A gradient based iterative solutions for Sylvester tensor equations

This paper is concerned with the numerical solution of the Sylvester tensor equation, which includes the Sylvester matrix equation as special case. By applying hierarchical identification principle proposed by Ding and Chen, 2005, and by using tensor arithmetic concepts, an iterative algorithm and its modification are established to solve the Sylvester tensor equation. Convergence analysis indicates that the iterative solutions always converge to the exact solution for arbitrary initial value. Finally, some examples are provided to show that the proposed algorithms are effective

A Gradient Based Iterative Solutions for Sylvester Tensor Equations

This paper is concerned with the numerical solution of the Sylvester tensor equation, which includes the Sylvester matrix equation as special case. By applying hierarchical identification principle proposed by Ding and Chen, 2005, and by using tensor arithmetic concepts, an iterative algorithm and its modification are established to solve the Sylvester tensor equation. Convergence analysis indicates that the iterative solutions always converge to the exact solution for arbitrary initial value. Finally, some examples are provided to show that the proposed algorithms are effective

Real Fast Structure-Preserving Algorithm for Eigenproblem of Complex Hermitian Matrices

It is well known that the flops for complex operations are usually 4 times of real cases. In the paper, using real operations instead of complex, a real fast structure-preserving algorithm for eigenproblem of complex Hermitian matrices is given. We make use of the real symmetric and skew-Hamiltonian structure transformed by Wilkinson's way, focus on symplectic orthogonal similarity transformations and their structure-preserving property, and then reduce it into a two-by-two block tridiagonal symmetric matrix. Finally a real algorithm can be quickly obtained for eigenvalue problems of the original Hermitian matrix. Numerical experiments show that the fast algorithm can solve real complex Hermitian matrix efficiently, stably, and with high precision

Inverse Eigenvalue Problem of Unitary Hessenberg Matrices

Let H∈ℂn×n be an n×n unitary upper Hessenberg matrix whose subdiagonal elements are all positive, let Hk be the kth leading principal submatrix of H, and let H˜k be a modified submatrix of Hk. It is shown that when the minimal and maximal eigenvalues of H˜k (k=1,2,…,n) are known, H can be constructed uniquely and efficiently. Theoretic analysis, numerical algorithm, and a small example are given

New method for square root of non-singular M-matrix

Square root of a matrix play an important role in many applications of matrix theory. In this paper, we propose a new iterative method for square root of a non-singular M-matrix. We first transform the matrix equation X2 – A=0 into special form of a non-symmetric algebraic Riccati equation (NARE), and then solve this special NARE by Newton method. Efficiency and effectiveness proved by theoretical analysis and numerical experiments. Keywords: - Matrix square root, M-matrix, Non-symmetric algebraic Riccati equation, Newton method

Non-negative Tucker decomposition with double constraints for multiway dimensionality reduction

Nonnegative Tucker decomposition (NTD) is one of the renowned techniques in feature extraction and representation for nonnegative high-dimensional tensor data. The main focus behind the NTD-like model was how to factorize the data to get ahold of a high quality data representation from multidimensional directions. However, existing NTD-like models do not consider relationship and properties between the factor matrix of columns while preserving the geometric structure of the data space. In this paper, we managed to capture nonlinear local features of data space and further enhance expressiveness of the NTD clustering method by syncretizing organically approximately orthogonal constraint and graph regularized constraint. First, based on the uni-side and bi-side approximate orthogonality, we flexibly proposed two novel approximately orthogonal NTD with graph regularized models, which not only in part make the factor matrix tend to be orthogonality, but also preserve the geometrical information from high-dimensional tensor data. Second, we developed the iterative updating algorithm dependent on the multiplicative update rule to solve the proposed models, and provided its convergence and computational complexity. Finally, we used numerical experimental results to demonstrate the effectiveness, robustness, and efficiency of the proposed new methods on the real-world image datasets

Perron complement and Perron root

AbstractFor a nonnegative irreducible matrix A, this paper is concerned with the estimation and determination of the unique Perron root or spectral radius of A. We present a new method that utilizes the relation between Perron roots of the nonnegative matrix and its (generalized) Perron complement. Several numerical examples are given to show that our method is effective, at least, for some classes of nonnegative matrices