The outer inverse of tensors plays increasingly significant roles in computational mathematics, numerical analysis, and other generalized inverses of tensors. In this paper, we compute outer inverses with prescribed ranges and kernels of a given tensor through tensor QR decomposition and hyperpower iterative method under the t-product structure, which is a family of tensor–tensor products, generalization of the t-product and t-product, allows us to suit the physical interpretations across those different modes. We discuss a theoretical analysis of the nineteen-order convergence of the proposed tensor-based iterative method. Further, we design effective tensor-based algorithms for computing outer inverses using t-QR decomposition and hyperpower iterative method. The theoretical results are validated with numerical examples demonstrating the appropriateness of the proposed methods. In addition, we examine the application of t-QR decomposition and hyperpower iterative methods to improve image compression efficiency and improve deblurring results in digital image processing.

The tensor eigenvalue problem has been widely studied in recent years. In this paper, several new properties of eigenvalues and determinants of tensors are explored. We also proposed a formula to compute the determinant of a tensor as a mimic of the matrix determinant. The Perron-Frobenius theorem, one of the most important results in non-negative matrix theory, is proposed for the class of non-negative tensors in the Einstein product framework. Further, the power method, a widely used matrix iterative method for finding the largest eigenvalue, is framed for tensors using the Einstein product. The proposed higher-order power method is applied to calculate the largest eigenvalue of the Laplacian tensors associated with hyper-stars and hyper-trees. The numerical results show that the higher-order power method with the Einstein product is stable.

The theory and computation of tensors with different tensor products play increasingly important roles in scientific computing and machine learning. Different products aim to preserve different algebraic properties from the matrix algebra, and the choice of the tensor product determines the algorithms that can be applied directly. This study introduced a novel full-rank decomposition and M-QDR decomposition for third-order tensors based on the M-product. Then we designed algorithms for computing these two decompositions along with the Moore-Penrose inverse and outer inverse of the tensors. Numerical examples are discussed in support of these theoretical results. In addition, we derive exact expressions for the outer inverses of tensors using a symbolic tensor (tensors with polynomial entries) computation. We designed efficient algorithms to compute the Moore-Penrose inverse of symbolic tensors. The prowess of the proposed M-QDR decomposition for third-order tensors is applied to compress lossy color images.

Let $S=left{s_1, s_2, ldots, s_nright}$ be an ordered set of $n$ distinct positive integers. The $m$ thorder $n$-dimensional tensor $mathscr{T}_{[S]}=left(t_{i_1 i_2 ldots i_m}right)$, where $t_{i_1 i_2 ldots i_m}=operatorname{GCD}left(s_{i_1}, s_{i_2}, ldots, s_{i_m}right)$, the greatest common divisor (GCD) of $s_{i_1}, s_{i_2}, ldots$, and $s_{i_m}$ is called the GCD tensor on $S$. The earliest result on GCD tensors goes back to Smith [Proc. Lond. Math. Soc., 1976], who computed the determinant of GCD matrix on $S={1,2, ldots, n}$ using the Euler's totient function, followed by Beslin-Ligh [Linear Algebra Appl., 1989] who showed all GCD matrices are positive definite. In this note, we study the positivity of higher-order tensors in the $k$-mode product. We show that all GCD tensors are strongly completely positive (CP). We then show that GCD tensors are infinite divisible. In fact, we prove that for every positive real number $r$, the tensor $mathscr{T}_{[S]}^{o r}=left(t_{i_1 i_2, ldots i_m}^rright)$ is strongly CP. Finally, we obtain an interesting decomposition of GCD tensors using Euler's totient function $Phi$. Using this decomposition, we show that the determinant (also called hyperdeterminant) of the $m$ th-order GCD tensor $mathscr{T}_{[S]}$ on a factor-closed set $S=left{s_1, ldots, s_nright} text { is } prod_{i=1}^n Phileft(s_iright)^{(m-1)^{(n-1)}} .$

Theory of numerical range and numerical radius for tensors is not studied much in the literature. Ke et al. [Linear Algebra Appl. 508 (2016), 100-132: MR3542984] introduced first the notion of the numerical range of a tensor via the k-mode product. However, the convexity of the numerical range via the k-mode product was not proved by them. In this paper, the notion of numerical range and numerical radius for even-order square tensors via the Einstein product are introduced first. Using the notion of the numerical radius of a tensor, we provide some sufficient conditions for a tensor to be unitary. The convexity of the numerical range is also proved. We also provide an algorithm to plot the numerical range of a tensor. Furthermore, some properties of the numerical range for the Moore–Penrose inverse of a tensor are discussed.

Very recently, Piri et al. (Numer Algorithms 81:1129–1148, 2019) introduced an iterative process to approximate a fixed point of generalized alpha-nonexpansive mappings and discussed its convergence analysis. In this paper, we introduce an iteration process to approximate a fixed point of a contractive self-mapping that is faster than all of Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, and Thakur processes for contractive mappings including the recent one by Piri et al. We also obtain convergence and stability theorems of this iterative process for a contractive self-mapping. Numerical examples show that our iteration process for approximating a fixed point of a contractive self-mapping is faster than the method proposed by Piri et al. Based on this process, we finally present a new modified Newton–Raphson method for finding the roots of a function and generate some nice polynomiographs.

The notions of C-tensor, C0-tensor and ¯𝐶-tensor are introduced first. Different necessary and sufficient conditions for a tensor to be a C-tensor, C0-tensor and ¯𝐶-tensor are provided. We next show that the sum of two C-tensors (C0-tensors) is a C-tensor (C0-tensor) while the Hadamard product of two C-tensors (C0-tensors) is not a C-tensor (C0-tensor). We also present a result that illustrates the Hadamard product of two C-tensor is again a C-tensor under some sufficient conditions. As an application of these classes of tensors, an exclusion interval for the real eigenvalues of a real tensor is proposed. Finally, we provide a necessary and sufficient condition for the exclusion interval to be nonempty.