Symmetric word equations in two positive definite letters

For every symmetric (``palindromic") word in two positive definite letters and for each fixed -by- positive definite and , it is shown that the symmetric word equation has an -by- positive definite solution . Moreover, if and are real, there is a real solution . The notion of symmetric word is generalized to allow non-integer exponents, with certain limitations. In some cases, the solution is unique, but, in general, uniqueness is an open question. Applications and methods for finding solutions are also discussed.

     

A class of modified block SSOR preconditioners for symmetric positive definite systems of linear equations

A class of modified block SSOR preconditioners is presented for solving symmetric positive definite systems of linear equations, which arise in the hierarchical basis finite element discretizations of the second order self‐adjoint elliptic boundary value problems. This class of methods is strongly related to two level methods, standard multigrid methods, and Jacobi‐like hierarchical basis methods. The optimal relaxation factors and optimal condition numbers are estimated in detail. Theoretical analyses show that these methods are very robust, and especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes. This revised version was published online in June 2006 with corrections to the Cover Date.     

On real matrices with positive definite symmetric component

Inequalities concerning real square matrices A with positive definite symmetric component A+A^*are derived from certain inertia relations which hold for any complex (not necessarily real) square matrices A with positive definite

A+A^*     

Multisplitting preconditioners for a symmetric positive definite matrix

We study convergence of multisplitting method associated with a block diagonal conformable multisplitting for solving a linear system whose coefficient matrix is a symmetric positive definite matrix which is not an H-matrix. Next, we study the validity ofm-step multisplitting polynomial preconditioners which will be used in the preconditioned conjugate gradient method.     

Graphical characterization of positive definite non symmetric quasi-Cartan matrices

It is known that each positive definite quasi-Cartan matrix $A$ is $\mathbb{Z}$-equivalent to a Cartan matrix $A_{\Delta }$ called Dynkin type of $A$, the matrix $A_{\Delta }$ is uniquely determined up to conjugation by permutation matrices. However, in most of the cases, it is not possible to determine the Dynkin type of a given connected quasi-Cartan matrix by simple inspection. In this paper, we give a graph theoretical characterization of non-symmetric connected quasi-Cartan matrices. For this purpose, a special assemblage of blocks is introduced. This result complements the approach proposed by Barot (1999, 2001), for ${\mathbb{A}}_n$, ${\mathbb{D}}_n$ and ${\mathbb{E}}_m$ with $m=6,7,8$.     

Computing the Karcher mean of symmetric positive definite matrices

We present and analyze an iterative method for approximating the Karcher mean of a set of $n\times n$ positive definite matrices $A_i$, $i=1\text{,}\dots \text{,}k$, defined as the unique positive definite solution of the matrix equation ${\sum }_{i=1}^k\log (A_i^{-1}X)=0$.     

Nonstationary two-stage multisplitting methods for symmetric positive definite matrices

Nonstationary synchronous two-stage multisplitting methods for the solution of the symmetric positive definite linear system of equations are considered. The convergence properties of these methods are studied. Relaxed variants are also discussed. The main tool for the construction of the two-stage multisplitting and related theoretical investigation is the diagonally compensated reduction (cf. [1]).     

Verifying all eigenpairs in symmetric positive definite generalized eigenvalue problem

A fast method for enclosing all eigenpairs in symmetric positive definite generalized eigenvalue problem is proposed. Firstly theorems on verifying all eigenvalues are presented. Next a theorem on verifying all eigenvectors is presented. The proposed method is developed based on these theorems. Numerical results are presented showing the efficiency of the proposed method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Improved seed methods for symmetric positive definite linear equations with multiple right‐hand sides

We consider symmetric positive definite systems of linear equations with multiple right‐hand sides. The seed conjugate gradient (CG) method solves one right‐hand side with the CG method and simultaneously projects over the Krylov subspace thus developed for the other right‐hand sides. Then the next system is solved and used to seed the remaining ones. Rounding error in the CG method limits how much the seeding can improve convergence. We propose three changes to the seed CG method: only the first right‐hand side is used for seeding, this system is solved past convergence, and the roundoff error is controlled with some reorthogonalization. We will show that results are actually better with only one seeding, even in the case of related right‐hand sides. Controlling rounding error gives the potential for rapid convergence for the second and subsequent right‐hand sides. Polynomial preconditioning can help reduce storage needed for reorthogonalization. The new seed methods are applied to examples including matrices from quantum chromodynamics. Copyright © 2013 John Wiley & Sons, Ltd.     

