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.     

Series expansion of weighted pseudoinverse matrices and iterative methods for calculating weighted pseudoinverse matrices and weighted normal pseudosolutions

Weighted pseudoinverse matrices are expanded into matrix power series with negative exponents and arbitrary positive parameters. Based on this expansion, iterative methods for evaluating weighted pseudoinverse matrices and weighted normal pseudosolutions are designed and analyzed. The iterative methods for weighted normal pseudosolutions are extended to solving constrained least-squares problems. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 32–62, January–February 2006.     

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. 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.     

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.     

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.     

Application of Pseudoinverse and Projective Matrices to Studying Control, Observation, and Identification Problems

The method of perturbation of pseudoinverse matrices, developed earlier by the authors based on splitting matrices, is extended to projective matrices for their further application to identification, nonlinear regressive analysis, function approximation, and prediction problems.     

Pseudoinverse in clustering problems

The clustering of vector observations of hyperplanes is studied. Different cases of correspondence distances are proposed and investigated, including the algebraic Jack Knife one. The efficiency, constructivity, and explicit form of formulas are provided by using the pseudoinverse technique including the pseudoinverse-perturbation theory. Results important for the application of pseudoinverse and related operators are presented. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 73–92, July–August 2007.     

Analysis and synthesis of signal classification systems by perturbing pseudoinverse and projection operations

The paper considers the synthesis of signal classification systems using the theory of perturbation of pseudoreverse and projection matrices and proposes an algorithm to select (filter) separable subsets on a finite point set from R ⁿ . Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 124–134, July–August 2009.     

Synthesis of systems of neurofunctional transformations in classification problems

Optimal synthesis of linear and nonlinear transformations is used to synthesize pattern recognition systems. The necessary and sufficient conditions for the existence of robust dichotomous linear separability of sets in feature space are represented in terms of pseudoinverse operations. The synthesis of classification systems is reduced to searching for the best nonlinear transformations of the components of the feature vector or optimal linear combinations of its components. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 47–56, May–June 2007.     

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.     

Optimization of the synthesis of hyperplane clusters and neurofunctional transforms in signal classification systems

The paper continues the analysis of clustering and signal classification problems. For problems of point-set clustering in a feature space, an optimal algorithm to synthesize hyperplane clusters and an algorithm to solve a problem on linear separability of a finite point set are obtained, criteria for linear strip separability of points in a feature space into two classes are formulated, and methods of finding optimal nonlinear transform of a coordinate of the feature vector in given classes of functions and its index are developed. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 50–58, November–December 2008.     

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.     

Orthogonal Transforms in Bases of Slant Step Functions. II. Methods of Generating Orthogonal Slant Transforms with Fast Algorithms for Image Coding and Compression

Recurrent matrix methods and methods based on the Walsh transform and rotation matrices generating orthogonal slant transforms of high and low correlation are proposed. These methods are used to develop efficient fast slant-transform algorithms without multiplication and additional permutations of output data. Continued from Cybernetics and Systems Analysis, Vol. 41, No. 3, 2005. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 116–132, July–August 2005.     

Finding maximum (in modulus) complex-conjugate eigenvalues of arbitrarily high dimension

Algorithms of determining maximum (in modulus) complex-conjugate eigenvalues are considered as applied to finding eigenvalues of high-dimension matrices according to the Khilenko method. An advantage of the algorithms is that the amount of calculation does not increase exponentially with the dimension of matrices. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 104–110, March–April 2006.     

Efficient Chebyshev–Petrov–Galerkin Method for Solving Second-Order Equations

A new efficient Chebyshev–Petrov–Galerkin (CPG) direct solver is presented for the second order elliptic problems in square domain where the Dirichlet and Neumann boundary conditions are considered. The CPG method is based on the orthogonality property of the kth-derivative of the Chebyshev polynomials. The algorithm differs from other spectral solvers by the high sparsity of the coefficient matrices: the stiffness and mass matrices are reduced to special banded matrices with two and four nonzero diagonals respectively. The efficiency and the spectral accuracy of CPG method have been validated.     

Efficient solution of nonlinear elliptic problems using hierarchical matrices with Broyden updates

Summary The numerical solution of nonlinear problems is usually connected with Newton’s method. Due to its computational cost, variants (so-called inexact and quasi–Newton methods) have been developed in which the arising inverse of the Jacobian is replaced by an approximation. In this article we present a new approach which is based on Broyden updates. This method does not require to store the update history since the updates are explicitly added to the matrix. In addition to updating the inverse we introduce a method which constructs updates of the LU decomposition. To this end, we present an algorithm for the efficient multiplication of hierarchical and semi-separable matrices. Since an approximate LU decomposition of finite element stiffness matrices can be efficiently computed in the set of hierarchical matrices, the complexity of the proposed method scales almost linearly. Numerical examples demonstrate the effectiveness of this new approach. This work was supported by the DFG priority program SPP 1146 “Modellierung inkrementeller Umformverfahren”.     

A recursive,numerically stable,and efficient simulation algorithm for serial robots

Traditionally, the dynamic model, i.e., the equations of motion, of a robotic system is derived from Euler–Lagrange (EL) or Newton–Euler (NE) equations. The EL equations begin with a set of generally independent generalized coordinates, whereas the NE equations are based on the Cartesian coordinates. The NE equations consider various forces and moments on the free body diagram of each link of the robotic system at hand, and, hence, require the calculation of the constrained forces and moments that eventually do not participate in the motion of the coupled system. Hence, the principle of elimination of constraint forces has been proposed in the literature. One such methodology is based on the Decoupled Natural Orthogonal Complement (DeNOC) matrices, reported elsewhere. It is shown in this paper that one can also begin with the EL equations of motion based on the kinetic and potential energies of the system, and use the DeNOC matrices to obtain the independent equations of motion. The advantage of the proposed approach is that a computationally more efficient forward dynamics algorithm for the serial robots having slender rods is obtained, which is numerically stable. The typical six-degree-of-freedom PUMA robot is considered here to illustrate the advantages of the proposed algorithm.     

图的赋权路径矩阵与所有点对最短路径问题

给出了二维元素矩阵的概念,对于赋权图对应的赋权矩阵,定义了二维元素初始赋权路径矩阵和二维元素一般赋权路径矩阵,在通常赋权矩阵“乘法”运算基础上定义了路径“乘法”运算,从而得到了二维元素一般赋权路径矩阵的“乘法”运算,通过其“乘法”运算来求出所有点对的最短距离与对应路径,在得到最短距离的同时也得到对应的路径,结果显示在最终的一般赋权路径矩阵上。该算法易于通过计算机编程实现,对于大规模有向图或无向图,更有优势。     

Efficiently solvable cases of quadratic assignment problem with generalized monotonic and incomplete anti-monge matrices

The paper describes new sufficient conditions that provide optimum to a quadratic assignment problem for a given substitution. The conditions separate out sets of matrices in the matrix space. The sets are formed by incomplete Anti-Monge matrices, which permit opposite ordering of rows and columns, and by additive monotone matrices, which are outside the class of Toeplitz matrices. The study was partially supported by INTAS (Project 03-51-5501). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 135–151, January–February 2007.