Solving polynomial nonlinear matrix equations by a linearization method

This paper deals with some modification of a matrix linearization method. The scheme proposed makes it possible to find tuples of solutions for systems of polynomial nonlinear equations defined on a commutative matrix ring. The matrix linearization method reduces an initial polynomial nonlinear problem to a linear one with respect to matrices of solutions. Then, the method of elimination of unknowns is used to obtain a generalized eigenvalue problem. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 60–69, May–June 2006.     

Solving inverse eigenvalue problems by a projected Newton method

The inverse eigenvalue problem for symmetric matrices (IEP) can be formulated as a system of two matrix equations. For solving the system a variation of Newton's method is used which has been proposed by Fusco and Zecca [Calcolo XXIII (1986), pp. 285–303] for the simultaneous computation of eigenvalues and eigenvectors of a given symmetric matrix. An iteration step of this method consists of a Newton step followed by an orthonormalization with the consequence that each iterate satisfies one of the given equations. The method is proved to convergence locally quadratically to regular solutions. The algorithm and some numerical examples are presented. In addition, it is shown that the so-called Method III proposed by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634–667] for solving IEP may be constructed similarly to the method presented here.     

Supervised inductive learning with Lotka–Volterra derived models

We present a classification algorithm built on our adaptation of the Generalized Lotka–Volterra model, well-known in mathematical ecology. The training algorithm itself consists only of computing several scalars, per each training vector, using a single global user parameter and then solving a linear system of equations. Construction of the system matrix is driven by our model and based on kernel functions. The model allows an interesting point of view of kernels’ role in the inductive learning process. We describe the model through axiomatic postulates. Finally, we present the results of the preliminary validation experiments.     

Modeling and simulation of drug release from multi-layer biodegradable polymer microstructure in three dimensions

Multi-layer biodegradable polymer drug delivery microstructures with micro-chambers have many exceptional advantages in a long-term controlled drug delivery. In this paper, a mathematical model for the drug release from this type of three-dimensional (3D) drug delivery microstructures is presented by using 3D cellular automata and discrete iterations. The validity of the model is proved by comparing with the Gőpferich’s simulated profiles for the same composite polymer cylinder matrices. Based on this model, furthermore, the simulations are done to describe the dynamic behavior of drug release from this type of microstructures. The simulation results show that this 3D mathematical model can describe accurately the drug release behavior and therefore can provide a new optimal design tool for this kind of multi-layer biodegradable polymer drug delivery microstructure.     

Analysis of the properties of a linear system using the method of artificial basis matrices

Formulas relating elements of the method in adjacent basis matrices are used to solve a system of linear algebraic equations and to represent analytically the general solutions to the corresponding system of linear algebraic inequalities for a nondegenerate constraint matrix. Sponsored by the ICS NATO program of April 18 2006, in line with the Project “Optimal replacement of information technologies and stable development (in Kazakhstan, Ukraine, and the USA),” NATO Grant CLG 982209. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 119–127, July–August 2007.     

Parallel direct method for linear equations

A simple direct method and its variants, based on matrix multiplications, to invert square non-singular matrices (need not be positive definite) and to solve systems of linear equations are developed. Since the method and its variants involve only matrix multiplications they are straightforward to parallelise and hence have an advantage over some existing well known direct methods. Theoretical background, analysis of the proposed method and the complexity of the algorithm for sparse and dense matrices are given. Two significantly different parallel algorithms for dense and sparse matrices are developed. In the case of the dense matrix standard parallelisation of matrix multiplications was undertaken. However, for sparse matrices the standard parallel matrix multiplication is not effective which led us to develop a parallel algorithm different from that for the dense matrix. The performances of our algorithms are tested via numerical examples.     

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.     

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.     

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.     

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.     

Complement of Source Equation of Elastic Line

A new notion of joint, defined in terms of the state of motor (active or locked) and type of the elastic or rigid element, gear and/or link that follows after the motor, is introduced. Special attention is paid to the motion of the flexible links in the robotic configuration. The paper deals with the relationship between the equation of elastic line equilibrium, the “Euler–Bernoulli approach” (EBA), and equation of motion at the point of elastic line tip, the “Lumped-mass approach” (LMA). The Euler–Bernoulli equations (which have for a long time been used in the literature) should be expanded according to the requirements of the motion complexity of elastic robotic systems. The Euler–Bernoulli equation (based on the known laws of dynamics) should be supplemented with all the forces that are participating in the formation of the elasticity moment of the considered mode. This yields the difference in the structure of Euler–Bernoulli equations for each mode. The stiffness matrix is a full matrix. Mathematical model of the actuators also comprises coupling between elasticity forces. Particular integral of Daniel Bernoulli should be supplemented with the stationary character of elastic deformation of any point of the considered mode, caused by the present forces. General form of the elastic line is a direct outcome of the system motion dynamics, and cannot be described by one scalar equation but by three equations for position and three equations for orientation of every point on that elastic line. The choice of reference trajectory is analyzed. Simulation results are shown for a selected robotic example involving the simultaneous presence of elasticity of the gear and of the link (two modes), as well as the environment force dynamics.     

Discovering Dynamic Characteristics of Biochemical Pathways using Geometric Patterns among Parameter-Parameter Dependencies in Differential Equations

This paper proposes a novel approach to the analysis and validation of mathematical models using two-dimensional geometrical patterns representing parameter-parameter dependencies (PPD) in dynamic systems. A geometrical pattern is obtained by calculating moment values, such as the area under the curve (AUC), area under the moment curve (AUMC), and mean residence time (MRT), for a series of simulations with a wide range of parameter values. In a mathematical model of the metabolic pathways of the cancer drug irinotecan (CPT11), geometrical patterns can be classified into three major categories: “independent,” “hyperbolic,” and “complex.” These categories characterize substructures arising in differential equations, and are helpful for understanding the behavior of large-scale mathematical models. The Open Bioinformatics Grid (OBIGrid) provides a cyber-infrastructure for users to share these data as well as computational resources.     

