A NEW ALGORITHM FOR COMPUTING THE INVERSE AND GENERALIZED INVERSE OF THE SCALED FACTOR CIRCULANT MATRIX

A new algorithm for finding the inverse of a nonsingular scaled factor circulant matrix is presented by the Euclid＇s algorithm. Extension is made to compute the group inverse and the Moore-Penrose inverse of the singular scaled factor circulant matrix. Numerical examples are presented to demonstrate the implementation of the proposed algorithm.     

A homogeneous smoothing-type algorithm for symmetric cone linear programs

In this paper, we investigate a smoothing-type algorithm for solving the symmetric cone linear program （（SCLP） for short） by making use of an augmented system of its optimality conditions. The algorithm only needs to solve one system of linear equations and to perform one line search at each iteration. It is proved that the algorithm is globally convergent without assuming any prior knowledge of feasibility/infeasibility of the problem. In particular, the algorithm may correctly detect solvability of （SCLP）. Furthermore, if （SCLP） has a solution, then the algorithm will generate a solution of （SCLP）, and if the problem is strongly infeasible, the algorithm will correctly detect infeasibility of （SCLP）.     

An algorithm for decomposing a polynomial system into normal ascending sets

We present an algorithm to decompose a polynomial system into a finite set of normal ascending sets such that the set of the zeros of the polynomial system is the union of the sets of the regular zeros of the normal ascending sets.If the polynomial system is zero dimensional,the set of the zeros of the polynomials is the union of the sets of the zeros of the normal ascending sets.     

A MODIFIED LEVENBERG-MARQUARDT ALGORITHM FOR SINGULAR SYSTEM OF NONLINEAR EQUATIONS

Based on the work of paper,we propose a modified Levenberg-Marquardt algoithm for solving singular system of nonlinear equations F(x)=0,where F(x):R^n→R^n is continuously differentiable and F‘(x)is Lipschitz continuous.The algorithm is equivalent to a trust region algorithm in some sense,and the global convergence result is given.The sequence generated by the algorithm converges to the solution quadratically,if ||F(x)||2 provides a local error bound for the system of nonlinear equations.Numerical results show that the algorithm performs well.     

ASYMPTOTIC NORMALITY OF THE CONTINUOUS-TIME STOCHASTIC APPROXIMATION ALGORITHM

In this paper the continuous-time stochastic approximation algorithm seeking for the zero of aregression function is considered when the measurement error is a stochastic process generated by anIto integral as the input of a linear system.The conditions are given to guarantee the asymptoticnormality of the algorithm which is modified from the Robbins-Monro procedure proposed for thecase where the measurement error is a process of independent increment.     

解线约束凸规划的势下降算法

A potential reduction algorithm is proposed for optimization of a convex function subject to linear constraints. At each step of the algorithm,a system of linear equations is solved toget a search direction and the Armijo‘s rule is used to determine a stepsize. It is proved that thealgorithm is globally convergent. Computational results are reported.     

A new sequential systems of linear equations algorithm of feasible descent for inequality constrained optimization

Based on a new efficient identification technique of active constraints introduced in this paper, a new sequential systems of linear equations （SSLE） algorithm generating feasible iterates is proposed for solving nonlinear optimization problems with inequality constraints. In this paper, we introduce a new technique for constructing the system of linear equations, which recurs to a perturbation for the gradients of the constraint functions. At each iteration of the new algorithm, a feasible descent direction is obtained by solving only one system of linear equations without doing convex combination. To ensure the global convergence and avoid the Maratos effect, the algorithm needs to solve two additional reduced systems of linear equations with the same coefficient matrix after finite iterations. The proposed algorithm is proved to be globally and superlinearly convergent under some mild conditions. What distinguishes this algorithm from the previous feasible SSLE algorithms is that an improving direction is obtained easily and the computation cost of generating a new iterate is reduced. Finally, a preliminary implementation has been tested.     

A SUPERLINEARLY CONVERGENT TRUST REGION ALGORITHM FOR LC^1 CONSTRAINED OPTIMIZATION PROBLEMS

In this paper, a new trust region algorithm for nonlinear equality constrained LC^1 optimization problems is given. It obtains a search direction at each iteration not by solving a quadratic programming subproblem with a trust region bound, but by solving a system of linear equations. Since the computational complexity of a QP-Problem is in general much larger than that of a system of linear equations, this method proposed in this paper may reduce the computational complexity and hence improve computational efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm is globally and super-linearly convergent to a solution of the original problem. Some numerical examples are reported, showing the proposed algorithm can be beneficial from a computational point of view.     

PARTIAL ELIMINTATION ALGORITHM FOR A LINEAR RECURRENCE SYSTEM R(n, m) OF ORDER m

This paper presents an algorithm for computing a linear recurrence system R(n, m) of order m for n equations on MIMD parallel system. This algorithm is not only easy to be programmed on a parallel computer system, but also reduces the data-waiting time due to compute-ahead strategy. The paper analyses how to achieve maximal load balancing when the algorithm is implemented on MIMD parallel system. By the end of the paper, an analysis on the speedup and parallel efficiency are given. The results indicate that the new parallel elimination algorithm has great improvement compared with the old ones.     

