Accuracy of the Kogbetliantz method for scaled diagonally dominant triangular matrices

The paper proves that Kogbetliantz method computes all singular values of a scaled diagonally dominant triangular matrix, which can be well scaled from both sides symmetrically, to high relative accuracy. Special attention is paid to deriving sharp accuracy bounds for one step, one batch and one sweep of the method. By a simple numerical test it is shown that the methods based on bidiagonalization are generally not accurate on that class of well-behaved matrices.     

对角占优矩阵奇异-非奇异的充分必要判据

本文研究对角占优矩阵奇异-非奇异的充分必要条件.基于Taussky定理,本文得出,可约对角占优矩阵的奇异性由其独立Frobenius块的奇异性决定,从而将这一问题化为不可约对角占优矩阵的奇异-非奇异性问题;运用Taussky定理研究奇异不可约对角占优矩阵的相似性和酉相似性,获得这类矩阵元素辐角间的关系;并与Taussky定理给出的这类矩阵元素模之间的关系结合在一起,研究不可约对角占优矩阵奇异的充分必要条件;最后给出不可约对角占优矩阵奇异-非奇异性的判定方法.     

Pivoting strategies leading to diagonal dominance by rows

Pivoting strategies for Gaussian elimination leading to upper triangular matrices which are diagonally dominant by rows are studied. Forward error analysis of triangular systems whose coefficient matrices are diagonally dominant by rows is performed. We also obtain small bounds of the backward errors for the pivoting strategies mentioned above. Our examples of matrices include H-matrices and some generalizations of diagonally dominant matrices, and scaled partial pivoting for the 1-norm is an example of these pivoting strategies. In the case of an M-matrix, a pivoting strategy of computational complexity is proposed, which satisfies all the results of the paper. Received June 6, 1997 / Revised version received October 27, 1997     

奇异M—矩阵和广义对角占成阵的实用判定准则

1　引言和符号首先对本文所采用的符号和术语作一约定和说明,而不再重申.N表示前面n个自然数的集合,而分别用Mn(C)和Mn(R)表示所有n阶复方阵和n阶实方阵的集合,Rn表示n维实列向量.Zn={A|A=(aij)∈Mn(R),aij≤0,i≠j,i,j∈N}.若A∈Zn则称A为Z-矩阵,有时也简记为A∈Z.I恒表示适当阶的单位矩阵.设α和β为N的非空子集,对于A∈Mn(C),把由A中行标属于α而列标属于β的元素按照原来相对位置所构成的子矩阵记为A(α,β),特别地,把主子阵A(α,α)简记为A(α)、当A(α)可逆时,其逆阵记为A(α)-1,此时称矩阵A/A(α)=A(α)-A(α,α).…     

Perturbation theory for the LDU factorization and accurate computations for diagonally dominant matrices

We present a structured perturbation theory for the LDU factorization of (row) diagonally dominant matrices and we use this theory to prove that a recent algorithm of Ye (Math Comp 77(264):2195–2230, 2008) computes the L, D and U factors of these matrices with relative errors less than 14n ³ u, where u is the unit roundoff and n × n is the size of the matrix. The relative errors for D are componentwise and for L and U are normwise with respect the “max norm” ||A||_M = max_ij |a_ij|{\|A\|_M = \max_{ij} |a_{ij}|}. These error bounds guarantee that for any diagonally dominant matrix A we can compute accurately its singular value decomposition and the solution of the linear system Ax = b for most vectors b, independently of the magnitude of the traditional condition number of A and in O(n ³) flops.     

非广义对角占优矩阵的判定准则

给出了判定非广义对角占优矩阵的充要条件,从理论上彻底解决了不可约非广义对角占优矩阵的判定问题,并给出了判定不可约非广义对角占优矩阵的具体算法.     

On diagonal-Schur complements of block diagonally dominant matrices

It is known that the diagonal-Schur complements of strictly diagonally dominant matrices are strictly diagonally dominant matrices [J.Z. Liu, Y.Q. Huang, Some properties on Schur complements of H-matrices and diagonally dominant matrices, Linear Algebra Appl. 389 (2004) 365-380], and the same is true for nonsingular H-matrices [J.Z. Liu, J.C. Li, Z.T. Huang, X. Kong, Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices, Linear Algebra Appl. 428 (2008) 1009-1030]. In this paper, we research the properties on diagonal-Schur complements of block diagonally dominant matrices and prove that the diagonal-Schur complements of block strictly diagonally dominant matrices are block strictly diagonally dominant matrices, and the same holds for generalized block strictly diagonally dominant matrices.     

