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1.
L. Yu. Kolotilina 《Journal of Mathematical Sciences》1997,86(4):2803-2827
In the symmetric positive definite case, two-sided eigenvalue bounds for block Jacobi scaled matrices and upper eigenvalue
bounds for matrices preconditioned with an incomplete block factorization are derived. A quantitative characterization of
block matrix partitionings is also suggested, which can be used when analyzing various block preconditioning methods. Bibliography:
13 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 5–41. 相似文献
2.
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
3.
We give an upper bound for the least eigenvalue of a principal submatrix of a real symmetric matrix with zero diagonal, from which we establish an upper bound for the least eigenvalue of a graph when some vertices are removed using the components of the least eigenvector(s). We give lower and upper bounds for the least eigenvalue of a graph when some edges are removed. We also establish bounds for the components of the least eigenvector(s) of a real symmetric matrix and a graph. 相似文献
4.
L. Yu. Kolotilina 《Journal of Mathematical Sciences》2006,137(3):4794-4800
The paper presents new upper and lower bounds for the singular values of rectangularmatrices explicitly involving the matrix
sparsity pattern. These bounds are based on an upper bound for the Perron root of a nonnegative matrix and on the sparsity-dependent
version of the Ostrowski-Brauer theorem on eigenvalue inclusion regions. Bibliography: 7 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 57–68. 相似文献
5.
本文讨论离散时间代数Riccati方程ATXA-X-(ATXB+L)(R+BTXB)^-1(LT+BTXA)+Q=0的唯一对称正定解的上界和下界。 相似文献
6.
Zhong-xiao Jia 《计算数学(英文版)》1999,17(3):257-274
1.IntroductionTheLanczosalgorithm[Zo]isaverypowerfultoolforextractingafewextremeeigenvaluesandassociatedeigenvectorsoflargesymmetricmatrices[4'5'22].Sincethe1980's,considerableattentionhasbeenpaidtogeneralizingittolargeunsymmetricproblems.Oneofitsgen... 相似文献
7.
N. Mastronardi M. Van Barel R. Vandebril 《Numerical Linear Algebra with Applications》2008,15(4):327-337
Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of a symmetric positive‐definite (SPD) Toeplitz matrix. An algorithm for computing upper and lower bounds to the smallest eigenvalue of a SPD Toeplitz matrix has been recently derived (Linear Algebra Appl. 2007; DOI: 10.1016/j.laa.2007.05.008 ). The algorithm relies on the computation of the R factor of the QR factorization of the Toeplitz matrix and the inverse of R. The simultaneous computation of R and R?1 is efficiently accomplished by the generalized Schur algorithm. In this paper, exploiting the properties of the latter algorithm, a numerical method to compute the smallest eigenvalue and the corresponding eigenvector of SPD Toeplitz matrices in an accurate way is proposed. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
8.
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
9.
The Markov–Bernstein inequalities for the Jacobi measure remained to be studied in detail. Indeed the tools used for obtaining
lower and upper bounds of the constant which appear in these inequalities, did not work, since it is linked with the smallest
eigenvalue of a five diagonal positive definite symmetric matrix. The aim of this paper is to generalize the qd algorithm
for positive definite symmetric band matrices and to give the mean to expand the determinant of a five diagonal symmetric
matrix. After that these new tools are applied to the problem to produce effective lower and upper bounds of the Markov–Bernstein
constant in the Jacobi case. In the last part we com pare, in the particular case of the Gegenbauer measure, the lower and
upper bounds which can be deduced from this paper, with those given in Draux and Elhami (Comput J Appl Math 106:203–243, 1999) and Draux (Numer Algor 24:31–58, 2000).
相似文献
10.
In this paper we consider the problem of estimating the largest eigenvalue and the corresponding eigenvector of a symmetric matrix. In particular, we consider iterative methods, such as the power method and the Lanczos method. These methods need a starting vector which is usually chosen randomly. We analyze the behavior of these methods when the initial vector is chosen with uniform distribution over the unitn-dimensional sphere. We extend and generalize the results reported earlier. In particular, we give upper and lower bounds on the pnorm of the randomized error, and we improve previously known bounds with a detailed analysis of the role of the multiplicity of the largest eigenvalue. 相似文献
11.