A TRANSFORMATION PATH ALGORITHM FOR UNCONSTRAINED SIGNOMIAL GEOMETRIC PROGRAMMING

In this paper we present a transformation path algorithm for UnconstrainedSignomial Geometric Programming (USGP). The algorithm is proposed from a newpoint of view based on exploring the characteristics of USGP problem. Firstly by somestable transformations, a particular subproblem is derived which is very easy to solve.Secondly, a special path is formed conveniently. And then the step of the algorithmconsists in finding a “good” point to the current iterate by choosing it along the specialpath and within a trust region. It is proved that the algorithm is globally convergent.     

SOME PROPERTIES OF CENTROSYMMETRIC MATRICES AND ITS APPLICATIONS

In this paper, some properties of centrosymmetric matrices, which often appear in the construction of orthonormal wavelet basis in wavelet analysis, are investigated. As an application, an algorithm which is tightly related to a so-called Lawton matrix is presented. In this algorithm, about only half of memory units are required and quarter of computational cost is needed by exploiting the property of the Lawton matrix and using a compression technique, it is compared to one for the original Lawton matrix.     

Convergence analysis of a nonlinear Lagrange algorithm for general nonlinear constrained optimization problems

The convergence analysis of a nonlinear Lagrange algorithm for solving nonlinear constrained optimization problems with both inequality and equality constraints is explored in detail. The estimates for the derivatives of the multiplier mapping and the solution mapping of the proposed algorithm are discussed via the technique of the singular value decomposition of matrix. Based on the estimates, the local convergence results and the rate of convergence of the algorithm are presented when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions. Furthermore, the condition number of the Hessian of the nonlinear Lagrange function with respect to the decision variables is analyzed, which is closely related to efficiency of the algorithm. Finally, the preliminary numericM results for several typical test problems are reported.     

一类非单调线性互补问题的高阶仿射尺度算法

In this paper, a new interior point algorithm-high-order atone scaling for a class of nonmonotonic linear complementary problems is developed. On the basis of idea of primal-dual affine scaling method for linear programming , the search direction of our algorithm is obtained by a linear system of equation at each step . We show that, by appropriately choosing the step size, the algorithm has polynomial time complexity. We also give the numberical results of the algorithm for two test problems.     

A NEW ALGORITHM FOR COMPUTING LARGEST REAL PART EIGENVALUE OF MATRICES：COLLATZ ＆ PERRON-FROBERNIUS＇ APPROACH

This paper describes a new method and algorithm for the numerical solution of eigenvalues with the largest real part of positive matrices.The method is based on a numerical implementation of Collatz’s eigenvalue inclusion theorem for non-negative irreducible matrices.Eigenvalues are analyzed for the studies of the stability of linear systems.Finally, a numerical discussion is given to derive the required number of mathematical operations of the new algorithm. Comparisons between the new algorithm and several well known ones, such as Power, and QR methods, are discussed.     

Domain decomposition method for a system of quasivariational inequalities

We propose a domain decomposition method for a system of quasivariational inequalities related to the HJB equation. The monotone convergence of the algorithm is also established.     

A FAST SINE TRANSFORM ALGORITHM FOR TOEPLITZ MATRICES AND ITS APPLICATIONS

In this paper, a fast algorithm for the discrete sine transform(DST) of a Toeplitz matrix of order N is derived. Only O(N log N) O(M) time is needed for the computation of M elements. The auxiliary storage requirement is O(N). An application of the new fast algorithm is also discussed.     

STABLE DOUBLE LR ALGORITHM AND ITS ERROR ANALYSIS

In this paper, the normative matrices and their double LR transformation with origin shifts are defined, and the essential relationship between the double LR transformation of a normative matrix and the QR transformation of the related symmetric tridiagonal matrix is proved. We obtain a stable double LR algorithm for double LR transformation of normative matrices and give the error analysis of our algorithm. The operation number of the stable double LR algorithm for normative matrices is only four sevenths of the rational QR algorithm for reed symmetric tridiagonal matrices.     

一类非最小相位随机系统的离散自适应滑动模态控制

This paper presents the discrete adaptive sliding mode control of input-output non-minimum phase system in the presence of the stochastic disturbance. The non-minimum phase system can be transformed into a minimum phase system by a operator. According to the minimum phase system, the controller and the adaptive algorithm we designed ensures the stability of system and holds that the mean-square deviation from the sliding surface is minimized.     

Reconstruction of the Linear Ordinary Differential System Based on Discrete Points

In this paper, we discuss an inverse problem, i.e., the reconstruction of a linear differential dynamic system from the given discrete data of the solution. We propose a model and a corresponding algorithm to recover the coefficient matrix of the differential system based on the normal vectors from the given discrete points, in order to avoid the problem of parameterization in curve fitting and approximation. We also give some theoretical analysis on our algorithm. When the data points are taken from the solution curve and the set composed of these data points is not degenerate, the coefficient matrix $A$ reconstructed by our algorithm is unique from the given discrete and noisefree data. We discuss the error bounds for the approximate coefficient matrix and the solution which are reconstructed by our algorithm. Numerical examples demonstrate the effectiveness of the algorithm.