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1.
Summary. The paper deals with eigenvalue estimates for block incomplete factorization methods for symmetric matrices. First, some
previous results on upper bounds for the maximum eigenvalue of preconditioned matrices are generalized to each eigenvalue.
Second, upper bounds for the maximum eigenvalue of the preconditioned matrix are further estimated, which presents a substantial
improvement of earlier results. Finally, the results are used to estimate bounds for every eigenvalue of the preconditioned
matrices, in particular, for the maximum eigenvalue, when a modified block incomplete factorization is used to solve an elliptic
equation with variable coefficients in two dimensions. The analysis yields a new upper bound of type for the condition number of the preconditioned matrix and shows clearly how the coefficients of the differential equation
influence the positive constant .
Received March 27, 1996 / Revised version received December 27, 1996 相似文献
2.
L. Yu. Kolotilina 《Journal of Mathematical Sciences》1996,79(3):1043-1047
The paper presents upper bounds for the largest eigenvalue of a block Jacobi scaled symmetric positive-definite matrix which
depend only on such parameters as the block semibandwidth of a matrix and its block size. From these bounds we also derive
upper bounds for the smallest eigenvalue of a symmetric matrix with identity diagonal blocks. Bibliography: 4 titles.
Translated by L. Yu. Kolotilina.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 18–25. 相似文献
3.
For two Hermitian matrices A and B, at least one of which is positive semidefinite, we give upper and lower bounds for each eigenvalue of AB in terms of the eigenvalues of A and B. For two complex matrices A,B with known singular values, upper and lower bounds are deduced for each singular value of AB. 相似文献
4.
对最大特征值的上下界进行估计是非负矩阵理论的重要部分,借助两个新的矩阵,从而得到一个判定非负矩阵最大特征值范围的界值定理,其结果比有关结论更加精确. 相似文献
5.
L. Yu. Kolotilina 《Journal of Mathematical Sciences》2011,176(1):68-77
The paper refines the classical Ostrowski disk theorem and suggests lower bounds for the smallest-in-modulus eigenvalue and
the smallest singular value of a square matrix under certain diagonal dominance conditions. A lower bound for the smallest-in-modulus
eigenvalue of a product of m ≥ 2 matrices satisfying joint diagonal dominance conditions is obtained. The particular cases of the bounds suggested that
correspond to the infinity norm are discussed separately and compared with some known results. Bibliography: 8 titles. 相似文献
6.
Yuji Nakatsukasa 《Applied Numerical Mathematics》2012,62(1):67-78
We derive new perturbation bounds for eigenvalues of Hermitian matrices with block tridiagonal structure. The main message of this paper is that an eigenvalue is insensitive to blockwise perturbation, if it is well-separated from the spectrum of the diagonal blocks nearby the perturbed blocks. Our bound is particularly effective when the matrix is block-diagonally dominant and graded. Our approach is to obtain eigenvalue bounds via bounding eigenvector components, which is based on the observation that an eigenvalue is insensitive to componentwise perturbation if the corresponding eigenvector components are small. We use the same idea to explain two well-known phenomena, one concerning aggressive early deflation used in the symmetric tridiagonal QR algorithm and the other concerning the extremal eigenvalues of Wilkinson matrices. 相似文献
7.
Luca Bergamaschi 《Numerical Linear Algebra with Applications》2012,19(4):754-772
This paper is devoted to the analysis of the eigenvalue distribution of two classes of block preconditioners for the generalized saddle point problem. Most of the bounds developed improve those appeared in previously published works. Numerical results onto a realistic test problem give evidence of the effectiveness of the estimates on the spectrum of preconditioned matrices. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
8.
Zhi-Ru Ren 《计算数学(英文版)》2012,30(5):533-543
We give general expressions, analyze algebraic properties and derive eigenvalue bounds for a sequence of Toeplitz matrices associated with the sinc discretizations of various orders of differential operators. We demonstrate that these Toeplitz matrices can be satisfactorily preconditioned by certain banded Toeplitz matrices through showing that the spectra of the preconditioned matrices are uniformly bounded. In particular, we also derive eigenvalue bounds for the banded Toeplitz preconditioners. These results are elementary in constructing high-quality structured preconditioners for the systems of linear equations arising from the sinc discretizations of ordinary and partial differential equations, and are useful in analyzing algebraic properties and deriving eigenvalue bounds for the corresponding preconditioned matrices. Numerical examples are given to show effectiveness of the banded Toeplitz preconditioners. 相似文献
9.
