arXiv subjects

Ranjan Kumar Das

Publications and source records attributed to Ranjan Kumar Das.

4 recordsLinked to original sources

arXiv2026

Solution Analysis of Tensor Equation $\mathcal{A} \ltimes \mathcal{X} \ltimes \mathcal{B}= \mathcal{C}$ via Semi Tensor Product with t-product

This paper focuses on the analysis of the tensor equation $\mathcal{A\ltimes X\ltimes B=C}$, formulated via the semi tensor product with t-product. For the unknown vector $\mathcal{X}$, we establish a necessary and sufficient condition that provides an equivalence criterion for the existence of solutions. For matrix valued and higher-order tensor valued unknown $\mathcal{X}$, solvability is determined by corresponding compatibility requirements. Moreover, the explicit structure (Toeplitz and Circulant) of $\mathcal{C}$ is characterized. The derived results are supported by several illustrative examples.

math.NA↗

arXiv2026

Solving Coupled Tensor Equation $\mathcal{A} \ltimes \mathcal{X} =\mathcal{B}, \ \mathcal{X}\ltimes \mathcal{C}=\mathcal{D}$ using Semi-Tensor Products in the t-product

This paper investigates the solution of coupled third-order tensor equation $\mathcal{A} \ltimes \mathcal{X} = \mathcal{B},\ \mathcal{X} \ltimes \mathcal{C} = \mathcal{D},$ of arbitrary dimensions by incorporating semi-tensor product (STP) within t-product framework, where the unknown $\mathcal{X}$ can take form of vector, matrix, or tensor. For the unknown $\mathcal{X}$, we establish a necessary and sufficient condition that provides an equivalence criterion for the existence of solutions. Moreover, the explicit structure (Toeplitz, Circulant) of $\mathcal{C}$ and $\mathcal{D}$ is characterized. Theoretical results are supported by several illustrative examples.

math.NA↗

arXiv2023

Unified framework for Fiedler-like strong linearizations of polynomial and rational matrices

Linearization is a widely used method for solving polynomial eigenvalue problems (PEPs) and rational eigenvalue problem (REPs) in which the PEP/REP is transformed to a generalized eigenproblem and then solve this generalized eigenproblem with algorithms available in the literature. Fiedler-like pencils (Fiedler pencils (FPs), generalized Fiedler pencils (GFPs), Fiedler pencils with repetition (FPRs) and generalized Fiedler pencils with repetition (GFPRs)) are well known classes of strong linearizations. GFPs are an intriguing family of linearizations, and GF pencils are the fundamental building blocks of FPRs and GFPRs. As a result, FPRs and GFPRs have distinctive features and they provide structure-preserving linearizations for structured matrix polynomials. But GFPRs do not use the full potential of GF pencils. Indeed, not all the GFPs are FPRs or GFPRs, and vice versa. The main aim of this paper is two-fold. First, to build a unified framework for all the Fiedler-like pencils FPs, GFPs, FPRs and GFPRs. To that end, we construct a new family of strong linearizations (named as EGFPs) of a matrix polynomial $P(\lam)$ that subsumes all the Fiedler-like linearizations. A salient feature of the EGFPs family is that it allows the construction of structured preserving banded linearizations with low bandwidth for structured (symmetric, Hermitian, palindromic) matrix polynomial. Low bandwidth structured linearizations may be useful for numerical computations. Second, to utilize EGFPs directly to form a family of Rosenbrock strong linearizations of an $n \times n$ rational matrix $G(\lam)$ associated with a realization. We describe the formulas for the construction of low bandwidth linearizations for $P(\lam)$ and $G(\lam)$. We show that the eigenvectors, minimal bases/indices of $P(\lam)$ and $G(\lam)$ can be easily recovered from those of the linearizations of $P(\lam)$ and $G(\lam)$.

math.NA↗

arXiv2020

Structured strong linearizations of structured rational matrices

Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational matrices have been introduced recently for computing poles, eigenvalues, eigenvectors, minimal bases and minimal indices of rational matrices. For structured rational matrices, it is desirable to construct structure-preserving linearizations so as to preserve the symmetry in the eigenvalues and poles of the rational matrices. With a view to constructing structure-preserving linearizations of structured rational matrices, we propose a family of Fiedler-like pencils and show that the family of Fiedler-like pencils is a rich source of structure-preserving strong linearizations of structured rational matrices. We construct symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, skew-Hermitian, para-Hermitian and para-skew-Hermitian strong linearizations of a rational matrix $G(λ)$ when $G(λ)$ has the same structure. Further, when $G(λ)$ is real and symmetric, we show that the transfer functions of real symmetric linearizations of $G(λ)$ preserve the Cauchy-Maslov index of $G(λ).$ We describe the recovery of eigenvectors, minimal bases and minimal indices of $G(λ)$ from those of the linearizations of $G(λ)$ and show that the recovery is operation-free.

math.NA↗