共查询到20条相似文献,搜索用时 46 毫秒
1.
本文研究了一类具有特殊结构的无限维二次型, 得到这类二次型的对称矩阵是符号为多项式的模的平方的Laurent 矩阵, 进一步得到了这类二次型是强正定的判断标准以及一类Weyl-Heisenberg 框架的构造. 本文还研究了这类二次型的矩阵的所有有限维主对角子矩阵的强正定性, 并由此得到一类子空间Weyl-Heisenberg 框架的构造. 最后举例说明本文的主要结果及其应用. 本文建立了两个看似不相关的领域间的联系. 相似文献
2.
Positive definite (p.d.) matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications. In modern high-dimensional applications, a common approach to finding sparse positive definite matrices is to threshold their small off-diagonal elements. This thresholding, sometimes referred to as hard-thresholding, sets small elements to zero. Thresholding has the attractive property that the resulting matrices are sparse, and are thus easier to interpret and work with. In many applications, it is often required, and thus implicitly assumed, that thresholded matrices retain positive definiteness. In this paper we formally investigate the algebraic properties of p.d. matrices which are thresholded. We demonstrate that for positive definiteness to be preserved, the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of complete components. This result rigorously demonstrates that, except in special cases, positive definiteness can be easily lost. We then proceed to demonstrate that the class of diagonally dominant matrices is not maximal in terms of retaining positive definiteness when thresholded. Consequently, we derive characterizations of matrices which retain positive definiteness when thresholded with respect to important classes of graphs. In particular, we demonstrate that retaining positive definiteness upon thresholding is governed by complex algebraic conditions. 相似文献
3.
一类含参数的分块对称矩阵的正定性及应用 总被引:3,自引:0,他引:3
金朝永 《高校应用数学学报(A辑)》2001,16(1):107-113
首先给出一种判断分块对称矩阵正定的方法,提供了确定一组尽可能小的参数,使一类含参数的分块对称矩阵正定的简单算法,然后,将其结果用于研究线性定常大系统的分散镇定性,得到了一类可分散镇定的线性大系统,并给出了相应的分散镇定算法,同文献中提供的方法相比,该算法不仅扩大了所考虑的系统范围,而且不会引起过高的反馈增益,同时还简单易算。 相似文献
4.
Filter approaches, initially presented by Fletcher and Leyffer in 2002, are attractive methods for nonlinear programming. In this paper, we propose an interior-point barrier projected Hessian updating algorithm with line search filter method for nonlinear optimization. The Lagrangian function value instead of the objective function value is used in the filter. The damped BFGS updating is employed to maintain the positive definiteness of the matrices in projected Hessian updating algorithm. The numerical experiments are reported to show the effectiveness of the proposed algorithm. 相似文献
5.
The efficiency of hierarchical matrices is based on the approximate evaluation of usual matrix operations. The introduced approximation error may, however, lead to a loss of important matrix properties. In this article we present a technique which preserves the positive definiteness of a matrix independently of the approximation quality. The importance of this technique is illustrated by an elliptic mixed boundary value problem with tiny Dirichlet part. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
6.
G. Rousseau 《Linear and Multilinear Algebra》1998,44(2):179-185
It is shown that a matrix with non-negative entries has non-negative determinant if in each row the elements decrease, by steadily smaller amounts, as one proceeds (in either direction) away from the main diagonal. This condition suffices to establish non- negativity of the determinant for certain matrices to which the familiar Minkowski- Hadamard-Ostrowski dominance conditions do not apply. In the symmetric case it provides a sufficient condition for non-negative definiteness. This may be applied to establish the positive definiteness of certain real symmetric Toeplitz matrices. 相似文献
7.
We present a sufficient regularity condition for interval matrices which generalizes two previously known ones. It is formulated
in terms of positive definiteness of a certain point matrix, and can also be used for checking positive definiteness of interval
matrices. Comparing it with Beeck’s strong regularity condition, we show by counterexamples that none of the two conditions
is more general than the other one. 相似文献
8.
Zhong-Zhi Bai. 《Mathematics of Computation》2006,75(254):791-815
For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.
9.
In this paper, we propose a fast algorithm for computing the spectral radii of symmetric nonnegative tensors. In particular, by this proposed algorithm, we are able to obtain the spectral radii of weakly reducible symmetric nonnegative tensors without requiring the partition of the tensors. As we know, it is very costly to determine the partition for large‐sized weakly reducible tensors. Numerical results are reported to show that the proposed algorithm is efficient and also able to compute the spectral radii of large‐sized tensors. As an application, we present an algorithm for testing the positive definiteness of Z‐tensors. By this algorithm, it is guaranteed to determine the positive definiteness for any Z‐tensor. 相似文献
10.
11.
一类新的非单调信赖域算法 总被引:1,自引:0,他引:1
提出了一类带线性搜索的非单调信赖域算法.算法将非单调Armijo线性搜索技术与信赖域方法相结合,使算法不需重解子问题.而且由于采用了MBFGS校正公式,使矩阵Bk能较好地逼近目标函数的Hesse矩阵并保持正定传递.在较弱的条件下,证明了算法的全局收敛性.数值结果表明算法是有效的. 相似文献
12.