10.
Schur定理规定了半正定矩阵的Hadamard乘积的所有特征值的整体界限,Eric Iksoon lm在同样的条件下确定了每个特征值的特殊的界限,本文给出了Hermitian矩阵的Hadamard乘积的每个特征值的估计,改进和推广了I.Schur和Eric Iksoon Im的相应结果。 相似文献
11.
The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we consider an interval eigenvalue problem with symmetric tridiagonal matrices. A theoretical result is obtained that under certain assumptions the upper and lower bounds of interval eigenvalues of the problem must be achieved just at some vertex matrices of the interval matrix. Then a sufficient condition is provided to guarantee the assumption to be satisfied. The conclusion is illustrated also by a numerical example. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
12.
L. Yu. Kolotilina 《Journal of Mathematical Sciences》2006,137(3):4815-4834
The paper presents new pseudoblock diagonal-dominance conditions for matrices with fixed block partitioning, which generalize
the pointwise kth-order diagonal-dominance conditions and pointwise circuit diagonal-dominance conditions. For matrices satisfying
pseudoblock conditions, the singularity/nonsingularity problem is considered. In particular, for block 2 × 2 matrices, certain
known results are improved and generalized. Some eigenvalue inclusion regions corresponding to pseudoblock nonsingularity
conditions are presented. Bibliography: 11 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 94–131. 相似文献
13.
Lilia Yu. Kolotilina 《Numerical Algorithms》2006,42(3-4):247-280
Using a unified approach based on the monotonicity property of the Perron root and its circuit extension, a series of exact two-sided bounds for the Perron root of a nonnegative matrix in terms of paths in the associated directed graph is obtained. A method for deriving the so-called mixed upper bounds is suggested. Based on the upper bounds for the Perron root, new diagonal dominance type conditions for matrices are introduced. The singularity/nonsingularity problem for matrices satisfying such conditions is analyzed, and the associated eigenvalue inclusion sets are presented. In particular, a bridge connecting Gerschgorin disks with Brualdi eigenvalue inclusion sets is found. Extensions to matrices partitioned into blocks are proposed. 相似文献
14.
Several “distances” between the spectra of two matrices are discussed and compared. Optimal bounds are given, which enable us to reduce certain bounds on the eigenvalue variation of matrices by a factor of about two. The results of Bhatia and Mukherjea and of Bhatia and Friedland on the eigenvalue variation are derived in an elementary way using results of Henrici on the spectral variation. 相似文献
15.
Moritz Schulze Darup Martin Kastsian Stefan Mross Martin Mönnigmann 《Journal of Global Optimization》2014,58(4):631-652
We compare two established and a new method for the calculation of spectral bounds for Hessian matrices on hyperrectangles by applying them to a large collection of 1,522 objective and constraint functions extracted from benchmark global optimization problems. Both the tightness of the spectral bounds and the computational effort of the three methods, which apply to $C^2$ functions ${\varphi }:\mathbb{R }^n\rightarrow \mathbb{R }$ that can be written as codelists, are assessed. Specifically, we compare eigenvalue bounds obtained with the interval variant of Gershgorin’s circle criterion (Adjiman et al. in Comput Chem Eng 22(9):1137–1158, 1998; Gershgorin in Izv. Akad. Nauk SSSR, Ser. fizmat. 6:749–754, 1931), Hertz (IEEE Trans Autom Control 37:532–535, 1992) and Rohn’s (SIAM J Matrix Anal Appl 15(1):175–184, 1994) method for tight bounds of interval matrices, and a recently proposed Hessian matrix eigenvalue arithmetic (Mönnigmann in SIAM J. Matrix Anal. Appl. 32(4): 1351–1366, 2011), which deliberately avoids the computation of interval Hessians. The eigenvalue arithmetic provides tighter, as tight, and less tight bounds than the interval variant of Gershgorin’s circle criterion in about 15, 61, and 24 % of the examples, respectively. Hertz and Rohn’s method results in bounds that are always as tight as or tighter than those from Gershgorin’s circle criterion, and as tight as or tighter than those from the eigenvalue arithmetic in 96 % of the cases. In 4 % of the examples, the eigenvalue arithmetic results in tighter bounds than Hertz and Rohn’s method. This result is surprising, since Hertz and Rohn’s method provides tight bounds for interval matrices. The eigenvalue arithmetic provides tighter bounds in these cases, since it is not based on interval matrices. 相似文献
16.
