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1.
Joris Chau Hernando Ombao Rainer von Sachs 《Journal of computational and graphical statistics》2019,28(2):427-439
Nondegenerate covariance, correlation, and spectral density matrices are necessarily symmetric or Hermitian and positive definite. This article develops statistical data depths for collections of Hermitian positive definite matrices by exploiting the geometric structure of the space as a Riemannian manifold. The depth functions allow one to naturally characterize most central or outlying matrices, but also provide a practical framework for inference in the context of samples of positive definite matrices. First, the desired properties of an intrinsic data depth function acting on the space of Hermitian positive definite matrices are presented. Second, we propose two pointwise and integrated data depth functions that satisfy each of these requirements and investigate several robustness and efficiency aspects. As an application, we construct depth-based confidence regions for the intrinsic mean of a sample of positive definite matrices, which is applied to the exploratory analysis of a collection of covariance matrices in a multicenter clinical trial. Supplementary materials and an accompanying R-package are available online. 相似文献
2.
Christian Wagner 《Numerische Mathematik》1997,78(1):119-142
Summary. The tangential frequency filtering decomposition (TFFD) is introduced. The convergence theory of an iterative scheme based
on the TFFD for symmetric matrices is the focus of this paper. The existence of the TFFD and the convergence of the induced
iterative algorithm is shown for symmetric and positive definite matrices. Convergence rates independent of the number of
unknowns are proven for a smaller class of matrices. Using this framework, the convergence independent of the number of unknowns
is shown for Wittum's frequency filtering decomposition. Some characteristic properties of the TFFD are illustrated and results
of several numerical experiments are presented.
Received April 1, 1996 / Revised version July 4, 1996 相似文献
3.
Summary. In this work we address the issue of integrating
symmetric Riccati and Lyapunov matrix differential equations. In
many cases -- typical in applications -- the solutions are positive
definite matrices. Our goal is to study when and how this property
is maintained for a numerically computed solution.
There are two classes of solution methods: direct and
indirect algorithms. The first class consists of the schemes
resulting from direct discretization of the equations. The second
class consists of algorithms which recover the solution by
exploiting some special formulae that these solutions are known to
satisfy.
We show first that using a direct algorithm -- a one-step scheme or
a strictly stable multistep scheme (explicit or implicit) -- limits
the order of the numerical method to one if we want to guarantee
that the computed solution stays positive definite. Then we show two
ways to obtain positive definite higher order approximations by
using indirect algorithms. The first is to apply a symplectic
integrator to an associated Hamiltonian system. The other uses
stepwise linearization.
Received April 21, 1993 相似文献
4.
5.
Ronaldo Gregório Paulo Roberto Oliveira 《Journal of Mathematical Analysis and Applications》2009,355(2):469-478
In this work, we propose a proximal algorithm for unconstrained optimization on the cone of symmetric semidefinite positive matrices. It appears to be the first in the proximal class on the set of methods that convert a Symmetric Definite Positive Optimization in Nonlinear Optimization. It replaces the main iteration of the conceptual proximal point algorithm by a sequence of nonlinear programming problems on the cone of diagonal definite positive matrices that has the structure of the positive orthant of the Euclidian vector space. We are motivated by results of the classical proximal algorithm extended to Riemannian manifolds with nonpositive sectional curvature. An important example of such a manifold is the space of symmetric definite positive matrices, where the metrics is given by the Hessian of the standard barrier function −lndet(X). Observing the obvious fact that proximal algorithms do not depend on the geodesics, we apply those ideas to develop a proximal point algorithm for convex functions in this Riemannian metric. 相似文献
6.
This paper introduces a robust preconditioner for general sparse matrices based on low‐rank approximations of the Schur complement in a Domain Decomposition framework. In this ‘Schur Low Rank’ preconditioning approach, the coefficient matrix is first decoupled by a graph partitioner, and then a low‐rank correction is exploited to compute an approximate inverse of the Schur complement associated with the interface unknowns. The method avoids explicit formation of the Schur complement. We show the feasibility of this strategy for a model problem and conduct a detailed spectral analysis for the relation between the low‐rank correction and the quality of the preconditioner. We first introduce the SLR preconditioner for symmetric positive definite matrices and symmetric indefinite matrices if the interface matrices are symmetric positive definite. Extensions to general symmetric indefinite matrices as well as to nonsymmetric matrices are also discussed. Numerical experiments on general matrices illustrate the robustness and efficiency of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
7.
In this paper, an algorithm based on a shifted inverse power iteration for computing generalized eigenvalues with corresponding eigenvectors of a large scale sparse symmetric positive definite matrix pencil is presented. It converges globally with a cubic asymptotic convergence rate, preserves sparsity of the original matrices and is fully parallelizable. The algebraic multilevel itera-tion method (AMLI) is used to improve the efficiency when symmetric positive definite linear equa-tions need to be solved. 相似文献
8.