本文推广了文[1]的主要定理,给出了用低阶矩阵判定高阶矩阵正定的判定定理,同时给出了矩阵方程AX=B的反问题在正定矩阵类中解存在的充要条件及解的一般形式. 相似文献
13.
Mathieu Bouchard Anne-Laure Jousselme Pierre-Emmanuel Doré 《International Journal of Approximate Reasoning》2013,54(5):615-626
In this paper we provide a proof for the positive definiteness of the Jaccard index matrix used as a weighting matrix in the Euclidean distance between belief functions defined in Jousselme et al. [13]. The idea of this proof relies on the decomposition of the matrix into an infinite sum of positive semidefinite matrices. The proof is valid for any size of the frame of discernment but we provide an illustration for a frame of three elements. The Jaccard index matrix being positive definite guaranties that the associated Euclidean distance is a full metric and thus that a null distance between two belief functions implies that these belief functions are strictly identical. 相似文献
14.
Iwona Skalna 《Linear and Multilinear Algebra》2017,65(12):2472-2482
Several verifiable sufficient and necessary conditions for strong regularity of parametric interval matrices are given. The conditions are formulated in terms of the spectral radius, positive definiteness and eigenvalues of certain real matrices and a scaled maximum norm. 相似文献
15.
Stefano Serra-Capizzano 《Numerische Mathematik》2007,107(1):175-179
In a recent paper, motivated by the analysis of some BVM methods, Aceto and Trigiante consider tests for proving the positive
definiteness of real banded Toeplitz matrices. Here we furnish a new test and we show that the analysis provided by Aceto
and Trigiante can be simplified by using known facts on the symbol. 相似文献
16.
This paper concerns the memoryless quasi-Newton method, that is precisely the quasi-Newton method for which the approximation to the inverse of Hessian, at each step, is updated from the identity matrix. Hence its search direction can be computed without the storage of matrices. In this paper, a scaled memoryless symmetric rank one (SR1) method for solving large-scale unconstrained optimization problems is developed. The basic idea is to incorporate the SR1 update within the framework of the memoryless quasi-Newton method. However, it is well-known that the SR1 update may not preserve positive definiteness even when updated from a positive definite matrix. Therefore we propose the memoryless SR1 method, which is updated from a positive scaled of the identity, where the scaling factor is derived in such a way that positive definiteness of the updating matrices are preserved and at the same time improves the condition of the scaled memoryless SR1 update. Under very mild conditions it is shown that, for strictly convex objective functions, the method is globally convergent with a linear rate of convergence. Numerical results show that the optimally scaled memoryless SR1 method is very encouraging. 相似文献
17.
Kenneth Nordström 《Linear algebra and its applications》2011,434(6):1489-1512
Convexity properties of the inverse of positive definite matrices and the Moore-Penrose inverse of nonnegative definite matrices with respect to the partial ordering induced by nonnegative definiteness are studied. For the positive definite case null-space characterizations are derived, and lead naturally to a concept of strong convexity of a matrix function, extending the conventional concept of strict convexity. The positive definite results are shown to allow for a unified analysis of problems in reproducing kernel Hilbert space theory and inequalities involving matrix means. The main results comprise a detailed study of the convexity properties of the Moore-Penrose inverse, providing extensions and generalizations of all the earlier work in this area. 相似文献
18.
Summary It is known that the matrices defining the discrete problem generated by a k-step Boundary Value Method (BVM) have a quasi-Toeplitz band structure [7]. In particular, when the boundary conditions are skipped, they become Toeplitz matrices. In this paper, by introducing a characterization of positive definiteness for such matrices, we shall prove that the Toeplitz matrices which arise when using the methods in the classes of BVMs known as Generalized BDF and Top Order Methods have such property.
Mathematics Subject Classification (2000):65L06, 47B35, 15A48Work supported by G.N.C.S. 相似文献
19.
Variable-fidelity modeling (VFM), sometimes also termed multi-fidelity modeling, refers to the utilization of two or more data layers of different accuracy in order to construct an inexpensive emulator of a given numerical high-fidelity model. In practical applications, this situation arises when simulators of different accuracy for the same physical process are given and it is assumed that many low-fidelity sample points are affordable, but the high-fidelity model is extremely costly to assess. More precisely, the VFM objective is to construct a predictor function, which interpolates the primary sample data but is driven by the trend indicated by the secondary data. In view of the computational effort, the objective is to use as few high-fidelity sample points as possible to achieve a desired level of accuracy. A widely used VFM method is Cokriging. The technique yields the best linear unbiased estimator based on the given data, taking spatial correlation into account. One way to model spatial correlation is via positive definite correlation kernels that yield positive definite correlation matrices for distinct input sample points. In this contribution, we will address the positive definiteness of Cokriging correlation matrices which is necessary for the method but not granted, due to the modeling of the cross-correlations. We discuss both the cases of distinct and coincident low- and high-fidelity sample points. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
20.
In this paper, we generalize the algorithm described by Rump and Graillat to compute verified and narrow error bounds such that a slightly perturbed matrix is guaranteed to have an eigenvalue with geometric multiplicity q within computed error bounds. The corresponding invariant subspace can be directly obtained by our algorithm. Our verification method is based on border matrix technique. We demonstrate the performance of our algorithm for matrices of dimension up to hundreds with non-defective and defective eigenvalues. 相似文献