Solving Some Tensor Equations

The problem of solving matrix equations is widely derived from signal processing,structural design,stability and control theory.Since tensors are higher order generalizations of matrices,the research of solving tensor equations becomes a hot topic.The work of this thesis is divided into three parts: the first part is concerned with numerical algorithms for solving two kinds of tensor equations via Einstein product;the second part is devoted to the numerical solutions of several tensor equations under mode product;the third part investigates the analytical solution of a class of tensor equations.It mainly includes:1.The numerical solutions of two kinds of tensor equations via Einstein product.The relations between Einstein multiplication and the usual matrix multiplication are founded in Chapter 2.By using the idea of linear search in tensor space,some numerical algorithms are presented to solve two tensor equations via Einstein product and the corresponding least squares problems.These methods solely use tensor computations,i.e.,no matricizations are involved.Theoretical analysis shows that,for any initial tensor,the exact solution can be obtained within finite steps in the absence of roundoff errors when the considered tensor equation is consistent.2.The numerical solution of the Sylvester tensor equation.The Sylvester tensor equation is the higher order form of Sylvester matrix equation.In Chapter 3,a linear mapping on tensor space is first introduced,and then two algorithms are derived to solve the Sylvester tensor equation.Moreover,the BiCOR and CORS methods are extended to solve the Sylvester tensor equation.Their convergent properties are also given.Numerical examples are provided to confirm the theoretical results.Comparing with other existing methods,these presented methods are more effective in terms of the required CPU times for the convergence and relative errors of approximate solutions.3.Least squares solution of the quaternion Sylvester tensor equation.The quaternion Sylvester tensor equation is proposed in Chapter 4.By the real and complex representations of tensors,some equivalent forms for the quaternion Sylvester tensor equation are given.The least squares problem for quaternion Sylvester tensor equation and the related best approximate problem are considered.The conjugate gradient least squares method based on tensor format is presented to solve these problems.The convergence properties of the proposed method are studied.Some numerical examples are provided to show the feasibility and effectiveness of the proposed method.4.Numerical methods for solving the Stein tensor equation.As a natural generalization of Stein matrix equation,the Stein tensor equation is proposed in Chapter 5.The equivalent linear system of the Stein tensor equation is given,and under special conditions the solution with the series representation is obtained.In addition,the BiCG and BiCR methods based on tensor format are developed to solve the Stein tensor equation.The convergent properties of the extended algorithms are studied.5.The solutions to a class of tensor equations and their applications.The analytic solutions of a class of tensor equations are investigated in Chapter 6.The Sylvester tensor equation,Stein tensor equation,continuous-time Lyapunov tensor equation and discrete-time Lyapunov tensor equation can be regarded as special cases of this kind of tensor equations.The solution of this kind of tensor equation is obtained under the condition of diagonalization.By some fundamental results of spectral theory,some conclusions given by Lancaster are extended.Moreover,some results of Wimmer and Ziebur are also developed.The connection between Lancaster's and Wimmer and Ziebur's results is discovered.Furthermore,some solution expressions are employed to prove Lyapunov and Stein stability theorems for tensors.