An efficient algorithm for the generalized centro-symmetric solution of matrix equation <Emphasis Type="Italic">A X B</Emphasis> = <Emphasis Type="Italic">C</Emphasis>

In this paper, an iterative algorithm is constructed for solving linear matrix equation AXB = C over generalized centro-symmetric matrix X. We show that, by this algorithm, a solution or the least-norm solution of the matrix equation AXB = C can be obtained within finite iteration steps in the absence of roundoff errors; we also obtain the optimal approximation solution to a given matrix X ₀ in the solution set of which. In addition, given numerical examples show that the iterative method is efficient.     

New method for general Kennaugh’s pseudo-eigenvalue equation in radar polarimetry

Kennaugh’s pseudo-eigenvalue equation is a basic equation that plays an extremely important role in radar polarimetry. In this paper, by means of real representation, we first present a necessary and sufficient condition for the general Kennaugh’s pseudo-eigenvalue equation having a solution, characterize the explicit form of the solution, and then study the solution of Kennaugh’s pseudo-eigenvalue equation. At last, we propose a new technique for finding the coneigenvalues and coneigenvectors of a complex matrix under appropriate conditions in radar polarimetry.     

Compound model of the generalized oscillator. I

We study the problem of realization of a given generalized oscillator by a system of N generalized oscillators of a different type. We consider a generalized oscillator related to a fixed system of orthogonal polynomials that are determined by three-term recurrent relations and the corresponding three-diagonal Jacobi matrix J. The case N =2 was considered in a previous paper. It was shown that in this case the orthogonality measure is symmetric with respect to rotation at angle π. In this paper, we consider the case N =3. We prove that such a problem has a solution only in two cases. In the first case, the Jacobi matrix related to the given “composite” generalized oscillator has block-diagonal form and consists of similar 3×3 blocks. In the second (more interesting) possible case, the Jacobi matrix is not block-diagonal. For this matrix, we construct the corresponding system of orthogonal polynomials. This system decomposes into three series which are related to Chebyshev polynomials of the second kind. The main result of the paper is a solution of the moment problem for the corresponding Jacobi matrix. In this case, the constructed measure is symmetric with respect to rotation at angle 2π/3. Bibliography: 6 titles.     

Discrete symmetries,Darboux transformation,and exact solutions of the Wess–Zumino–Novikov–Witten model

The matrix Darboux transformation is applied to an auxiliary problem of the classical Wess–Zumino–Novikov–Witten model. One- and two-soliton solutions are written explicitly, and a matrix expression for the N-soliton solution is given. Discrete symmetries of the WZNW model are analyzed, and a solution of the linearized equation of motion is obtained. Bibliography: 19 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 139–152.     

A numerical algorithm for solving the matrix equation AX + XTB = C1

An algorithm of the Bartels-Stewart type for solving the matrix equation AX + X ^T B = C is proposed. By applying the QZ algorithm, the original equation is reduced to an equation of the same type having triangular matrix coefficients A and B. The resulting matrix equation is equivalent to a sequence of low-order systems of linear equations for the entries of the desired solution. Through numerical experiments, the situation where the conditions for unique solvability are “nearly” violated is simulated. The loss of the quality of the computed solution in this situation is analyzed.     

On quasi-solution to infeasible linear complementarity problem obtained by Lemke’s method

For a linear complementarity problem with inconsistent system of constraints a notion of quasi-solution of Tschebyshev type is introduced. It’s shown that this solution can be obtained automatically by Lemke’s method if the constraint matrix of the original problem is copositive plus or belongs to the intersection of matrix classes P ₀ and Q ₀.     

The quaternion matrix equation ΣA i XB i =E

LetH _F be the generalized quaternion division algebra over a fieldF with charF#2. In this paper, the adjoint matrix of anyn×n matrix overH _F [γ] is defined and its properties is discussed. By using the adjoint matrix and the method of representation matrix, this paper obtains several necessary and sufficient conditions for the existence of a solution or a unique solution to the matrix equation Σ _i=0 ^k A ⁱ XB _i =E overH _F , and gives some explicit formulas of solutions. Supported by the National Natural Science Foundation of China and Human     

Multi-parameter regularization techniques for ill-conditioned linear systems

Summary.  When a system of linear equations is ill-conditioned, regularization techniques provide a quite useful tool for trying to overcome the numerical inherent difficulties: the ill-conditioned system is replaced by another one whose solution depends on a regularization term formed by a scalar and a matrix which are to be chosen. In this paper, we consider the case of several regularizations terms added simultaneously, thus overcoming the problem of the best choice of the regularization matrix. The error of this procedure is analyzed and numerical results prove its efficiency. Received January 15, 2002 / Revised version received July 31, 2002 / Published online October 29, 2002 Mathematics Subject Classification (1991): 65F05 – 65F22     

Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise

Quantum stochastic differential equations of the form

Accumulating Jacobians as chained sparse matrix products

 The chain rule – fundamental to any kind of analytical differentiation - can be applied in various ways to computational graphs representing vector functions. These variants result in different operations counts for the calculation of the corresponding Jacobian matrices. The minimization of the number of arithmetic operations required for the calculation of the complete Jacobian leads to a hard combinatorial optimization problem. We will describe an approach to the solution of this problem that builds on the idea of optimizing chained matrix products using dynamic programming techniques. Reductions by a factor of 3 and more are possible regarding the operations count for the Jacobian accumulation. After discussing the mathematical basics of Automatic Differentiation we will show how to compute Jacobians by chained sparse matrix products. These matrix chains can be reordered, must be pruned, and are finally subject to a dynamic programming algorithm to reduce the number of scalar multiplications performed. Received: January 17, 2002 / Accepted: May 29, 2002 Published online: February 14, 2003 Key words. chained matrix product – combinatorial optimization – dynamic programming – edge elimination in computational graphs     

由Smith标准形求解矩阵方程∑AiXBi=C

By using the Smith normal form of polynomial matrix and algebraic methods, this paper discusses the solvability for the linear matrix equation ∑AiXBi = C over a field, and obtains the explicit formulas of general solution or unique solution.     