In this article, we use known bounds on the smallest eigenvalue of a symmetric matrix and Schoenberg's theorem to provide both necessary as well as sufficient trace inequalities that guarantee a matrix D is a Euclidean distance matrix, EDM . We also provide necessary and sufficient trace inequalities that guarantee a matrix D is an EDM generated by a regular figure. 相似文献
12.
G. V. Savinov 《Journal of Mathematical Sciences》1984,24(1):95-98
For the determination of the minimal eigenvalue of a symmetric positive-definite matrix one obtains an estimate of the asymptotic rate of convergence of a generalized method of conjugate gradients, based on the method of the symmetric upper relaxation. One establishes the asymptotic value of the relaxation parameter.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instita im. V. A. Steklova AN SSSR, Vol. 111, pp. 145–150, 1981. 相似文献
13.
We study Lanczos and polynomial algorithms with random start for estimating an eigenvector corresponding to the largest eigenvalue of an n × n large symmetric positive definite matrix. We analyze the two error criteria: the randomized error and the randomized residual error. For the randomized error, we prove that it is not possible to get distribution-free bounds, i.e., the bounds must depend on the distribution of eigenvalues of the matrix. We supply such bounds and show that they depend on the ratio of the two largest eigenvalues. For the randomized residual error, distribution-free bounds exist and are provided in the paper. We also provide asymptotic bounds, as well as bounds depending on the ratio of the two largest eigenvalues. The bounds for the Lanczos algorithm may be helpful in a practical implementation and termination of this algorithm. © 1998 John Wiley & Sons, Ltd. 相似文献
14.
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
15.
Huiqiu Lin 《Linear and Multilinear Algebra》2013,61(4):442-447
Let D(G) denote the distance matrix of a connected graph G. The largest eigenvalue of D(G) is called the distance spectral radius of a graph G, denoted by ?(G). In this article, we give sharp upper and lower bounds for the distance spectral radius and characterize those graphs for which these bounds are best possible. 相似文献
16.
Using the equivalent block two-by-two real linear systems and relaxing technique, we establish a new block preconditioner for a class of complex symmetric indefinite linear systems. The new preconditioner is much closer to the original block two-by-two coefficient matrix than the Hermitian and skew-Hermitian splitting (HSS) preconditioner. We analyze the spectral properties of the new preconditioned matrix, discuss the eigenvalue distribution and derive an upper bound for the degree of its minimal polynomial. Finally, some numerical examples are provided to show the effectiveness and robustness of our proposed preconditioner. 相似文献
17.
Jian Wang 《数学学报(英文版)》2012,28(10):1995-2010
By adopting a nice auxiliary transform of Markov operators, we derive new bounds for the first eigenvalue of the generator corresponding to symmetric Markov processes. Our results not only extend the related topic in the literature, but also are efficiently used to study the first eigenvalue of birth-death processes with killing and that of elliptic operators with killing on half line. In particular, we obtain two approximation procedures for the first eigenvalue of birth-death processes with killing, and present qualitatively sharp upper and lower bounds for the first eigenvalue of elliptic operators with killing on half line. 相似文献
18.
A. N. Borzykh 《Journal of Mathematical Sciences》2008,150(2):1917-1925
A new optimization algorithm for computing the largest eigenvalue of a real symmetric matrix is considered. The algorithm
is based on a sequence of plane rotations increasing the sum of the matrix entries. It is proved that the algorithm converges
linearly and it is shown that it may be regarded as a relaxation method for the Rayleigh quotient. Bibliography: 6 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 346, 2007, pp. 5–20. 相似文献
19.
We present an upper bound O(n
2
) for the mixing time of a simple random walk on upper triangular matrices. We show that this bound is sharp up to a constant,
and find tight bounds on the eigenvalue gap. We conclude by applying our results to indicate that the asymmetric exclusion
process on a circle indeed mixes more rapidly than the corresponding symmetric process.
Received: 25 January 1999 / Revised version: 17 September 1999 / Published online: 14 June 2000 相似文献
20.
In this note, we present upper matrix bounds for the solution of the discrete algebraic Riccati equation (DARE). Using the matrix bound of Theorem 2.2, we then give several eigenvalue upper bounds for the solution of the DARE and make comparisons with existing results. The advantage of our results over existing upper bounds is that the new upper bounds of Theorem 2.2 and Corollary 2.1 are always calculated if the stabilizing solution of the DARE exists, whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions. Finally, we give numerical examples to demonstrate the effectiveness of the derived results. 相似文献