Massively parallel preconditioners for symmetric positive definite linear systems

In this paper, we address the problem of solving sparse symmetric linear systems on parallel computers. With further restrictive assumptions on the matrix (e.g., bidiagonal or tridiagonal structure), several direct methods may be used. These methods give ideas for constructing efficient data parallel preconditioners for general positive definite symmetric matrices. We describe two examples of such preconditioners for which the factorization (i.e., the construction of the preconditioning matrix) turns out to be parallel. This revised version was published online in June 2006 with corrections to the Cover Date.     

Convergence of multisplitting method for a symmetric positive definite matrix

The auxiliary principle is used to suggest and analyze some iterative methods for solving solving hemivariational inequalities under mild conditions. The results obtained in this paper can be considered as a novel application of the auxiliary principle technique. Since hemivariational inequalities include variational inequalities and nonlinear optimization problems as special cases, our results continue to hold-for these problems.     

Computational complexity of LCPs associated with positive definite symmetric matrices

Murty in a recent paper has shown that the computational effort required to solve a linear complementarity problem (LCP), by either of the two well known complementary pivot methods is not bounded above by a polynomial in the size of the problem. In that paper, by constructing a class of LCPs—one of ordern forn 2—he has shown that to solve the problem of ordern, either of the two methods goes through 2 ⁿ pivot steps before termination.However that paper leaves it as an open question to show whether or not the same property holds if the matrix,M, in the LCP is positive definite and symmetric. The class of LCPs in whichM is positive definite and symmetric is of particular interest because of the special structure of the problems, and also because they appear in many practical applications.In this paper, we study the computational growth of each of the two methods to solve the LCP, (q, M), whenM is positive definite and symmetric and obtain similar results.This research is partially supported by Air Force Office of Scientific Research, Air Force Number AFOSR-78-3646. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon.     

A hybrid iterative method for symmetric positive definite linear systems

A hybrid iterative scheme that combines the Conjugate Gradient (CG) method with Richardson iteration is presented. This scheme is designed for the solution of linear systems of equations with a large sparse symmetric positive definite matrix. The purpose of the CG iterations is to improve an available approximate solution, as well as to determine an interval that contains all, or at least most, of the eigenvalues of the matrix. This interval is used to compute iteration parameters for Richardson iteration. The attraction of the hybrid scheme is that most of the iterations are carried out by the Richardson method, the simplicity of which makes efficient implementation on modern computers possible. Moreover, the hybrid scheme yields, at no additional computational cost, accurate estimates of the extreme eigenvalues of the matrix. Knowledge of these eigenvalues is essential in some applications.Research supported in part by NSF grant DMS-9409422.Research supported in part by NSF grant DMS-9205531.     

Integro-differential equations of hyperbolic type with positive definite kernels

The main purpose of this work is to study the damping effect of memory terms associated with singular convolution kernels on the asymptotic behavior of the solutions of second order evolution equations in Hilbert spaces. For kernels that decay exponentially at infinity and possess strongly positive definite primitives, the exponential stability of weak solutions is obtained in the energy norm. It is also shown that this theory applies to several examples of kernels with possibly variable sign, and to a problem in nonlinear viscoelasticity.     

子空间上的对称正定及对称半正定阵的左右特征值反问题

文[1][2][3]中讨论AX＝B的对称阵逆特征值问题，文[4][5][6]中讨论了半正定阵的逆特征值问题。本文讨论了空间了子空间上的对称正定及对称半正定阵的左右特征值反问题，给出了解存在的充分条件及解的表达式。     

Convergence of block two-stage iterative methods for symmetric positive definite systems

Summary. We study the convergence of two-stage iterative methods for solving symmetric positive definite (spd) systems. The main tool we used to derive the iterative methods and to analyze their convergence is the diagonally compensated reduction (cf. [1]). Received December 11, 1997 / Revised version received March 25, 1999 / Published online May 30, 2001     

Factorizing complex symmetric matrices with positive definite real and imaginary parts

Complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for LU factorization. This result adds to the classes of matrix for which it is known to be safe not to pivot in LU factorization. Block factorization with the pivoting strategy of Bunch and Kaufman is also considered, and it is shown that for such matrices only pivots are used and the same growth factor bound of 2 holds, but that interchanges that destroy band structure may be made. The latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in LINPACK and LAPACK.

     

New factorization codes for sparse,symmetric and positive definite matrices

Suitable techniques for storing the matrix pattern during the factorization of sparse, symmetric and positive definite matrices are considered. Especially we discuss the consequences of switching from a sparse factorization code to a full code when the uneliminated part of the matrix is full or almost full. The resulting codes seem to be among the most efficient for solving one-off problems regarding both execution time and storage requirements.This work has been supported by the Danish Natural Science Research Council, Grant No. 511-8476.     

Calculation of zonal polynomials of 3×3 positive definite symmetric matrices

A zonal polynomial identity is derived and is used to construct algorithms for the calculation of the zonal polynomials of 2×2 and 3×3 positive definite symmetric matrices.