Numerical Methods For Two Kinds Of Matrix Singular Value Problems

In 1843,William Rowan Hamilton,an Irish physicist and mathematician,originally proposed the quaternion.It has a history of more than one hundred years,but the research on the quaternion matrix eigenvalue problems(qEPs)and singular value decomposition(qSVD)has been attracting lots of attention by scholars in recent years.On the one hand,the quaternion and qEPs are widely used in quantum mechanics,color face recognition and other fields.On the other hand,the qSVD is widely used in color image compression,image completion,image denoising,signal processing,rolling bearing fault diagnosis,electroencephalography,and so forth.Moreover,there are many methods to solve the right eigenvalue problems of quaternion matrices,including pure topology method,quaternion QR method,implicit Jacobi method,quaternion power method,and a series of real structure-preserving algorithms emerging in recent years.Therefore,the study on the qEPs and the algorithms of the qSVD has important scientific significance and application value.In addition,more attentions have been paid on the inverse eigenvalue problems(IEPs)because of its wide application in structural dynamics,geophysics,spectroscopy,and so on.As a natural extension of IEPs,inverse singular value problems(ISVPs)have been widely used in structural health monitoring,transient circuit simulation,computed tomography,code division multiple access system.In particular,the transfer probability matrix can be accurately obtained by solving the stochastic inverse singular value problem(StISVP)in the parameter estimation of hidden Markov model.According to different structural ISVPs,the existing algorithms include constructive methods,recursive algorithm,Newton-type methods,alternating projection method and Riemannian Newton's method.Therefore,the research on the algorithms of ISVPs is also full of certain theoretical significance and application value.This thesis is concerned with numerical methods for qSVD and StISVP,which consists of five chapters in total.Chapter 1 reviews the background and the research progress of qEPs(qSVD)and IEPs(ISVPs).The main contributions and contents of this thesis are also included.In Chapter 2,a structure-preserving one-sided Jacobi method is proposed for qSVD.In this method,by using a sequence of orthogonal JRS-symplectic transformations for the real counterpart of a quaternion matrix,the updated matrix has sufficiently orthogonal columns,which makes the columns of the corresponding updated quaternion matrix are sufficiently orthogonal.In this case,the column scaling of the updated quaternion matrix leads to the SVD of the original quaternion matrix.The quadratic convergence analysis of the proposed method is also established.Finally,the proposed algorithm is applied to color image compression and the numerical results show the effectiveness of the proposed method.In Chapter 3,the preconditioned one-sided and two-sided Jacobi methods are presented for solving the qSVD.First,the classical rank-revealing QR factorization is extended to a quaternion matrix.Then,using this as the preconditioner,the structurepreserving one-sided and two-sided Jacobi methods are proposed.In the quaternion rank-revealing QR factorization,a permutation matrix is found by using a recursive way and the obtained upper triangular factor reveals the numerical rank of the original quaternion matrix.This preconditioner reduce a high-dimensional qSVD to a lowdimensional qSVD problem.The efficiency of preprocessing one-sided and two-sided Jacobi methods is higher than that of the real structure preserving one-sided Jacobi method proposed in Chapter 2,especially for the low rank quaternion matrix.In the numerical simulations,the proposed methods are compared with classical methods for qSVD,and the numerical results show the superiority of the algorithm.In Chapter 4,a Riemannian Newton-CG method is proposed for solving the StISVPs.The considered problem is reformulated as a matrix equation over a product matrix manifold.Then a Riemannian Newton-CG method is proposed for solving the matrix equation.Under some assumptions,the proposed method is proved to converge linearly or superlinearly.This method is also extended to the case of prescribed entries.Finally,some numerical examples are reported to demonstrate the effectiveness of the proposed method.Chapter 5 summarizes the whole research of this paper and looks forward to the future work.