Fast inversion algorithms for diagonal plus semiseparable matrices

Here are considered matrices represented as a sum of diagonal and semiseparable ones. These matrices belong to the class of structured matrices which arises in numerous applications. FastO(N) algorithms for their inversion were developed before under additional restrictions which are a source of instability. Our aim is to eliminate these restrictions and to develop reliable and stable numerical algorithms. In this paper we obtain such algorithms with the only requirement that the considered matrix is invertible and its determinant is not close to zero. The case of semiseparable matrices of order one was considered in detail in an earlier paper of the authors.     

Using the Sherman-Morrison-Woodbury inversion formula for a fast solution of tridiagonal block Toeplitz systems

A fast numerical algorithm for solving systems of linear equations with tridiagonal block Toeplitz matrices is presented. The algorithm is based on a preliminary factorization of the generating quadratic matrix polynomial associated with the Toeplitz matrix, followed by the Sherman-Morrison-Woodbury inversion formula and solution of two bidiagonal and one diagonal block Toeplitz systems. Tight estimates of the condition numbers are provided for the matrix system and the main matrix systems generated during the preliminary factorization. The emphasis is put on rigorous stability analysis to rounding errors of the Sherman-Morrison-Woodbury inversion. Numerical experiments are provided to illustrate the theory.     

Componentwise error analysis for the block LU factorization of totally nonnegative matrices

For the first time, perturbation bounds including componentwise perturbation bounds for the block LU factorization have been provided by Dopico and Molera (2005) [5]. In this paper, componentwise error analysis is presented for computing the block LU factorization of nonsingular totally nonnegative matrices. We present a componentwise bound on the equivalent perturbation for the computed block LU factorization. Consequently, combining with the componentwise perturbation results we derive componentwise forward error bounds for the computed block factors.     

H-矩阵和S-双对角占优矩阵

In this paper, the concept of the s-doubly diagonally dominant matrices is introduced and the properties of these matrices are discussed. With the properties of the s-doubly diagonally dominant matrices and the properties of comparison matrices, some equivalent conditions for H-matrices are presented. These conditions generalize and improve existing results about the equivalent conditions for H-matrices. Applications and examples using these new equivalent conditions are also presented, and a new inclusion region of k-multiple eigenvalues of matrices is obtained.     

Accurate computations with Said-Ball-Vandermonde matrices

A generalization of the Vandermonde matrices which arise when the power basis is replaced by the Said-Ball basis is considered. When the nodes are inside the interval (0,1), then those matrices are strictly totally positive. An algorithm for computing the bidiagonal decomposition of those Said-Ball-Vandermonde matrices is presented, which allows us to use known algorithms for totally positive matrices represented by their bidiagonal decomposition. The algorithm is shown to be fast and to guarantee high relative accuracy. Some numerical experiments which illustrate the good behaviour of the algorithm are included.     

Displacement structure approach to Chebyshev-Vandermonde and related matrices

In this paper we use the displacement structure concept to introduce a new class of matrices, designated asChebyshev-Vandermonde-like matrices, generalizing ordinary Chebyshev-Vandermonde matrices, studied earlier by different authors. Among other results the displacement structure approach allows us to give a nice explanation for the form of the Gohberg-Olshevsky formulas for the inverses of ordinary Chebyshev-Vandermonde matrices. Furthermore, the fact that the displacement structure is inherited by Schur complements leads to a fastO(n ²) implementation of Gaussian elimination withpartial pivoting for Chebyshev-Vandermonde-like matrices.     

An eigenvalue localization theorem for pentadiagonal symmetric Toeplitz matrices

We express the eigenvalues of a pentadiagonal symmetric Toeplitz matrix as the zeros of explicitly given rational functions.     

Fast inversion of Chebyshev--Vandermonde matrices

Summary. This paper contains two fast algorithms for inversion of Chebyshev--Vandermonde matrices of the first and second kind. They are based on special representations of the Bezoutians of Chebyshev polynomials of both kinds. The paper also contains the results of numerical experiments which show that the algorithms proposed here are not only much faster, but also more stable than other algorithms available. It is also efficient to use the above two algorithms for solving Chebyshev--Vandermode systems of equations with preprocessing. Received January 23, 1993/Revised version received May 21, 1993     

Factoring matrices with a tree-structured sparsity pattern

Let A be a matrix whose sparsity pattern is a tree with maximal degree d_max. We show that if the columns of A are ordered using minimum degree on |A|+|A|^∗, then factoring A using a sparse LU with partial pivoting algorithm generates only O(d_maxn) fill, requires only O(d_maxn) operations, and is much more stable than LU with partial pivoting on a general matrix. We also propose an even more efficient and just-as-stable algorithm called sibling-dominant pivoting. This algorithm is a strict partial pivoting algorithm that modifies the column preordering locally to minimize fill and work. It leads to only O(n) work and fill. More conventional column pre-ordering methods that are based (usually implicitly) on the sparsity pattern of |A|^∗|A| are not as efficient as the approaches that we propose in this paper.     

