Cauchy型矩阵及其逆矩阵的快速三角分解

Abstract, In this paper,algorithms for determining the triangular factorization of Cauchy typematrices and their inverses are derived by using O(n2) operations.     

An implicit QR algorithm for symmetric semiseparable matrices

The QR algorithm is one of the classical methods to compute the eigendecomposition of a matrix. If it is applied on a dense n × n matrix, this algorithm requires O(n³) operations per iteration step. To reduce this complexity for a symmetric matrix to O(n), the original matrix is first reduced to tridiagonal form using orthogonal similarity transformations. In the report (Report TW360, May 2003) a reduction from a symmetric matrix into a similar semiseparable one is described. In this paper a QR algorithm to compute the eigenvalues of semiseparable matrices is designed where each iteration step requires O(n) operations. Hence, combined with the reduction to semiseparable form, the eigenvalues of symmetric matrices can be computed via intermediate semiseparable matrices, instead of tridiagonal ones. The eigenvectors of the intermediate semiseparable matrix will be computed by applying inverse iteration to this matrix. This will be achieved by using an O(n) system solver, for semiseparable matrices. A combination of the previous steps leads to an algorithm for computing the eigenvalue decompositions of semiseparable matrices. Combined with the reduction of a symmetric matrix towards semiseparable form, this algorithm can also be used to calculate the eigenvalue decomposition of symmetric matrices. The presented algorithm has the same order of complexity as the tridiagonal approach, but has larger lower order terms. Numerical experiments illustrate the complexity and the numerical accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

A block Cholesky-LU-based QR factorization for rectangular matrices

The Householder method provides a stable algorithm to compute the full QR factorization of a general matrix. The standard version of the algorithm uses a sequence of orthogonal reflections to transform the matrix into upper triangular form column by column. In order to exploit (level 3 BLAS or structured matrix) computational advantages for block-partitioned algorithms, we develop a block algorithm for the QR factorization. It is based on a well-known block version of the Householder method which recursively divides a matrix columnwise into two smaller matrices. However, instead of continuing the recursion down to single matrix columns, we introduce a novel way to compute the QR factors in implicit Householder representation for a larger block of several matrix columns, that is, we start the recursion at a block level instead of a single column. Numerical experiments illustrate to what extent the novel approach trades some of the stability of Householder's method for the computational efficiency of block methods.     

Asymptotic properties of the QR factorization of banded Hessenberg–Toeplitz matrices

We consider Givens QR factorization of banded Hessenberg–Toeplitz matrices of large order and relatively small bandwidth. We investigate the asymptotic behaviour of the R factor and Givens rotation when the order of the matrix goes to infinity, and present some interesting convergence properties. These properties can lead to savings in the computation of the exact QR factorization and give insight into the approximate QR factorizations of interest in preconditioning. The properties also reveal the relation between the limit of the main diagonal elements of R and the largest absolute root of a polynomial. Copyright © 2005 John Wiley & Sons, Ltd.     

On the perturbation of the Q‐factor of the QR factorization

This paper gives normwise and componentwise perturbation analyses for the Q‐factor of the QR factorization of the matrix A with full column rank when A suffers from an additive perturbation. Rigorous perturbation bounds are derived on the projections of the perturbation of the Q‐factor in the range of A and its orthogonal complement. These bounds overcome a serious shortcoming of the first‐order perturbation bounds in the literature and can be used safely. From these bounds, identical or equivalent first‐order perturbation bounds in the literature can easily be derived. When A is square and nonsingular, tighter and simpler rigorous perturbation bounds on the perturbation of the Q‐factor are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

QR factorization for row or column symmetric matrix

The problem of fast computing the QR factorization of row or column symmetric matrix is considered. We address two new algorithms based on a correspondence of Q and R matrices between the row or column symmetric matrix and its mother matrix. Theoretical analysis and numerical evidence show that, for a class of row or column symmetric matrices, the QR factorization using the mother matrix rather than the row or column symmetric matrix per se can save dramatically the CPU time and memory without loss of any numerical precision.     

基于反Krylov矩阵正交分解的半可分矩阵

在本文中，我们证明了对一个反Krylov矩阵作QR分解后，利用得到的正交矩阵可以将一个具有互异特征值的对称矩阵转化为一个半可分矩阵的形式，这个结果表明了反Krylov矩阵与半可分矩阵之间的联系．另外，我们还证明了这类对称半可分矩阵在QR达代下矩阵结构保持不变性．     

Divide and conquer algorithms for computing the eigendecomposition of symmetric diagonal-plus-semiseparable matrices

Three fast and stable divide and conquer algorithms to compute the eigendecomposition of symmetric diagonal-plus-semiseparable matrices are considered. AMS subject classification 15A18, 15A23, 65F15The research of the second and the third author was supported by the Research Council K.U. Leuven, to13.25cmproject OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research –     

Perturbation analysis for the symplectic QR factorization

In this paper, the perturbation analysis for the symplectic QR factorization is considered. Some first-order and rigorous normwise perturbation bounds with normwise or componentwise perturbations in the given matrix are presented.     

