Gradient based iterative solutions for general linear matrix equations

AbstractIn this paper, we present a gradient based iterative algorithm for solving general linear matrix equations by extending the Jacobi iteration and by applying the hierarchical identification principle. Convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices. Two numerical examples are provided to show that the proposed algorithm is effective

Least Squares Based Iterative Algorithm for the Coupled Sylvester Matrix Equations

By analyzing the eigenvalues of the related matrices, the convergence analysis of the least squares based iteration is given for solving the coupled Sylvester equations AX+YB=C and DX+YE=F in this paper. The analysis shows that the optimal convergence factor of this iterative algorithm is 1. In addition, the proposed iterative algorithm can solve the generalized Sylvester equation AXB+CXD=F. The analysis demonstrates that if the matrix equation has a unique solution then the least squares based iterative solution converges to the exact solution for any initial values. A numerical example illustrates the effectiveness of the proposed algorithm

Toward Solution of Matrix Equation X=Af(X)B+C

This paper studies the solvability, existence of unique solution, closed-form solution and numerical solution of matrix equation $X=Af(X) B+C$ with $f(X) =X^{\mathrm{T}},$ $f(X) =\bar{X}$ and $f(X) =X^{\mathrm{H}},$ where $X$ is the unknown. It is proven that the solvability of these equations is equivalent to the solvability of some auxiliary standard Stein equations in the form of $W=\mathcal{A}W\mathcal{B}+\mathcal{C}$ where the dimensions of the coefficient matrices $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$ are the same as those of the original equation. Closed-form solutions of equation $X=Af(X) B+C$ can then be obtained by utilizing standard results on the standard Stein equation. On the other hand, some generalized Stein iterations and accelerated Stein iterations are proposed to obtain numerical solutions of equation equation $X=Af(X) B+C$. Necessary and sufficient conditions are established to guarantee the convergence of the iterations

Positive Definite Solutions of the Nonlinear Matrix Equation X+AHXˉ−1A=IX+A^{\mathrm{H}}\bar{X}^{-1}A=I

This paper is concerned with the positive definite solutions to the matrix equation $X+A^{\mathrm{H}}\bar{X}^{-1}A=I$ where $X$ is the unknown and $A$ is a given complex matrix. By introducing and studying a matrix operator on complex matrices, it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation $W+B^{\mathrm{T}}W^{-1}B=I$ which has been extensively studied in the literature, where $B$ is a real matrix and is uniquely determined by $A.$ It is also shown that if the considered nonlinear matrix equation has a positive definite solution, then it has the maximal and minimal solutions. Bounds of the positive definite solutions are also established in terms of matrix $A$. Finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed

WavePacket: A Matlab package for numerical quantum dynamics. II: Open quantum systems, optimal control, and model reduction

WavePacket is an open-source program package for numeric simulations in quantum dynamics. It can solve time-independent or time-dependent linear Schr\"odinger and Liouville-von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows, e.g., to simulate molecular quantum dynamics beyond the Born-Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semi-classical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry. Being highly versatile and offering visualization of quantum dynamics 'on the fly', WavePacket is well suited for teaching or research projects in atomic, molecular and optical physics as well as in physical or theoretical chemistry. Building on the previous Part I which dealt with closed quantum systems and discrete variable representations, the present Part II focuses on the dynamics of open quantum systems, with Lindblad operators modeling dissipation and dephasing. This part also describes the WavePacket function for optimal control of quantum dynamics, building on rapid monotonically convergent iteration methods. Furthermore, two different approaches to dimension reduction implemented in WavePacket are documented here. In the first one, a balancing transformation based on the concepts of controllability and observability Gramians is used to identify states that are neither well controllable nor well observable. Those states are either truncated or averaged out. In the other approach, the H2-error for a given reduced dimensionality is minimized by H2 optimal model reduction techniques, utilizing a bilinear iterative rational Krylov algorithm

Iterative Solutions of a Set of Matrix Equations by Using the Hierarchical Identification Principle

This paper is concerned with iterative solution to a class of the real coupled matrix equations. By using the hierarchical identification principle, a gradient-based iterative algorithm is constructed to solve the real coupled matrix equations A1XB1+A2XB2=F1 and C1XD1+C2XD2=F2. The range of the convergence factor is derived to guarantee that the iterative algorithm is convergent for any initial value. The analysis indicates that if the coupled matrix equations have a unique solution, then the iterative solution converges fast to the exact one for any initial value under proper conditions. A numerical example is provided to illustrate the effectiveness of the proposed algorithm

LSMR Iterative Method for General Coupled Matrix Equations

By extending the idea of LSMR method, we present an iterative method to solve the general coupled matrix equations ∑k=1qAikXkBik=Ci, i=1,2,…,p, (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over some constrained matrix groups (X1,X2,…,Xq), such as symmetric, generalized bisymmetric, and (R,S)-symmetric matrix groups. By this iterative method, for any initial matrix group (X1(0),X2(0),…,Xq(0)), a solution group (X1*,X2*,…,Xq*) can be obtained within finite iteration steps in absence of round-off errors, and the minimum Frobenius norm solution or the minimum Frobenius norm least-squares solution group can be derived when an appropriate initial iterative matrix group is chosen. In addition, the optimal approximation solution group to a given matrix group (X¯1,X¯2,…,X¯q) in the Frobenius norm can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the presented method

A gradient based iterative solutions for Sylvester tensor equations

This paper is concerned with the numerical solution of the Sylvester tensor equation, which includes the Sylvester matrix equation as special case. By applying hierarchical identification principle proposed by Ding and Chen, 2005, and by using tensor arithmetic concepts, an iterative algorithm and its modification are established to solve the Sylvester tensor equation. Convergence analysis indicates that the iterative solutions always converge to the exact solution for arbitrary initial value. Finally, some examples are provided to show that the proposed algorithms are effective