Cycles of Nonzero Elements in Low Rank Matrices

Dedicated to the memory of Paul Erdős We consider the problem of finding some structure in the zero-nonzero pattern of a low rank matrix. This problem has strong motivation from theoretical computer science. Firstly, the well-known problem on rigidity of matrices, proposed by Valiant as a means to prove lower bounds on some algebraic circuits, is of this type. Secondly, several problems in communication complexity are also of this type. The special case of this problem, where one considers positive semidefinite matrices, is equivalent to the question of arrangements of vectors in euclidean space so that some condition on orthogonality holds. The latter question has been considered by several authors in combinatorics [1, 4]. Furthermore, we can think of this problem as a kind of Ramsey problem, where we study the tradeoff between the rank of the adjacency matrix and, say, the size of a largest complete subgraph. In this paper we show that for an real matrix with nonzero elements on the main diagonal, if the rank is o(n), the graph of the nonzero elements of the matrix contains certain cycles. We get more information for positive semidefinite matrices. Received September 9, 1999 RID="*" ID="*" Partially supported by grant A1019901 of the Academy of Sciences of the Czech Republic and by a cooperative research grant INT-9600919/ME-103 from the NSF (USA) and the MŠMT (Czech Republic).     

Smooth Schur factorizations in the continuation of separatrices

Numerical computation of separatrices as general connecting orbits in dynamical systems is performed, and their continuation as problem parameters vary is approached via a direct application of smooth block Schur factorizations of Jacobians and monodromy matrix functions. Several numerical examples are presented to illustrate the effectiveness of the algorithms used.     

关于实方阵的几个问题

本文讨论如下内容：１．把有关对称正定（半正定）的一些性质推广到广义正定（半正定）。２．给定ｘ∈Ｒｍ×ｍ，∧为对角阵，求ＡＸ＝ｘ∧在对称半正定矩阵类中解存在的充要条件及一般形式，并讨论了对任意给定的对称正定（半正定）矩阵Ａ，在上述解的集合中求得Ａ，使得     

On the modification of an eigenvalue problem that preserves an eigenspace

Eigenvalue problems arise in many application areas ranging from computational fluid dynamics to information retrieval. In these fields we are often interested in only a few eigenvalues and corresponding eigenvectors of a sparse matrix. In this paper, we comment on the modifications of the eigenvalue problem that can simplify the computation of those eigenpairs. These transformations allow us to avoid difficulties associated with non-Hermitian eigenvalue problems, such as the lack of reliable non-Hermitian eigenvalue solvers, by mapping them into generalized Hermitian eigenvalue problems. Also, they allow us to expose and explore parallelism. They require knowledge of a selected eigenvalue and preserve its eigenspace. The positive definiteness of the Hermitian part is inherited by the matrices in the generalized Hermitian eigenvalue problem. The position of the selected eigenspace in the ordering of the eigenvalues is also preserved under certain conditions. The effect of using approximate eigenvalues in the transformation is analyzed and numerical experiments are presented.     

Weighted median algorithms for L1 approximation

The weighted median problem arises as a subproblem in certain multivariate optimization problems, includingL ₁ approximation. Three algorithms for the weighted median problem are presented and the relationships between them are discussed. We report on computational experience with these algorithms and on their use in the context of multivariateL ₁ approximation.This work was supported in part by National Science Foundation Grant CCR-8713893 and in part by a grant from The City University of New York PSC-CUNY Research Award program.     

A note on properties of splittings of singular symmetric positive semidefinite matrices

Summary. Recently, Benzi and Szyld have published an important paper [1] concerning the existence and uniqueness of splittings for singular matrices. However, the assertion in Theorem 3.9 on the inheriting property of P-regular splitting for singular symmetric positive semidefinite matrices seems to be incorrect. As a complement of paper [1], in this short note we point out that if a matrix T is resulted from a P-regular splitting of a symmetric positive semidefinite matrix A, then splittings induced by T are not all P-regular. Received January 7, 1999 / Published online December 19, 2000     

A regularized strong duality for nonsymmetric semidefinite least squares problem

The nonsymmetric semidefinite least squares problem (NSDLS) is to find a nonsymmetric semidefinite matrix which is closest to a given matrix in Frobenius norm. It is an extension of the semidefinite least squares problem (SDLS) and has important application in the area of robotics and automation. In this note, by developing the minimal representation of the underlying cone with the linear constraints, we obtain a regularized strong duality with low-dimensional projection for NSDLS. Further, we study the generalized differential properties and nonsingularity of the first order optimality system about the dual problem. These theoretical results demonstrate that we can solve NSDLS as good as the current Lagrangian dual approaches to SDLS.     