Wilkinson's inertia‐revealing factorization and its application to sparse matrices

We propose a new inertia‐revealing factorization for sparse symmetric matrices. The factorization scheme and the method for extracting the inertia from it were proposed in the 1960s for dense, banded, or tridiagonal matrices, but they have been abandoned in favor of faster methods. We show that this scheme can be applied to any sparse symmetric matrix and that the fill in the factorization is bounded by the fill in the sparse QR factorization of the same matrix (but is usually much smaller). We describe our serial proof‐of‐concept implementation and present experimental results, studying the method's numerical stability and performance.     

A Björck–Pereyra-type algorithm for Szegö–Vandermonde matrices based on properties of unitary Hessenberg matrices

In this paper we carry over the Björck-Pereyra algorithm for solving Vandermonde linear systems to what we suggest to call Szegö-Vandermonde systems V_Φ(x), i.e., polynomial-Vandermonde systems where the corresponding polynomial system Φ is the Szegö polynomials. The properties of the corresponding unitary Hessenberg matrix allow us to derive a fast O(n²) computational procedure. We present numerical experiments that indicate that for ill-conditioned matrices the new algorithm yields better forward accuracy than Gaussian elimination.     

A small note on the scaling of symmetric positive definite semiseparable matrices

In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix satisfies property A, one can easily construct a diagonal matrix such that has the lowest condition number over all matrices , for any choice of diagonal matrix . Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices. Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form. *The research was partially supported by the Research Council K.U. Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann–Hilbert problems, random matrices and Padé–Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling). The scientific responsibility rests with the authors. The second author participates in the SCCM program, Gates 2B, Stanford University, CA, USA and is also partially supported by the NSF. The first author visited the second one with a grant by the Fund for Scientific Research–Flanders (Belgium).     

New perturbation bounds and condition numbers for the hyperbolic QR factorization

Using the modified matrix-vector equation approach, the technique of Lyapunov majorant function and the Banach fixed point theorem, we obtain some new rigorous perturbation bounds for R factor of the hyperbolic QR factorization under normwise perturbation. These bounds are always tighter than the one given in the literature. Moreover, the optimal first-order perturbation bounds and the normwise condition numbers for the hyperbolic QR factorization are also presented.     

A generalization of Bartlett’s decomposition

Bartlett’s decomposition provides the distributional properties of the elements of the Cholesky factor of A=G^TG where the elements of G are i.i.d. standard Gaussian random variables. In this paper the most general case where the elements of G have a joint multivariate Gaussian density is considered.     

A new parallel chasing algorithm for transforming arrowhead matrices to tridiagonal form

Rutishauser, Gragg and Harrod and finally H.Y. Zha used the same class of chasing algorithms for transforming arrowhead matrices to tridiagonal form. Using a graphical theoretical approach, we propose a new chasing algorithm. Although this algorithm has the same sequential computational complexity and backward error properties as the old algorithms, it is better suited for a pipelined approach. The parallel algorithm for this new chasing method is described, with performance results on the Paragon and nCUBE. Comparison results between the old and the new algorithms are also presented.

     

Universal similarity factorization equalities for commutative quaternions and their matrices

In this work, we have established universal similarity factorization equalities over the commutative quaternions and their matrices. Based on these equalities, real matrix representations of commutative quaternions and their matrices have been derived, and their algebraic properties and fundamental equations have been determined. Moreover, illustrative examples are provided to support our results.     

Wiener–Hopf factorization for a group of exponentials of nilpotent matrices

A complete study of the generalized factorization for a group of 2×2 matrix functions of the form G=I+γN, where , I denotes the 2×2 identity matrix and N represents a rational nilpotent matrix function, is presented. A closely related class involving the same matrix N is also studied. The canonical and non-canonical factorizations are considered and explicit formulas are obtained for the partial indices and the factors in such factorizations. It is shown in particular that only one of the columns in the factors needs to be determined, as a solution to a homogeneous linear Riemann–Hilbert problem, the other column being expressed in terms of the first. Necessary and sufficient conditions for existence of a canonical factorization within the same class are established, as well as explicit formulas for the factors in this case.