Detecting exponential dichotomy on the real line: SVD and QR algorithms

In this paper we propose and implement numerical methods to detect exponential dichotomy on the real line. Our algorithms are based on the singular value decomposition and the QR factorization of a fundamental matrix solution. The theoretical justification for our methods was laid down in the companion paper: “Exponential Dichotomy on the real line: SVD and QR methods.”     

Diophantine Quadratic Equation and Smith Normal Form Using Scaled Extended Integer Abaffy–Broyden–Spedicato Algorithms

Classes of integer Abaffy–Broyden–Spedicato (ABS) methods have recently been introduced for solving linear systems of Diophantine equations. Each method provides the general integer solution of the system by computing an integer solution and an integer matrix, named Abaffian, with rows generating the integer null space of the coefficient matrix. The Smith normal form of a general rectangular integer matrix is a diagonal matrix, obtained by elementary nonsingular (unimodular) operations. Here, we present a class of algorithms for computing the Smith normal form of an integer matrix. In doing this, we propose new ideas to develop a new class of extended integer ABS algorithms generating an integer basis for the integer null space of the matrix. For the Smith normal form, having the need to solve the quadratic Diophantine equation, we present two algorithms for solving such equations. The first algorithm makes use of a special integer basis for the row space of the matrix, and the second one, with the intention of controlling the growth of intermediate results and making use of our given conjecture, is based on a recently proposed integer ABS algorithm. Finally, we report some numerical results on randomly generated test problems showing a better performance of the second algorithm in controlling the size of the solution. We also report the results obtained by our proposed algorithm on the Smith normal form and compare them with the ones obtained using Maple, observing a more balanced distribution of the intermediate components obtained by our algorithm.     

Degenerate Nevanlinna-Pick problem

A general solution of the degenerate Nevanlinna-Pick problem is described in terms of fractional-linear transformations. A resolvent matrix of the problem is obtained in the form of a J-expanding matrix of full rank. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1334–1343, October, 2005.     

The uniqueness of a solution to the renewal type system of integral equations on the line

We study the uniqueness of a solution to a renewal type system of integral equations z=g+F * z on the line ℝ; here z is the unknown vector function, g is a known vector function, and F is a nonlattice matrix of finite measures on ℝ such that the matrix F(ℝ) is of spectral radius 1 and indecomposable. We show that in a certain class of functions each solution to the corresponding homogeneous system coincides almost everywhere with a right eigenvector of F(ℝ) with eigenvalue 1.     

Gauge theories as matrix models

We discuss the relation between the Seiberg-Witten prepotentials, Nekrasov functions, and matrix models. On the semiclassical level, we show that the matrix models of Eguchi-Yang type are described by instantonic contributions to the deformed partition functions of supersymmetric gauge theories. We study the constructed explicit exact solution of the four-dimensional conformal theory in detail and also discuss some aspects of its relation to the recently proposed logarithmic beta-ensembles. We also consider “quantizing” this picture in terms of two-dimensional conformal theory with extended symmetry and stress its difference from the well-known picture of the perturbative expansion in matrix models. Instead, the representation of Nekrasov functions using conformal blocks or Whittaker vectors provides a nontrivial relation to Teichmüller spaces and quantum integrable systems.     

Realistic Eigenvalue Bounds for the Galerkin Mass Matrix

In time-dependent finite-element calculations, a mass matrixnaturally arises. To avoid the solution of the correspondingalgebraic equation system at each time step, ‘mass lumping’is widely used, even though this pragmatic diagonalization ofthe mass matrix often reduces accuracy. We show how the unassembled form of finite-element equationscan be used to establish (in an element-by-element manner) realisticupper and lower bounds on the eigenvalues of the fully consistentmass matrix when preconditioned by its diagonal entries. Weuse this technique to give specific results for a number ofdifferent types of finite elements in one, two, and three dimensions.The bounds are found by independent calculations on the elements,and, for certain element types, are independent of mesh irregularity.We give examples of when some of the bounds are attained. These results indicate that the preconditioned conjugate-gradientmethod is appropriate and very rapid for the solution of Galerkinmass-matrix equations.     

Algorithms for solving matrix polynomial equations of special form

In this paper we consider a series of algorithms for calculating radicals of matrix polynomial equations. A particular aspect of this problem arise in author’s work, concerning parameter identification of linear dynamic stochastic system. Special attention is given to searching the solution of an equation in a neighbourhood of some initial approximation. The offered approaches and algorithms allow us to receive fast and quite exact solution. We give some recommendations for application of given algorithms.