Numerical exterior algebra and the compound matrix method

Summary. The compound matrix method, which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k=2,3 dimensional subspaces of , by using compound matrices as coordinates, is reformulated in a coordinate-free way using exterior algebra spaces, . This formulation leads to a general framework for studying systems of differential equations on k-dimensional subspaces. The framework requires the development of several new ideas: the role of Hodge duality and the Hodge star operator in the construction, an efficient strategy for constructing the induced differential equations on , general formulation of induced boundary conditions, the role of geometric integrators for preserving the manifold of k-dimensional subspaces – the Grassmann manifold, , and a formulation for induced systems on an unbounded interval. The numerical exterior algebra framework is most advantageous for numerical solution of differential eigenvalue problems on unbounded domains, where there are significant difficulties in setting up matrix discretizations. The formulation is presented for k-dimensional subspaces of systems on with k and n arbitrary, and examples are given for the cases of k=2 and n=4, and k=3 and n=6, with an indication of implementation details for systems of larger dimension. The theory is illustrated by application to four differential eigenvalue problems on unbounded intervals: hydrodynamic stablity of boundary-layer flow past a compliant surface, the eigenvalue problem associated with the stability of solitary waves, the stability of Bickley jet in oceanography, and the eigenvalue problem associated with the stability of the Ekman layer in atmospheric dynamics. Received February 2, 2001 / Revised version received May 28, 2001 / Published online October 17, 2001     

A penalty method for rank minimization problems in symmetric matrices

The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be cast equivalently as a semidefinite program with complementarity constraints (SDCMPCC). The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization problem. We investigate calmness of locally optimal solutions to the SDCMPCC formulation and hence show that any locally optimal solution is a KKT point. We develop a penalty formulation of the problem. We present calmness results for locally optimal solutions to the penalty formulation. We also develop a proximal alternating linearized minimization (PALM) scheme for the penalty formulation, and investigate the incorporation of a momentum term into the algorithm. Computational results are presented.     

A local least-squares method for solving nonlinear partial differential equations of second order

In this paper a mesh-free method for the treatment of time-independent and time-dependent nonlinear PDEs of second order is presented. The basic idea of the discretization is a local least-squares approximation, similar to the moving least-squares approach in data approximation. However, in our approach the PDE is incorporated as an additional minimization constraint. The discretization leads to a fixed-point problem, which is solved by iteration. Because of the local nature of the method only small dimensional matrix inversions have to be done. The approximation error of the discretization—even on unstructured meshes—is comparable to respective versions of finite elements. As a by-product the method provides an a posteriori measure for the local approximation error. We discuss implementational aspects and present numerical simulations.     

On the convergence of splittings for semidefinite linear systems

Recently, Lee et al. [Young-ju Lee, Jinbiao Wu, Jinchao Xu, Ludmil Zikatanov, On the convergence of iterative methods for semidefinite linear systems, SIAM J. Matrix Anal. Appl. 28 (2006) 634-641] introduce new criteria for the semi-convergence of general iterative methods for semidefinite linear systems based on matrix splitting. The new conditions generalize the classical notion of P-regularity introduced by Keller [H.B. Keller, On the solution of singular and semidefinite linear systems by iterations, SIAM J. Numer. Anal. 2 (1965) 281-290]. In view of their results, we consider here stipulations on a splitting A=M-N, which lead to fixed point systems such that, the iterative scheme converges to a weighted Moore-Penrose solution to the system Ax=b. Our results extend the result of Lee et al. to a more general case and we also show that it requires less restrictions on the splittings than Keller’s P-regularity condition to ensure the convergence of iterative scheme.     

Preserving monotonicity in the numerical solution of Riccati differential equations

Summary. Solutions of symmetric Riccati differential equations (RDEs for short) are in the usual applications positive semidefinite matrices. Moreover, in the class of semidefinite matrices, solutions of different RDEs are also monotone, with respect to properly ordered data. Positivity and monotonicity are essential properties of RDEs. In Dieci and Eirola (1994), we showed that, generally, a direct discretization of the RDE cannot maintain positivity, and be of order greater than one. To get higher order, and to maintain positivity, we are thus forced to look into indirect solution procedures. Here, we consider the problem of how to maintain monotonicity in the numerical solutions of RDEs. Naturally, to obtain order greater than one, we are again forced to look into indirect solution procedures. Still, the restrictions imposed by monotonicity are more stringent that those of positivity, and not all of the successful indirect solution procedures of Dieci and Eirola (1994) maintain monotonicity. We prove that by using symplectic Runge-Kutta (RK) schemes with positive weights (e.g., Gauss schemes) on the underlying Hamiltonian matrix, we eventually maintain monotonicity in the computed solutions of RDEs. Received May 2, 1995     

Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion

A basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix inequalities. The sparsity, characterized with a chordal graph structure, can be detected in the variable matrix or in a linear or nonlinear matrix-inequality constraint of the problem. We classify the sparsity in two types, the domain-space sparsity (d-space sparsity) for the symmetric matrix variable in the objective and/or constraint functions of the problem, which is required to be positive semidefinite, and the range-space sparsity (r-space sparsity) for a linear or nonlinear matrix-inequality constraint of the problem. Four conversion methods are proposed in this framework: two for exploiting the d-space sparsity and the other two for exploiting the r-space sparsity. When applied to a polynomial semidefinite program (SDP), these conversion methods enhance the structured sparsity of the problem called the correlative sparsity. As a result, the resulting polynomial SDP can be solved more effectively by applying the sparse SDP relaxation. Preliminary numerical results on the conversion methods indicate their potential for improving the efficiency of solving various problems.     

