Expansion of weighted pseudoinverse matrices with positive definite weights into matrix power products. Iterative methods

Weighted pseudoinverse matrices with positive definite weights are expanded into matrix power products with negative exponents and arbitrary positive parameters. These expansions are used to develop and analyze iterative methods for evaluating weighted pseudoinverse matrices and weighted normal pseudosolutions and solving constrained least-squares problems. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 45–64, January–February 2007.     

Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. I. Positive definite weights

The paper reviews studies on the representations and expansions of weighted pseudoinverse matrices with positive definite weights and on iterative methods and regularized problems for calculation of weighted pseudoinverse matrices and weighted normal pseudosolutions. The use of these methods to solve constrained least-squares problems is examined. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 47–73, January–February 2008.     

Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. II. Singular weights

The paper reviews studies on the representations and expansions of weighted pseudoinverse matrices with positive semidefinite weights and on the construction of iterative methods and regularized problems for the calculation of weighted pseudoinverses and weighted normal pseudosolutions based on these representations and expansions. The use of these methods to solve constrained least squares problems is examined. Continued from Cybernetics and Systems Analysis, 44, No. 1, 36–55 (2008). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 75–102, May–June 2008.     

Iterative methods with different rates of convergence for calculating weighted pseudoinverse matrices and weighted normal pseudosolutions with positive definite weights

The authors develop and analyze iterative methods with different (linear, quadratic, or of p (p2) order) rates of convergence. The methods are used to calculate weighted pseudoinverse matrices with positive defined weights. To find weighted normal pseudosolutions with positive defined weights, iterative methods with a quadratic rate of convergence are developed and analyzed. The iterative methods for calculation of weighted normal pseudosolutions are used to solve least-square problems with constraints.Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 20–44, September–October 2004.     

Methods of computing weighted pseudoinverse matrices with singular weights

Three iterative processes are constructed and investigated for computing weighted pseudoinverse matrices with singular weights and ML-weighted pseudoinverse matrices. Two of them are based on the decompositions of the weighted pseudoinverse matrix with singular weights into matrix power series, and the third is a generalization of the Schulz method to nonsingular square matrices. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 150–169, September–October, 1999.     

Limiting Representations of Weighted Pseudoinverse Matrices with Positive Definite Weights. Problem Regularization

Limiting representations for weighted pseudoinverse matrices with positive definite weights are derived. It is shown that regularized problems can be constructed based on such limiting representations intended for evaluation of weighted pseudoinverse matrices and weighted normal pseudosolutions with positive definite weights. The results obtained, concerning regularization of problems on evaluation of weighted normal pseudosolutions, are employed for regularization of least-squares problems with constraints.     

Existence and uniqueness theorems in the theory of weighted pseudoinverses with singular weights

Necessary and sufficient conditions for the existence and uniqueness of weighted pseudoinverses with singular weights are obtained. Pseudoinverses are expanded into matrix power series and power products. A relationship is found between weighted pseudoinverses and weighted normal pseudosolutions, and iterative methods are established for calculating pseudoinverses and pseudosolutions.     

Methods for Computing Weighted Pseudoinverses and Weighted Normal Pseudosolutions with Singular Weights

The paper surveys articles that construct and investigate direct and iterative methods for computing weighted pseudoinverses and weighted normal pseudosolutions with singular weights. The methods considered in the paper are mainly constructed based on the authors’ articles devoted to the development of the theory of weighted pseudoinversion aimed at investigating the characteristics of both weighted pseudoinverses and weighted normal pseudosolutions with singular weights. The paper uses the following results obtained and investigated by the authors: expansions of weighted pseudoinverses into matrix power series and products, limit representations of such matrices, and determination of decompositions of weighted pseudoinverses based on weighted singular value decompositions of matrices with singular weights.     

基于Moore-Penrose 逆的加权距离函数研究

"距离"是科学研究与工程技术领域中使用非常广泛的一种度量.在分析各种距离优、缺点的基础上,根据马氏距离不受量纲影响,能描述和处理相关性数据的性能优势,利用加权Moore-Penrose(WMP)广义逆定义了WMP马氏距离,并通过奇异值分解及矩阵的谱分解理论构造其数学形式和计算方法.理论分析和仿真实验表明,所提出的方法不仅保持了马氏距离和MP马氏距离的优点,而且克服了它们的缺点,同时又具有更好的独特性能.     

Iterative methods to compute weighted normal pseudosolutions with positive definite weights

We construct iterative processes to compute the weighted normal pseudosolution with positive definite weights (weighted least squares solutions with weighted minimum Euclidean norm) for systems of linear algebraic equations (SLAE) with an arbitrary rectangular real matrix. We examine two iterative processes based on the expansion of the weighted pseudoinversc matrix into matrix power series. The iterative processes are applied to solve constrained least squares problems that arise in mathematical programming and to findL-pseudosolutions. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 116–124, March–April, 1998.     