一类广义对角占优矩阵

对角占优型矩阵的研究一直是诸多领域中广泛关注的问题．本文讨论了一类广义对角占优矩阵,得到了其优良性质,以及与重要矩阵类拟对角占优矩阵和Ｍ矩阵的关系     

The Lidskii-Mirsky-Wielandt theorem – additive and multiplicative versions

Summary. We use a simple matrix splitting technique to give an elementary new proof of the Lidskii-Mirsky-Wielandt Theorem and to obtain a multiplicative analog of the Lidskii-Mirsky-Wielandt Theorem, which we argue is the fundamental bound in the study of relative perturbation theory for eigenvalues of Hermitian matrices and singular values of general matrices. We apply our bound to obtain numerous bounds on the matching distance between the eigenvalues and singular values of matrices. Our results strengthen and generalize those in the literature. Received November 20, 1996 / Revised version received January 27, 1998     

Estimates for the inverse elements of tridiagonal matrices

In this work, new upper and lower bounds for the inverse entries of the tridiagonal matrices are presented. The bounds improve the bounds in D. Kershaw [Inequalities on the elements of the inverse of a certain tridiagonal matrix, Math. Comput. 24 (1970) 155–158], P.N. Shivakumar, C.X. Ji [Upper and lower bounds for inverse elements of finite and infinite tridiagonal matrices, Linear Algebr. Appl. 247 (1996) 297–316], R. Nabben [Two-sided bounds on the inverse of diagonally dominant tridiagonal matrices, Linear Algebr. Appl. 287 (1999) 289–305] and R. Peluso, T. Politi [Some improvements for two-sided bounds on the inverse of diagonally dominant tridiagonal matrices, Linear. Algebr. Appl. 330 (2001) 1–14].     

A lower bound for the smallest singular value of a matrix

For a matrix A which is diagonally dominant both by rows and by columns, we give bounds for 6A^-16₁ and 6A^-16_∞, which then can be used to give a lower bound for the smallest singular value. We also show that these bounds can be attained, and show how the result can be extended to block matrices.     

局部双对角占优矩阵及应用

本文引进了局部双对角占优矩阵的概念，讨论了这类矩阵的性质，给出了局部双对角占优矩阵是广义严格对角占优矩阵的等价表征，得到了Ｍ－矩阵的新表征，推广了［１－１２］的相应结果。     

Computing singular values of diagonally dominant matrices to high relative accuracy

For a (row) diagonally dominant matrix, if all of its off-diagonal entries and its diagonally dominant parts (which are defined for each row as the absolute value of the diagonal entry subtracted by the sum of the absolute values of off-diagonal entries in that row) are accurately known, we develop an algorithm that computes all the singular values, including zero ones if any, with relative errors in the order of the machine precision. When the matrix is also symmetric with positive diagonals (i.e. a symmetric positive semi-definite diagonally dominant matrix), our algorithm computes all eigenvalues to high relative accuracy. Rounding error analysis will be given and numerical examples will be presented to demonstrate the high relative accuracy of the algorithm.

     

Perturbation Theory for Simultaneous Bases of Singular Subspaces

New perturbation theorems are proved for simultaneous bases of singular subspaces of real matrices. These results improve the absolute bounds previously obtained in [6] for general (complex) matrices. Unlike previous results, which are valid only for the Frobenius norm, the new bounds, as well as those in [6] for complex matrices, are extended to any unitarily invariant matrix norm. The bounds are complemented with numerical experiments which show their relevance for the algorithms computing the singular value decomposition. Additionally, the differential calculus approach employed allows to easily prove new sin perturbation theorems for singular subspaces which deal independently with left and right singular subspaces.     

Perturbation of the SVD in the presence of small singular values

This paper gives SVD perturbation bounds and expansions that are of use when an m × n, m ? n matrix A has small singular values. The first part of the paper gives subspace bounds that are closely related to those of Wedin but are stated so as to isolate the effect of any small singular values to the left singular subspace. In the second part first and second order approximations are given for perturbed singular values. The subspace bounds are used to show that all approximations retain accuracy when applied to small singular values. The paper concludes by deriving a subspace bound for multiplicative perturbations and using that bound to give a simple approximation to a singular value perturbed by a multiplicative perturbation.     

对角占优矩阵直积的注记

将对角占优矩阵的性质与矩阵的直积结合起来,给出了两矩阵的直积是对角占优矩阵的一些充分和必要条件,推广了近期的一些结果.最后用相应的数值例子说明了所得结果的有效性.     

Upper norm bounds for the inverse of locally doubly strictly diagonally dominant matrices with its applications in linear complementarity problems

Numerical Algorithms - In this paper, we present two error bounds for the linear complementarity problems (LCPs) of locally doubly strictly diagonally dominant (LDSDD) matrices. The error bounds...     

Accurate solutions of diagonally dominant tridiagonal linear systems

In this paper, we settle Higham’s conjecture for the LU factorization of diagonally dominant tridiagonal matrices. We establish a strong componentwise perturbation bound for the solution of a diagonally dominant tridiagonal linear system, independent of the traditional condition number of the coefficient matrix. We then accurately and efficiently solve the linear system by the GTH-like algorithm without pivoting, as suggested by the perturbation result.     

广义对角占优矩阵的充分条件

给出了一类局部双对角占优矩阵,进而获得了几个新的广义对角占优矩阵的充分条件.     

Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices

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.