Algebraic multigrid methods for Laplacians of graphs

Classical algebraic multigrid theory relies on the fact that the system matrix is positive definite. We extend this theory to cover the positive semidefinite case as well, by formulating semiconvergence results for these singular systems. For the class of irreducible diagonal dominant singular M-matrices we show that the requirements of the developed theory hold and that the coarse level systems are still of the same class, if the C/F-splitting is good enough. An important example for matrices that are irreducible diagonal dominant M-matrices are Laplacians of graphs. Recent shape optimizing methods for graph partitioning require to solve singular linear systems involving these Laplacians. We present convergence results as well as experimental results for numerous graphs arising from finite element discretizations with up to 10⁶ vertices.     

Direct and iterative solution of the generalized Dirichlet–Neumann map for elliptic PDEs on square domains

In this work we derive the structural properties of the Collocation coefficient matrix associated with the Dirichlet–Neumann map for Laplace’s equation on a square domain. The analysis is independent of the choice of basis functions and includes the case involving the same type of boundary conditions on all sides, as well as the case where different boundary conditions are used on each side of the square domain. Taking advantage of said properties, we present efficient implementations of direct factorization and iterative methods, including classical SOR-type and Krylov subspace (Bi-CGSTAB and GMRES) methods appropriately preconditioned, for both Sine and Chebyshev basis functions. Numerical experimentation, to verify our results, is also included.     

A framework for polynomial preconditioners based on fast transforms I: Theory

Optimal and superoptimal approximations of a complex square matrix by polynomials in a normal basis matrix are considered. If the unitary transform associated with the eigenvectors of the basis matrix is computable using a fast algorithm, the approximations may be utilized for constructing preconditioners. Theorems describing how the parameters of the approximations could be efficiently computed are given, and for special cases earlier results by other authors are recovered. Also, optimal and superoptimal approximations for block matrices are determined, and the same type of theorems as for the point case are proved. This research was supported by the Swedish National Board for Industrial and Technical Development (NUTEK) and by the U.S. National Science Foundation under grant ASC-8958544.     

Some Properties for the Euclidean Distance Matrix and Positive Semidefinite Matrix Completion Problems

The Euclidean distance matrix (EDM) completion problem and the positive semidefinite (PSD) matrix completion problem are considered in this paper. Approaches to determine the location of a point in a linear manifold are studied, which are based on a referential coordinate set and a distance vector whose components indicate the distances from the point to other points in the set. For a given referential coordinate set and a corresponding distance vector, sufficient and necessary conditions are presented for the existence of such a point that the distance vector can be realized. The location of the point (if it exists) given by the approaches in a linear manifold is independent of the coordinate system, and is only related to the referential coordinate set and the corresponding distance vector. An interesting phenomenon about the complexity of the EDM completion problem is described. Some properties about the uniqueness and the rigidity of the conformation for solutions to the EDM and PSD completion problems are presented.     

Numerical solution of one-dimensional Burgers’ equation using reproducing kernel function

This paper presents a numerical method for one-dimensional Burgers’ equation by the Hopf–Cole transformation and a reproducing kernel function, abbreviated as RKF. The numerical solution is given as explicit integral expressions with the RKF at each time step, so that the computation is fully parallel. The stability and error estimates are derived. Numerical results for some test problems are presented and compared with the exact solutions. Some numerical results are also compared with the results obtained by other methods. The present method is easily implemented and effective.     

Structure methods for solving the nearest correlation matrix problem

The nearest correlation matrix problem is to find a positive semidefinite matrix with unit diagonal, that is, nearest in the Frobenius norm to a given symmetric matrix A. This problem arises in the finance industry, where the correlations are between stocks. In this paper, we formulate this problem as a smooth unconstrained minimization problem, for which rapid convergence can be obtained. Other methods are also studied. Comparative numerical results are reported.     

Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods

Summary. Given a nonsingular matrix , and a matrix of the same order, under certain very mild conditions, there is a unique splitting , such that . Moreover, all properties of the splitting are derived directly from the iteration matrix . These results do not hold when the matrix is singular. In this case, given a matrix and a splitting such that , there are infinitely many other splittings corresponding to the same matrices and , and different splittings can have different properties. For instance, when is nonnegative, some of these splittings can be regular splittings, while others can be only weak splittings. Analogous results hold in the symmetric positive semidefinite case. Given a singular matrix , not for all iteration matrices there is a splitting corresponding to them. Necessary and sufficient conditions for the existence of such splittings are examined. As an illustration of the theory developed, the convergence of certain alternating iterations is analyzed. Different cases where the matrix is monotone, singular, and positive (semi)definite are studied. Received September 5, 1995 / Revised version received April 3, 1996