Stationary splitting iterative methods for the matrix equation AX B = C

Stationary splitting iterative methods for solving AXB = Care considered in this paper. The main tool to derive our new method is the induced splitting of a given nonsingular matrix A = M −N by a matrix H such that (I −H) invertible. Convergence properties of the proposed method are discussed and numerical experiments are presented to illustrate its computational efficiency and the effectiveness of some preconditioned variants. In particular, for certain surface fitting applications, our method is much more efficient than the progressive iterative approximation (PIA), a conventional iterative method often used in computer-aided geometric design (CAGD).The authors would like to thank the supports of the National Natural Science Foundation of China under Grant No. 11371075, the Hunan Key Laboratory of mathematical modeling and analysis in engineering, and the Portuguese Funds through FCT–Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013

(R, S) conjugate solution to coupled Sylvester complex matrix equations with conjugate of two unknowns

In this work, we are concerned with (R, S) – conjugate solutions to coupled Sylvester complex matrix equations with conjugate of two unknowns. When the considered two matrix equations are consistent, it is demonstrated that the solutions can be obtained by utilizing this iterative algorithm for any initial arbitrary (R,S) – conjugate matrices V1,W1. A necessary and sufficient condition is established to guarantee that the proposed method converges to the (R,S) – conjugate solutions. Finally, two numerical examples are provided to demonstrate the efficiency of the described iterative technique

A Preconditioned Iteration Method for Solving Sylvester Equations

A preconditioned gradient-based iterative method is derived by judicious selection of two auxil- iary matrices. The strategy is based on the Newton’s iteration method and can be regarded as a generalization of the splitting iterative method for system of linear equations. We analyze the convergence of the method and illustrate that the approach is able to considerably accelerate the convergence of the gradient-based iterative method

A sketch-and-project method for solving the matrix equation AXB = C

In this paper, based on an optimization problem, a sketch-and-project method for solving the linear matrix equation AXB = C is proposed. We provide a thorough convergence analysis for the new method and derive a lower bound on the convergence rate and some convergence conditions including the case that the coefficient matrix is rank deficient. By varying three parameters in the new method and convergence theorems, the new method recovers an array of well-known algorithms and their convergence results. Meanwhile, with the use of Gaussian sampling, we can obtain the Gaussian global randomized Kaczmarz (GaussGRK) method which shows some advantages in solving the matrix equation AXB = C. Finally, numerical experiments are given to illustrate the effectiveness of recovered methods.Comment: arXiv admin note: text overlap with arXiv:1506.03296, arXiv:1612.06013, arXiv:2204.13920 by other author

Invariant Bilinear Differential Pairings on Parabolic Geometries

This thesis is concerned with the theory of invariant bilinear differential pairings on parabolic geometries. It introduces the concept formally with the help of the jet bundle formalism and provides a detailed analysis. More precisely, after introducing the most important notations and definitions, we first of all give an algebraic description for pairings on homogeneous spaces and obtain a first existence theorem. Next, a classification of first order invariant bilinear differential pairings is given under exclusion of certain degenerate cases that are related to the existence of invariant linear differential operators. Furthermore, a concrete formula for a large class of invariant bilinear differential pairings of arbitrary order is given and many examples are computed. The general theory of higher order invariant bilinear differential pairings turns out to be much more intricate and a general construction is only possible under exclusion of finitely many degenerate cases whose significance in general remains elusive (although a result for projective geometry is included). The construction relies on so-called splitting operators examples of which are described for projective geometry, conformal geometry and CR geometry in the last chapter.Comment: This is a PhD thesis from the University of Adelaid