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.