is used to estimate the nonlinear parameters. The Jacobian matrix required by LM is calculated by the Golub and Pereyra, Kaufman, and Ruano methods. Combining the nonlinear and linear parameter estimation methods, three estimation models are obtained and the feasibility and stability of the model estimation are demonstrated. The model is validated by simulation data and real data. The experimental results also illustrate the feasibility and stability of the model.

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“…It is easy to perform the protocol. After that, quantum cryptography protocols were widely researched [Jiang, Jiang and Ling (2014); Qu, Chen, Zhou et al (2010); Qu, Wu, Wang et al (2017); Qu, Chen, Ji et al (2018)]. QKD protocols based on continuous variable were also investigated.…”

Section: Introductionmentioning

confidence: 99%

Rational Non-Hierarchical Quantum State Sharing Protocol

Dou¹,

Xu²,

Chen³

et al. 2019

Computers, Materials &Amp; Continua

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Rational participants want to maximize their benefits. The protocol with rational participants will be more realistic than the protocol with honest, semi-honest and dishonest participants. We research the rational non-hierarchical quantum state sharing in this paper. General steps of some known quantum state sharing protocol are summarized. Based on these steps, a new rational protocol is proposed. It means that lots of common protocols could be modified to rational protocols. Our protocol is widely applicable. Analyses show that the proposed protocol is rational and secure. It is also all-win for agents. Furthermore, number of deceiving agents is considered to redefine the utilities of agents.

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An algebraic method for quaternion and complex Least Squares coneigen-problem in quantum mechanics

Cited by 23 publications

References 15 publications

Efficient Parameters Estimation Method for the Separable Nonlinear Least Squares Problem

Efficient Parameters Estimation Method for the Separable Nonlinear Least Squares Problem

Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems

Rational Non-Hierarchical Quantum State Sharing Protocol

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