Functional Topology Classification of Biological Computing Networks

Current analyses of complex biological networks focus either on their global statistical connectivity properties (e.g. topological path lengths and nodes connectivity ranks) or the statistics of specific local connectivity circuits (motifs). Here we present a different approach – Functional Topology, to enable identification of hidden topological and geometrical fingerprints of biological computing networks that afford their functioning – the form-function fingerprints. To do so we represent the network structure in terms of three matrices: 1. Topological connectivity matrix – each row (i) is the shortest topological path lengths of node i with all other nodes; 2. Topological correlation matrix – the element (i,j) is the correlation between the topological connectivity of nodes (i) and (j); and 3. Weighted graph matrix – in this case the links represent the conductance between nodes that can be simply one over the geometrical length, the synaptic strengths in case of neural networks or other quantity that represents the strengths of the connections. Various methods (e.g. clustering algorithms, random matrix theory, eigenvalues spectrum etc.), can be used to analyze these matrices, here we use the newly developed functional holography approach which is based on clustering of the matrices following their collective normalization. We illustrate the approach by analyzing networks of different topological and geometrical properties: 1. Artificial networks, including – random, regular 4-fold and 5-fold lattice and a tree-like structure; 2. Cultured neural networks: A single network and a network composed of three linked sub-networks; and 3. Model neural network composed of two overlapping sub-networks. Using these special networks, we demonstrate the method’s ability to reveal functional topology features of the networks.     

Some observations on multigrid convergence for convection–diffusion equations

This paper is concerned with the convergence behaviour of multigrid methods for two- dimensional discrete convection-diffusion equations. In Elman and Ramage (BIT 46:283–299, 2006), we showed that for constant coefficient problems with grid-aligned flow and semiperiodic boundary conditions, the two-grid iteration matrix can be reduced via a set of orthogonal transformations to a matrix containing individual 4 × 4 blocks, enabling a trivial computation of the norm of the iteration matrix. Here we use a similar Fourier analysis technique to investigate the individual contributions from the smoothing and approximation property matrices which form the basis of many standard multigrid analyses. As well as the theoretical results in the semiperiodic case, we present numerical results for a corresponding Dirichlet problem and examine the correlation between the two cases.     

Solution of systems of linear algebraic equations with preliminary flattening

A method of improving computing properties of matrices of systems of linear algebraic equations is considered. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 144–149, September–October, 1999.     

A fast stationary iterative method for a partial integro-differential equation in pricing options

A jump-diffusion model for the pricing of options leads to a partial integro-differential equation (PIDE). Discretizing the PIDE by certain method, we get a sequence of systems of linear equations, where the coefficient matrices are Toeplitz matrices. In this paper, we decompose the coefficient matrix as the sum of a tridiagonal matrix and a near low-rank matrix, and approximate the near low-rank matrix by low-rank matrices. Then we introduce a stationary iterative method for the approximate systems of linear equations. Comparison of the performance of our algorithm to that proposed in Pang et al. (Linear Algebra Appl. 434:2325–2342, 2011) is presented.     

A note on double splittings of different monotone matrices

Comparison theorems between the spectral radii of different matrices are a useful tool for judging the efficiency of preconditioners. For single splittings of different monotone matrices, Elsner et al. (Linear Algebra Appl. 363:65–80, 2003) gave out comparison theorems for spectral radii. For double splittings, some convergence and comparison theorems of a monotone matrix are presented by Shen et al. (Comput. Math. Appl. 51:1751–1760, 2006). In this note we give the comparison theorem for the spectral radii of matrices arising from double splittings of different monotone matrices.     

A Note on the Stability of Gauss–Jordan Elimination for Diagonally Dominant Matrices

Peters and Wilkinson [2] state that “it is well known that Gauss–Jordan is stable” for a diagonally dominant matrix, but a proof does not seem to have been published [1]. The present note fills this gap. Gauss–Jordan elimination is backward stable for matrices diagonally dominant by rows and not for those diagonally dominant by columns. In either case it is forward stable. Received February 28, 2000     

Watson–Crick palindromes in DNA computing

This paper provides an overview of existing approaches to encoding information on DNA strands for biocomputing, with a focus on the notion of Watson–Crick (WK) palindromes. We obtain a closed form for, as well as several properties of WK palindromes: The set of WK-palindromes is dense, context-free, but not regular, and is in general not closed under catenation and insertion. We obtain some properties that link the WK palindromes to classical notions such as that of primitive words. For example we show that the set of WK-palindromic words that cannot be written as the product of two nonempty WK-palindromes equals the set of primitive WK-palindromes. We also investigate various simultaneous Watson–Crick conjugate equations of words and show that the equations have, in most cases, only Watson–Crick palindromic solutions. Our results hold for more general functions, such as arbitrary morphic and antimorphic involutions.     

Mathematical modeling and optimization of intratumor drug transport

A pseudohyperbolic problem of optimal control of intratumoral drug distribution is formulated. It takes into account the heterogeneity of tumor tissues and effects of convection diffusion in a fissured porous medium. A mathematical model constructed and the corresponding optimal control problem are shown to be correct. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 147–154, November–December 2007.