Computing the extremal eigenvalue bounds of interval matrices is non‐deterministic polynomial‐time (NP)‐hard. We investigate bounds on real eigenvalues of real symmetric tridiagonal interval matrices and prove that for a given real symmetric tridiagonal interval matrices, we can achieve its exact range of the smallest and largest eigenvalues just by computing extremal eigenvalues of four symmetric tridiagonal matrices. 相似文献
17.
THE EIGENVALUE PERTURBATION BOUND FOR ARBITRARY MATRICES 总被引:1,自引:0,他引:1
Wen Li Jian-xin Chen 《计算数学(英文版)》2006,24(2):141-148
In this paper we present some new absolute and relative perturbation bounds for theeigenvalue for arbitrary matrices, which improves some recent results. The eigenvalueinclusion region is also discussed. 相似文献
18.
Steven Delvaux 《Mathematische Nachrichten》2012,285(16):1935-1962
We consider banded block Toeplitz matrices Tn with n block rows and columns. We show that under certain technical assumptions, the normalized eigenvalue counting measure of Tn for n → ∞ weakly converges to one component of the unique vector of measures that minimizes a certain energy functional. In this way we generalize a recent result of Duits and Kuijlaars for the scalar case. Along the way we also obtain an equilibrium problem associated to an arbitrary algebraic curve, not necessarily related to a block Toeplitz matrix. For banded block Toeplitz matrices, there are several new phenomena that do not occur in the scalar case: (i) The total masses of the equilibrium measures do not necessarily form a simple arithmetic series but in general are obtained through a combinatorial rule; (ii) The limiting eigenvalue distribution may contain point masses, and there may be attracting point sources in the equilibrium problem; (iii) More seriously, there are examples where the connection between the limiting eigenvalue distribution of Tn and the solution to the equilibrium problem breaks down. We provide sufficient conditions guaranteeing that no such breakdown occurs; in particular we show this if Tn is a Hessenberg matrix. 相似文献
19.
Janez Povh 《Journal of Global Optimization》2010,48(3):447-463
Finding global optimum of a non-convex quadratic function is in general a very difficult task even when the feasible set is
a polyhedron. We show that when the feasible set of a quadratic problem consists of orthogonal matrices from
\mathbbRn×k{\mathbb{R}^{n\times k}} , then we can transform it into a semidefinite program in matrices of order kn which has the same optimal value. This opens new possibilities to get good lower bounds for several problems from combinatorial
optimization, like the Graph partitioning problem (GPP), the Quadratic assignment problem (QAP) etc. In particular we show how to improve significantly the well-known Donath-Hoffman eigenvalue lower bound for GPP
by semidefinite programming. In the last part of the paper we show that the copositive strengthening of the semidefinite lower
bounds for GPP and QAP yields the exact values. 相似文献
20.
In this paper we compare Krylov subspace methods with Chebyshev series expansion for approximating the matrix exponential operator on large, sparse, symmetric matrices. Experimental results upon negative‐definite matrices with very large size, arising from (2D and 3D) FE and FD spatial discretization of linear parabolic PDEs, demonstrate that the Chebyshev method can be an effective alternative to Krylov techniques, especially when memory bounds do not allow the storage of all Ritz vectors. We also discuss the sensitivity of Chebyshev convergence to extreme eigenvalue approximation, as well as the reliability of various a priori and a posteriori error estimates for both methods. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献