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1.
Yongzhong Song 《Numerische Mathematik》2002,92(3):563-591
Summary. This paper investigates the comparisons of asymptotic rates of convergence of two iteration matrices. On the basis of nonnegative
matrix theory, comparisons between two nonnegative splittings and between two parallel multisplitting methods are derived.
When the coefficient matrix A is Hermitian positive (semi)definite, comparison theorems about two P-regular splittings and
two parallel multisplitting methods are proved.
Received April 4, 1998 / Revised version received October 18, 1999 / Published online November 15, 2001 相似文献
2.
Yongzhong Song 《Numerische Mathematik》1993,65(1):245-252
Summary In a recent paper the author has proposed some theorems on the comparison of the asymptotic rates of convergence of two nonnegative splittings. They extended the corresponding result of Miller and Neumann and implied the earlier theorems of Varga, Beauwens, Csordas and Varga. An open question by Miller and Neumann, which additional and appropriate conditions should be imposed to obtain strict inequality, was also answered. This article continues to investigate the comparison theorems for nonnegative splittings. The new results extend and imply the known theorems by the author, Miller and Neumann.The Project Supported by the Natural Science Foundation of Jiangsu Province Education Commission 相似文献
3.
Summary Comparison theorems for weak splittings of bounded operators are presented. These theorems extend the classical comparison theorem for regular splittings of matrices by Varga, the less known result by Wonicki, and the recent results for regular and weak regular splittings of matrices by Neumann and Plemmons, Elsner, and Lanzkron, Rose and Szyld. The hypotheses of the theorems presented here are weaker and the theorems hold for general Banach spaces and rather general cones. Hypotheses are given which provide strict inequalities for the comparisons. It is also shown that the comparison theorem by Alefeld and Volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators.This work was supported by the National Science Foundation grants DMS-8807338 and INT-8918502 相似文献
4.
BIT Numerical Mathematics - Some variants of the (block) Gauss–Seidel iteration for the solution of linear systems with M-matrices in (block) Hessenberg form are discussed. Comparison results... 相似文献
5.
MP matrices are those real matrices which possess a nonnegative, nonsingular l-inverse. This paper characterizes the nonnegative MP matrices and hence, determines when a nonnegative matrix A has a convergent regular splitting M—Q which induces the linear stationary iterative scheme xk+1=M-1Qxk+M-1b to solve Ax=b. 相似文献
6.
We introduce the class of strongly close-to-convex mappings of order c~ in the unit ball of a complex Banach space, and then, we give the sharp distortion theorems for this class of mappings in the unit ball of a complex Hilbert space X or the unit polydisc in Cn. As an application, a sharp growth theorem for strongly close-to-convex mappings of order α is obtained. 相似文献
7.
Semiconvergence of nonnegative splittings for singular matrices 总被引:1,自引:0,他引:1
Yongzhong Song 《Numerische Mathematik》2000,85(1):109-127
Summary. In this paper, we discuss semiconvergence of the matrix splitting methods for solving singular linear systems. The concepts
that a splitting of a matrix is regular or nonnegative are generalized and we introduce the terminologies that a splitting
is quasi-regular or quasi-nonnegative. The equivalent conditions for the semiconvergence are proved. Comparison theorem on
convergence factors for two different quasi-nonnegative splittings is presented. As an application, the semiconvergence of
the power method for solving the Markov chain is derived. The monotone convergence of the quasi-nonnegative splittings is
proved. That is, for some initial guess, the iterative sequence generated by the iterative method introduced by a quasi-nonnegative
splitting converges towards a solution of the system from below or from above.
Received August 19, 1997 / Revised version received August 20, 1998 / Published online January 27, 2000 相似文献
8.
Summary In this article, new comparison theorems for regular splittings of matrices are derived. In so doing, the initial results of Varga in 1960 on regular splittings of matrices, and the subsequent unpublished results of Wonicki in 1973 on regular splittings of matrices, will be seen to be special cases of these new comparison theorems.Dedicated to Fritz Bauer on the occasion of his 60th birthdayResearch supported in part by the Air Force Office of Scientific Research, and by the Department of Energy 相似文献
9.
Kh. N. Narzullaev 《Journal of Mathematical Sciences》1984,25(2):1084-1089
One presents the proof of a theorem which gives the possibility to isolate DOTU matrices and allows us to use computers in the investigation of DOTU matrices of small size.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 159–166, 1981. 相似文献
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For real A, B such that −1 ? B < A ? 1, we denote by R(A,B) the class of functions, such that f′ ? (1 + Az)/(1 + Bz). Sharp distortion results for functions from RA,B are obtained. 相似文献
13.
We develop a new, simple technique of proof for density theorems (i.e.,for the sufficient conditions to guarantee that the proper efficient points of a set are dense in the efficient frontier) in an ordered topological vector space. The results are the following: (i) the set of proper efficient points of any compact setQ is dense in the set of efficient points with respect to the original topology of the space whenever the ordering coneK is weakly closed and admits strictly positive functionals; moreover, ifK is not weakly closed, then there exists a compact set for which the density statement fails; (ii) ifQ is weakly compact, then we have only weak density, but ifK has a closed bounded base, then we can assert the density with respect to the original topology, (iii) there exists a similar possibility to assert the strong density for weakly compactQ if additional restrictions are placed onQ instead ofK. These three results are obtained in a unified way as corollaries of the same statement. In this paper, we use the concept of proper efficiency due to Henig. We extend his definition to the setting of a Hausdorff topological vector space.Research of the first author was supported by the Foundation of Fundamental Research of the Republic of Belarus. Authors are grateful to Professor Valentin V. Gorokhovik for suggesting the problem studied in this paper and for numerous fruitful conversations. 相似文献
14.
M. S. Derevyagin 《Mathematical Notes》2005,77(3-4):587-591
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The rates of convergence of iterative methods with standard preconditioning techniques usually degrade when the skew-symmetric part S of the matrix is relatively large. In this paper, we address the issue of preconditioning matrices with such large skew-symmetric parts. The main idea of the preconditioner is to split the matrix into its symmetric and skew-symmetric parts and to invert the (shifted) skew-symmetric matrix. Successful use of the method requires the solution of a linear system with matrix I+S. An efficient method is developed using the normal equations, preconditioned by an incomplete orthogonal factorization.Numerical experiments on various systems arising in physics show that the reduction in terms of iteration count compensates for the additional work per iteration when compared to standard preconditioners. 相似文献
18.
Eric M. Rains 《Probability Theory and Related Fields》1998,112(3):411-423
Using the machinery of zonal polynomials, we examine the limiting behavior of random symmetric matrices invariant under conjugation
by orthogonal matrices as the dimension tends to infinity. In particular, we give sufficient conditions for the distribution
of a fixed submatrix to tend to a normal distribution. We also consider the problem of when the sequence of partial sums of
the diagonal elements tends to a Brownian motion. Using these results, we show that if O
n
is a uniform random n×n orthogonal matrix, then for any fixed k>0, the sequence of partial sums of the diagonal of O
k
n
tends to a Brownian motion as n→∞.
Received: 3 February 1998 / Revised version: 11 June 1998 相似文献
19.
We present sufficient conditions for the convergent splitting of a non-Hermitian positive definite matrix. These results are applicable to identify the convergence of iterative methods for solving large sparse system of linear equations. 相似文献
20.
Oskar Maria Baksalary Götz Trenkler 《International Journal of Mathematical Education in Science & Technology》2013,44(6):827-829
By considering a general representation of proper rotation matrices, the eigenvalues and eigenspaces of those matrices are identified. 相似文献