Spectral Radius of Nonnegative Centrosymmetric Matrices

In this paper, we present some results about the spectral radius of a kind of structured matrices, nonnegative centrosymmetric matrices. Furthermore, we constructrue a algorithm to compute the spectral radius of nonnegative centrosymmetric matrices

Centrosymmetric, Skew Centrosymmetric and Centrosymmetric Cauchy Tensors

Recently, Zhao and Yang introduced centrosymmetric tensors. In this paper, we further introduce skew centrosymmetric tensors and centrosymmetric Cauchy tensors, and discuss properties of these three classes of structured tensors. Some sufficient and necessary conditions for a tensor to be centrosymmetric or skew centrosymmetric are given. We show that, a general tensor can always be expressed as the sum of a centrosymmetric tensor and a skew centrosymmetric tensor. Some sufficient and necessary conditions for a Cauchy tensor to be centrosymmetric or skew centrosymmetric are also given. Spectral properties on H-eigenvalues and H-eigenvectors of centrosymmetric, skew centrosymmetric and centrosymmetric Cauchy tensors are discussed. Some further questions on these tensors are raised

Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

Recently, we used the Sinc collocation method with the double exponential transformation to compute eigenvalues for singular Sturm-Liouville problems. In this work, we show that the computation complexity of the eigenvalues of such a differential eigenvalue problem can be considerably reduced when its operator commutes with the parity operator. In this case, the matrices resulting from the Sinc collocation method are centrosymmetric. Utilizing well known properties of centrosymmetric matrices, we transform the problem of solving one large eigensystem into solving two smaller eigensystems. We show that only 1/(N+1) of all components need to be computed and stored in order to obtain all eigenvalues, where (2N+1) corresponds to the dimension of the eigensystem. We applied our result to the Schr\"odinger equation with the anharmonic potential and the numerical results section clearly illustrates the substantial gain in efficiency and accuracy when using the proposed algorithm.Comment: 11 pages, 4 figure

A constructive arbitrary-degree Kronecker product decomposition of tensors

We propose the tensor Kronecker product singular value decomposition~(TKPSVD) that decomposes a real $k$-way tensor $\mathcal{A}$ into a linear combination of tensor Kronecker products with an arbitrary number of $d$ factors $\mathcal{A} = \sum_{j=1}^R \sigma_j\, \mathcal{A}^{(d)}_j \otimes \cdots \otimes \mathcal{A}^{(1)}_j$. We generalize the matrix Kronecker product to tensors such that each factor $\mathcal{A}^{(i)}_j$ in the TKPSVD is a $k$-way tensor. The algorithm relies on reshaping and permuting the original tensor into a $d$-way tensor, after which a polyadic decomposition with orthogonal rank-1 terms is computed. We prove that for many different structured tensors, the Kronecker product factors $\mathcal{A}^{(1)}_j,\ldots,\mathcal{A}^{(d)}_j$ are guaranteed to inherit this structure. In addition, we introduce the new notion of general symmetric tensors, which includes many different structures such as symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors.Comment: Rewrote the paper completely and generalized everything to tensor

Counter Example: The Algorithm of Determinant of Centrosymmteric Matrix based on Lower Hessenberg Form

The algorithm for computing determinant of centrosymmetric matrix has been evaluated before. This algorithm shows the efficient computational determinant process on centrosymmetric matrix by working on block matrix only. One of block matrix at centrosymmetric matrix appearing on this algorithm is lower Hessenberg form. However, the other block matrices may possibly appear as block matrix for centrosymmetric matrix’s determinant. Therefore, this study is aimed to show the possible block matrices at centrosymmetric matrix and how the algorithm solve the centrosymmetric matrix’s determinant. Some numerical examples for different cases of block matrices on determinant of centrosymmetric matrix are given also. These examples are useful for more understanding for applying the algorithm with different cases

Atomic norm denoising with applications to line spectral estimation

Motivated by recent work on atomic norms in inverse problems, we propose a new approach to line spectral estimation that provides theoretical guarantees for the mean-squared-error (MSE) performance in the presence of noise and without knowledge of the model order. We propose an abstract theory of denoising with atomic norms and specialize this theory to provide a convex optimization problem for estimating the frequencies and phases of a mixture of complex exponentials. We show that the associated convex optimization problem can be solved in polynomial time via semidefinite programming (SDP). We also show that the SDP can be approximated by an l1-regularized least-squares problem that achieves nearly the same error rate as the SDP but can scale to much larger problems. We compare both SDP and l1-based approaches with classical line spectral analysis methods and demonstrate that the SDP outperforms the l1 optimization which outperforms MUSIC, Cadzow's, and Matrix Pencil approaches in terms of MSE over a wide range of signal-to-noise ratios.Comment: 27 pages, 10 figures. A preliminary version of this work appeared in the Proceedings of the 49th Annual Allerton Conference in September 2011. Numerous numerical experiments added to this version in accordance with suggestions by anonymous reviewer