Fourth-order tensor Riccati equations with the Einstein product
Yun Miao, Yimin Wei, Zhen Chen
Abstract
In this paper, the fourth-order tensor Riccati equation with the Einstein product is investigated. Using the Einstein product, we introduce the M-tensor, the spectrum and the block tensor multiplication. We also present an expression of the fourth-order permutation tensors and generalize the Riccati equation from the matrix to the tensor case. The existence and uniqueness of the solution for the fourth-order M-tensor Riccati equation are studied. Furthermore, we provide a perturbation analysis of the mixed and componentwise condition numbers for the M-tensor Riccati equation. An algorithm for solving the equation is proposed. Numerical examples are presented to show the effectiveness of our algorithm.
Topics & Concepts
Riccati equationMathematicsAlgebraic Riccati equationTensor (intrinsic definition)UniquenessSymmetric tensorCartesian tensorExact solutions in general relativityTensor productMathematical analysisApplied mathematicsTensor densityTensor fieldPure mathematicsDifferential equationTensor decomposition and applicationsPower System Optimization and Stability