Maximum Eigenvalue
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Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 211
Asuka Ohashi ◽  
Tomohiro Sogabe

We consider computing an arbitrary singular value of a tensor sum: T:=In⊗Im⊗A+In⊗B⊗Iℓ+C⊗Im⊗Iℓ∈Rℓmn×ℓmn, where A∈Rℓ×ℓ, B∈Rm×m, C∈Rn×n. We focus on the shift-and-invert Lanczos method, which solves a shift-and-invert eigenvalue problem of (TTT−σ˜2Iℓmn)−1, where σ˜ is set to a scalar value close to the desired singular value. The desired singular value is computed by the maximum eigenvalue of the eigenvalue problem. This shift-and-invert Lanczos method needs to solve large-scale linear systems with the coefficient matrix TTT−σ˜2Iℓmn. The preconditioned conjugate gradient (PCG) method is applied since the direct methods cannot be applied due to the nonzero structure of the coefficient matrix. However, it is difficult in terms of memory requirements to simply implement the shift-and-invert Lanczos and the PCG methods since the size of T grows rapidly by the sizes of A, B, and C. In this paper, we present the following two techniques: (1) efficient implementations of the shift-and-invert Lanczos method for the eigenvalue problem of TTT and the PCG method for TTT−σ˜2Iℓmn using three-dimensional arrays (third-order tensors) and the n-mode products, and (2) preconditioning matrices of the PCG method based on the eigenvalue and the Schur decomposition of T. Finally, we show the effectiveness of the proposed methods through numerical experiments.

Jian-Hui Li ◽  
Jin-Long Liu ◽  
Zu-Guo Yu ◽  
Bao-Gen Li ◽  
Da-Wen Huang

In this paper, we study the synchronizability of three kinds of dynamical weighted fractal networks (WFNs). These WFNs are weighted Cantor-dust networks, weighted Sierpinski networks and weighted Koch networks. We calculated some features of these WFNs, including average distance ([Formula: see text]), fractal dimension ([Formula: see text]), information dimension ([Formula: see text]), correlation dimension ([Formula: see text]). We analyze two representative types of synchronizable dynamical networks (the type-I and the type-II). There are two indexes ([Formula: see text] and [Formula: see text]) that can be used to characterize the synchronizability of the two types of dynamical network. Here, [Formula: see text] and [Formula: see text] are the minimum nonzero eigenvalue and the maximum eigenvalue of the Laplacian matrix of the network, respectively. We find that the larger scaling factor [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text] implies stronger synchronizability for the type-I dynamical WFNs.

Algorithms ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 222
Hailemariam Abebe Tekile ◽  
Michele Fedrizzi ◽  
Matteo Brunelli

Pairwise comparison matrices play a prominent role in multiple-criteria decision-making, particularly in the analytic hierarchy process (AHP). Another form of preference modeling, called an incomplete pairwise comparison matrix, is considered when one or more elements are missing. In this paper, an algorithm is proposed for the optimal completion of an incomplete matrix. Our intention is to numerically minimize a maximum eigenvalue function, which is difficult to write explicitly in terms of variables, subject to interval constraints. Numerical simulations are carried out in order to examine the performance of the algorithm. The results of our simulations show that the proposed algorithm has the ability to solve the minimization of the constrained eigenvalue problem. We provided illustrative examples to show the simplex procedures obtained by the proposed algorithm, and how well it fills in the given incomplete matrices.

2021 ◽  
Vol 3 (2) ◽  
pp. 138-149
B Vivekanandam

One of the most crucial roles of the cognitive radio (CR) is detection of spectrum ‘holes’. The ‘no a-priori knowledge required’ prospective of blind detection techniques has attracted the attention of researchers and industries, using simple Eigen values. Over the years, a number of study and research has been carried out to determine the impact of thermal noise in the performance of the detector. However, there has not been much work on the impact of man-made noise, which also hinders the performance of the detector. As a result, both man-made impulse noise and thermal Gaussian noise are examined in this proposed study to determine the performance of blind Eigen value-based spectrum sensing. Many studies have been conducted over long sample length by oversampling or increasing the duration of sensing. As a result, a research progress has been made on shorter sample lengths by using a novel algorithm. The proposed system utilizes three algorithms; they are contra-harmonic-mean minimum Eigen value, contra-harmonic mean Maximum Eigen value and maximum Eigenvalue harmonic mean. For smaller sample lengths, there is a substantial rise in the number of cooperative secondary users, as well as a low signal-to-noise ratio when employing the maximum Eigen value Harmonic mean. The experimental analysis of the proposed work with respect to impulse noise and Gaussian signal using Nakagami-m fading channel is observed and the results identified are tabulated.

T. Pham

This paper proposes a novel method to detect small sea targets, using polarimetric radar. The standard deviation of maximum eigenvalue of the polarization covariance matrix is ultilized for the detection. In order to evaluate the new algorithm in practical applications, the real data collected from polarimetric IPIX radar at MC Master University is tested with the new algorithm, and the result is displayed on the radar screen. The results of this method is also compared with those of the DoP and SP-GLR methods. Initial results show that the novel method significantly improves the detectability of small sea targets on the sea surface. 

Faten Mashta ◽  
Mohieddin Wainakh ◽  
Wissam Altabban

Spectrum sensing for cognitive radio requires speed and good detection performance at very low SNR ratios. There is no single-stage spectrum sensing technique that is perfect enough to be implemented in practical cognitive radio. In this paper, the authors propose a new parallel fully blind multistage detector. They assume the appropriate stage based on the estimated SNR values that are achieved from the SNR estimator. Energy detection is used in first stage for its simplicity and sensing accuracy at high SNR. For low SNRs, they adopt the maximum eigenvalues detector with different smoothing factor in higher stages. The sensing accuracy for the maximum eigenvalue detector technique improves with higher value of the smoothing factor. However, the computational complexity will increase significantly. They analyze the performance of two cases of the proposed detector: two-stage and three-stage schemes. The simulation results show that the proposed detector improves spectrum sensing in terms of accuracy and speed.

Dinghui Wu ◽  
Juan Zhang ◽  
Bo Wang ◽  
Tinglong Pan

Traditional static threshold–based state analysis methods can be applied to specific signal-to-noise ratio situations but may present poor performance in the presence of large sizes and complexity of power system. In this article, an improved maximum eigenvalue sample covariance matrix algorithm is proposed, where a Marchenko–Pastur law–based dynamic threshold is introduced by taking all the eigenvalues exceeding the supremum into account for different signal-to-noise ratio situations, to improve the calculation efficiency and widen the application fields of existing methods. The comparison analysis based on IEEE 39-Bus system shows that the proposed algorithm outperforms the existing solutions in terms of calculation speed, anti-interference ability, and universality to different signal-to-noise ratio situations.

2020 ◽  
Vol 07 (04) ◽  
pp. 2050051
Subhojit Biswas ◽  
Diganta Mukherjee ◽  
Indranil SenGupta

This paper proposes swaps on two important new measures of generalized variance, namely, the maximum eigenvalue and trace of the covariance matrix of the assets involved. We price these generalized variance swaps for Barndorff-Nielsen and Shephard model used in financial markets. We consider multiple assets in the portfolio for theoretical purpose and demonstrate our approach with numerical examples taking three stocks in the portfolio. The results obtained in this paper have important implications for the commodity sector where such swaps would be useful for hedging risk.

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