按对角阵加权信息融合Kalman滤波器

为了克服按矩阵加权信息融合非稳态Kalman滤波器的在线计算负担大的缺点,和按标量加权融合Kalman滤波器精度较低的缺点,应用现代时间序列分析方法,提出了按对角阵加权的线性最小方差多传感器信息融合稳态Kalman滤波器.它等价于状态分量按标量加权信息融合Kalman滤波器,实现了解耦信息融合Kalman滤波器.它的精度和计算负担介于按矩阵和按标量加权融合器两者之间,且便于实时应用.为了计算最优加权,提出了计算稳态滤波误差方差阵和协方差阵的Lyapunov方程.一个三传感器的雷达跟踪系统的仿真例子说明了其有效性.     

Torque distribution using a weighted pseudoinverse in a redundantly actuated mechanism

When mobility, the number of independent variables to describe system motion exactly, is greater than the degree-of-freedom of task space, the system is called a kinematically redundant system. On the other hand, redundant actuation indicates a situation when there are more actuators than a system's mobility. Redundant actuation yields many advantages. First, actuation redundancy can increase the force, velocity and acceleration of an end-effector. Second, if some actuators are out of order, the system can still work well. This fault-tolerant capability is useful for remote control robots in space or nuclear plants. Impulsive force can decrease when modulating arbitrary stiffness without feedback control. The performance of a system can be improved by optimizing redundancy. However, there are some issues of economic efficiency and minimization of a system, because redundant actuation may involve more actuators than non-redundant actuation. In addition, there are infinite torque sets of motors for the same task. We used a weighted pseudoinverse matrix for torque distribution. To reduce the maximum torque, we suggested the use of the minimum norm torque as the weighting values. This method allows for smaller motor capacity, and can contribute to economic efficiency and minimization of a system.     

Representing Weighted Pseudoinverse Matrices with Mixed Weights in Terms of Other Pseudoinverses

The paper considers weighted pseudoinverse where both weighted matrices are symmetric and one of them is positive definite matrix and the other is nonsingular and indefinite. Formulas are obtained to represent these matrices in terms of the Moore–Penrose pseudoinverse matrix and other weighted pseudoinverses.     

New quadratic models for the maximum weighted cut problem

New quadratic models are proposed to improve the upper-bound estimates in the maximum weighted cut problem. They are found by two original methods for deriving redundant quadratic constraints. A well-known linear model is shown to follow from the models proposed. Recommendations on how to develop its strengthened analogs are given. This study is partially sponsored from the grant UM2-2574-KV-03 (CRDF Cooperative Grants Program). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 63—75, January–February 2006.     

Reliability analysis of computer solutions of systems of linear algebraic equations with approximate initial data

A weighted least squares problem {ie863-01} with positive definite weights M and N is considered, where A ∈ R^m×n is a rank-deficient matrix, b ∈ R^m. The hereditary, computational, and global errors of a weighted normal pseudosolution are estimated for perturbed initial data, including the case where the rank of the perturbed matrix varies. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 83–95, November–December 2008.     

Order-recursive factorization of the pseudoinverse of a covariancematrix

A numerically reliable algorithm is developed to recursively update the square root of a pseudoinverse matrix using the square root of a lower dimension pseudoinverse matrix. The numerical computations are based on a generalized singular value decomposition which is used to do a canonical correlation analysis. An operation count is given for sequential and parallel implementation of a partitioned order-recursive algorithm. These methods are useful for covariance analysis to determine the contributions due to various modeling errors in the design of a Kalman filter     

Laplacian matrices of weighted digraphs represented as quantum states

Representing graphs as quantum states is becoming an increasingly important approach to study entanglement of mixed states, alternate to the standard linear algebraic density matrix-based approach of study. In this paper, we propose a general weighted directed graph framework for investigating properties of a large class of quantum states which are defined by three types of Laplacian matrices associated with such graphs. We generalize the standard framework of defining density matrices from simple connected graphs to density matrices using both combinatorial and signless Laplacian matrices associated with weighted directed graphs with complex edge weights and with/without self-loops. We also introduce a new notion of Laplacian matrix, which we call signed Laplacian matrix associated with such graphs. We produce necessary and/or sufficient conditions for such graphs to correspond to pure and mixed quantum states. Using these criteria, we finally determine the graphs whose corresponding density matrices represent entangled pure states which are well known and important for quantum computation applications. We observe that all these entangled pure states share a common combinatorial structure.     

Remarks on a formula for the pseudoinverse by the modified huang algorithm

We present and prove an alternative formula of the Moore-Penrose pseudoinverse for a matrix by the modified Huang algorithm. The results correct a formula in the book entitled ‘ABS projection Algorithms: mathematical techniques for linear and nonlinear equations’. Supported by the National Natural Science Foundation of China.     

On-line system identification algorithm combining stochastic approximation and pseudoinverse

A new algorithm for on-line system identification is presented combining stochastic approximation with the recursive pseudoinverse algorithm for least squares estimation. It is an extension of Clarke's generalized least squares method, and gives unbiased estimates of the parameters even in the presence of large noise. An example of a simulated second-order process compares the proposed method with three earlier methods.