The numerical solution of elliptic selfadjoint second-order boundary value problems leads to a class of linear systems of equations with symmetric, positive definite, large and sparse matrices which can be solved iteratively using a preconditioned version of some algorithm. Such differential equations originate from various applications such as heat conducting and electromagnetics. Systems of equations of similar type can also arise in the finite element analysis of structures. We discuss a recursive method constructing preconditioners to a symmetric, positive definite matrix. An algebraic multilevel technique based on partitioning of the matrix in two by two matrix block form, approximating some of these by other matrices with more simple sparsity structure and using the corresponding Schur complement as a matrix on the lower level, is considered. The quality of the preconditioners is improved by special matrix polynomials which recursively connect the preconditioners on every two adjoining levels. Upper and lower bounds for the degree of the polynomials are derived as conditions for a computational complexity of optimal order for each level and for an optimal rate of convergence, respectively. The method is an extended and more accurate algebraic formulation of a method for nine-point and mixed five- and nine-point difference matrices, presented in some previous papers. 相似文献
9.
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 相似文献
10.
Zhong‐Zhi Bai 《Numerical Linear Algebra with Applications》2010,17(6):917-933
For the large sparse linear complementarity problems, by reformulating them as implicit fixed‐point equations based on splittings of the system matrices, we establish a class of modulus‐based matrix splitting iteration methods and prove their convergence when the system matrices are positive‐definite matrices and H+‐matrices. These results naturally present convergence conditions for the symmetric positive‐definite matrices and the M‐matrices. Numerical results show that the modulus‐based relaxation methods are superior to the projected relaxation methods as well as the modified modulus method in computing efficiency. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
11.
Uses of the numerical radius in the analysis of basic iterative solution methods, of the SOR method for quasi-Hermitian positive definite matrices (not being consistently ordered) and of maximal eigenvalues of symmetric positive definite matrices using incomplete block-matrix factorizations are presented. 相似文献
12.
文[1-5]中研究了对称、对称半正定及流形上的对称半正定的反问题,并说明了其应用背景.本文研究线性流形上的正定及半正定阵的反问题,说明了文[1-3]中的一些结果为本文的特例. 相似文献
13.
14.
Uses of the numerical radius in the analysis of basic iterative solution methods, of the SOR method for quasi-Hermitian positive definite matrices (not being consistently ordered) and of maximal eigenvalues of symmetric positive definite matrices using incomplete block-matrix factorizations are presented. 相似文献
15.
A positive definite symmetric matrix is called a Stieltjes matrix provided that all its off diagonal elements are nonpositive. We characterize functions which preserve the class of Stieltjes matrices. 相似文献
16.
Maya Neytcheva 《Linear algebra and its applications》2011,434(11):2308-2324
We discuss a methodology to construct sparse approximations of Schur complements of two-by-two block matrices arising in Finite Element discretizations of partial differential equations. Earlier results from [2] are extended to more general symmetric positive definite matrices of two-by-two block form. The applicability of the method for general symmetric and nonsymmetric matrices is analysed. The paper demonstrates the applicability of the presented method providing extensive numerical experiments. 相似文献
17.
18.
Summary.
An adaptive Richardson iteration method is described for the solution of
large sparse symmetric positive definite linear systems of equations with
multiple right-hand side vectors. This scheme ``learns' about the linear
system to be solved by computing inner products of residual matrices during
the iterations. These inner products are interpreted as block modified moments.
A block version of the modified Chebyshev algorithm is presented which yields
a block tridiagonal matrix from the block modified moments and the recursion
coefficients of the residual polynomials. The eigenvalues of this block
tridiagonal matrix define an interval, which determines the choice of relaxation
parameters for Richardson iteration. Only minor modifications are necessary
in order to obtain a scheme for the solution of symmetric indefinite linear
systems with multiple right-hand side vectors. We outline the changes required.
Received April 22, 1993 相似文献
19.
Charles R. Johnson 《Linear and Multilinear Algebra》2013,61(12):1583-1590
Recently, a new class of matrices, called mixed matrices, that unifies the Z-matrices and symmetric matrices has been identified. They share the property that when the leading principal minors are positive, all principal minors are positive. It is natural to ask what other properties of M-matrices and positive definite matrices are enjoyed by mixed matrices as well. Here, we show that mixed P-matrices satisfy a broad family of determinantal inequalities, the Koteljanskii inequalities, previously known for those two classes. In the process, other properties of mixed matrices are developed, and consequences of the Koteljanskii inequalities are given. 相似文献
20.
《Journal of Computational and Applied Mathematics》2006,188(1):150-164
The present paper investigates the approach mentioned, but not developed, in [Benzi, Tuma, Numer. Linear Algebra Appl. 10 (2003)]. In this approach, using a conjugate Gram–Schmidt algorithm, an ILU preconditioner whose existence is guaranteed for any nonsingular (symmetric and nonsymmetric) real positive definite matrix is obtained. Numerical results on some nonsymmetric positive definite matrices demonstrate the efficiency of the method. 相似文献