Tensor properties of multilevel Toeplitz and related matrices

A general proposal is presented for fast algorithms for multilevel structured matrices. It is based on investigation of their tensor properties and develops the idea recently introduced by Kamm and Nagy in the block Toeplitz case. We show that tensor properties of multilevel Toeplitz matrices are related to separation of variables in the corresponding symbol, present analytical tools to study the latter, expose truncation algorithms preserving the structure, and report on some numerical results confirming advantages of the proposal.     

Parallel and superfast algorithms for Hankel systems of equations

Summary Utilizing kernel structure properties a unified construction of Hankel matrix inversion algorithms is presented. Three types of algorithms are obtained: 1)O(n ²) complexity Levinson type, 2)O (n) parallel complexity Schur-type, and 3)O(n log² n) complexity asymptotically fast ones. All algorithms work without additional assumption (like strong nonsingularity).     

Lagrange's formula for tangential interpolation with application to structured matrices

Lagrange's interpolation formula is generalized to tangential interpolation. This includes interpolation by vector polynomials and by rational vector functions with prescribed pole characteristics. The formula is applied to obtain representations of the inverses of Cauchy-Vandermonde matrices generalizing former results.     

Algorithms for inversion of diagonal plus semiseparable operator matrices

LetH be a Hilbert space andRHH be a bounded linear operator represented by an operator matrix which is a sum of a diagonal and of a semiseparable type of order one operator matrices. We consider three methods for solution of the operator equationRx=y. The obtained results yields new algorithms for solution of integral equations and for inversion of matrices.     

On nonsingularity of block two-by-two matrices

We derive necessary and sufficient conditions for guaranteeing the nonsingularity of a block two-by-two matrix by making use of the singular value decompositions and the Moore–Penrose pseudoinverses of the matrix blocks. These conditions are complete, and much weaker and simpler than those given by Decker and Keller [D.W. Decker, H.B. Keller, Multiple limit point bifurcation, J. Math. Anal. Appl. 75 (1980) 417–430], and may be more easily examined than those given by Bai [Z.-Z. Bai, Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks, J. Comput. Appl. Math. 237 (2013) 295–306] from the computational viewpoint. We also derive general formulas for the rank of the block two-by-two matrix by utilizing either the unitarily compressed or the orthogonally projected sub-matrices.     

A look-ahead Bareiss algorithm for general Toeplitz matrices

Summary. The Bareiss algorithm is one of the classical fast solvers for systems of linear equations with Toeplitz coefficient matrices. The method takes advantage of the special structure, and it computes the solution of a Toeplitz system of order~ with only~ arithmetic operations, instead of~ operations. However, the original Bareiss algorithm requires that all leading principal submatrices be nonsingular, and the algorithm is numerically unstable if singular or ill-conditioned submatrices occur. In this paper, an extension of the Bareiss algorithm to general Toeplitz systems is presented. Using look-ahead techniques, the proposed algorithm can skip over arbitrary blocks of singular or ill-conditioned submatrices, and at the same time, it still fully exploits the Toeplitz structure. Implementation details and operations counts are given, and numerical experiments are reported. We also discuss special versions of the proposed look-ahead Bareiss algorithm for Hermitian indefinite Toeplitz systems and banded Toeplitz systems. Received August 27, 1993 / Revised version received March 1994     

Stability analysis and fast algorithms for triangularization of Toeplitz matrices

Summary. A new algorithm for triangularizing an Toeplitz matrix is presented. The algorithm is based on the previously developed recursive algorithms that exploit the Toeplitz structure and compute each row of the triangular factor via updating and downdating steps. A forward error analysis for this existing recursive algorithm is presented, which allows us to monitor the conditioning of the problem, and use the method of corrected semi-normal equations to obtain higher accuracy for certain ill-conditioned matrices. Numerical experiments show that the new algorithm improves the accuracy significantly while the computational complexity stays in . Received April 30, 1995 / Revised version received February 12, 1996     

Perturbations on the subdiagonals of Toeplitz matrices

Let L be an Hermitian linear functional defined on the linear space of Laurent polynomials. It is very well known that the Gram matrix of the associated bilinear functional in the linear space of polynomials is a Toeplitz matrix. In this contribution we analyze some linear spectral transforms of L such that the corresponding Toeplitz matrix is the result of the addition of a constant in a subdiagonal of the initial Toeplitz matrix. We focus our attention in the analysis of the quasi-definite character of the perturbed linear functional as well as in the explicit expressions of the new monic orthogonal polynomial sequence in terms of the first one.     

A direct method to solve block banded block Toeplitz systems with non-banded Toeplitz blocks

A fast solution algorithm is proposed for solving block banded block Toeplitz systems with non-banded Toeplitz blocks. The algorithm constructs the circulant transformation of a given Toeplitz system and then by means of the Sherman-Morrison-Woodbury formula transforms its inverse to an inverse of the original matrix. The block circulant matrix with Toeplitz blocks is converted to a block diagonal matrix with Toeplitz blocks, and the resulting Toeplitz systems are solved by means of a fast Toeplitz solver.The computational complexity in the case one uses fast Toeplitz solvers is equal to ξ(m,n,k)=O(mn³)+O(k³n³) flops, there are m block rows and m block columns in the matrix, n is the order of blocks, 2k+1 is the bandwidth. The validity of the approach is illustrated by numerical experiments.