A new recursive algorithm for inverting Hessenberg matrices

In the current article, the authors present a new recursive symbolic computational Hessenberg matrix algorithm, for inverting general Hessenberg matrices.     

Efficient parallel solution to large‐size sparse eigenproblems with block FSAI preconditioning

The choice of the preconditioner is a key factor to accelerate the convergence of eigensolvers for large‐size sparse eigenproblems. Although incomplete factorizations with partial fill‐in prove generally effective in sequential computations, the efficient preconditioning of parallel eigensolvers is still an open issue. The present paper describes the use of block factorized sparse approximate inverse (BFSAI) preconditioning for the parallel solution of large‐size symmetric positive definite eigenproblems with both a simultaneous Rayleigh quotient minimization and the Jacobi–Davidson algorithm. BFSAI coupled with a block diagonal incomplete decomposition proves a robust and efficient parallel preconditioner in a number of test cases arising from the finite element discretization of 3D fluid‐dynamical and mechanical engineering applications, outperforming FSAI even by a factor of 8 and exhibiting a satisfactory scalability. Copyright © 2011 John Wiley & Sons, Ltd.     

Efficient and stable solution of structured Hessenberg linear systems arising from difference equations

This paper is concerned with the efficient solution of (block) Hessenberg linear systems whose coefficient matrix is a Toeplitz matrix in (block) Hessenberg form plus a band matrix. Such problems arise, for instance, when we apply a computational scheme based on the use of difference equations for the computation of many significant special functions and quantities occurring in engineering and physics. We present a divide‐and‐conquer algorithm that combines some recent techniques for the numerical treatment of structured Hessenberg linear systems. Our approach is computationally efficient and, moreover, in many practical cases it can be shown to be componentwise stable. Copyright © 2000 John Wiley & Sons, Ltd.     

Hessenberg matrices and the Pell and Perrin numbers

In this paper, we investigate the Pell sequence and the Perrin sequence and we derive some relationships between these sequences and permanents and determinants of one type of Hessenberg matrices.     

Rounding Errors in Solving Block Hessenberg Systems

A rounding error analysis is presented for a divide-and-conquer algorithm to solve linear systems with block Hessenberg matrices. Conditions are derived under which the algorithm computes a stable solution. The algorithm is shown to be stable for block diagonally dominant matrices and for M-matrices.

     

Rounding Error Analysis in Solving M-Matrix Linear Systems of Block Hessenberg Form

The problem of solving large M-matrix linear systems with sparse coefficient matrix in block Hessenberg form is here addressed. In previous work of the authors a divide-and-conquer strategy was proposed and a backward error analysis of the resulting algorithm was presented showing its effectiveness for the solution of computational problems of queueing theory and Markov chains. In particular, it was shown that for block Hessenberg M-matrices the algorithm is weakly backward stable in the sense that the computed solution is the exact solution of a nearby linear system, where the norm of the perturbation is proportional to the condition number of the coefficient matrix. In this note a better error estimate is given by showing that for block Hessenberg M-matrices the algorithm is even backward stable.     

Infinite classes of anti‐mitre and 5‐sparse Steiner triple systems

In this paper, we present three constructions for anti‐mitre Steiner triple systems and a construction for 5‐sparse ones. The first construction for anti‐mitre STSs settles two of the four unsettled admissible residue classes modulo 18 and the second construction covers such a class modulo 36. The third construction generates many infinite classes of anti‐mitre STSs in the remaining possible orders. As a consequence of these three constructions we can construct anti‐mitre systems for at least 13/14 of the admissible orders. For 5‐sparse STS(υ), we give a construction for υ ≡ 1, 19 (mod 54) and υ ≠ 109. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 237–250, 2006     

求解欠定线性方程组稀疏解的算法

针对欠定线性方程组稀疏解的求解问题,文中提出两个改进的迭代重加权最小范数解算法(IRMNS)及一个光滑的0函数算法.其中,第一个算法基于 q(q∈(0,1])范数提出的,当q较小的时候,算法可以增强恢复稀疏解的能力;第二个算法是直接由0范数最小化问题提出的,它可以看做是第一个算法在q =0时的拓展;第三个算法是通过用一个光滑函数来近似0范数从而将原问题进行转化求解的.数值例子表明这三种算法都是快速有效的.     

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 block incomplete orthogonalization method for large nonsymmetric eigenproblems

The incomplete orthogonalization method (IOM) proposed by Saad for computing a few eigenpairs of large nonsymmetric matrices is generalized into a block incomplete orthogonalization method (BIOM). It is studied how the departure from symmetry A – A ^H affects the conditioning of the block basis vectors generated by BIOM, and some relationships are established between the approximate eigenpairs obtained by BIOM and Ritz pairs. It is proved that BIOM behaves much like generalized block Lanczos methods if the basis vectors of the block Krylov subspace generated by it are strongly linearly independent. However, it is shown that BIOM may generate a nearly linearly dependent basis for a general nonsymmetric matrix. Numerical experiments illustrate the convergence behavior of BIOM.This work was supported in part by the Graduiertenkolleg at the University of Bielefeld, Germany.     

A Solution of Inverse Eigenvalue Problems for Unitary Hessenberg Matrices

Let H∈Cn×n be an n×n unitary upper Hessenberg matrix whose subdiagonal elements are all positive. Partition H as H=[H11 H12 H21 H22],(0.1) where H11 is its k×k leading principal submatrix; H22 is the complementary matrix of H11. In this paper, H is constructed uniquely when its eigenvalues and the eigenvalues of (H|^)11 and (H|^)22 are known. Here (H|^)11 and (H|^)22 are rank-one modifications of H11 and H22 respectively.     

Hessenberg matrix for sums of Hermitian positive definite matrices and weighted shifts

In this work, we introduce an algebraic operation between bounded Hessenberg matrices and we analyze some of its properties. We call this operation m-sum and we obtain an expression for it that involves the Cholesky factorization of the corresponding Hermitian positive definite matrices associated with the Hessenberg components.This work extends a method to obtain the Hessenberg matrix of the sum of measures from the Hessenberg matrices of the individual measures, introduced recently by the authors for subnormal matrices, to matrices which are not necessarily subnormal.Moreover, we give some examples and we obtain the explicit formula for the m-sum of a weighted shift. In particular, we construct an interesting example: a subnormal Hessenberg matrix obtained as the m-sum of two not subnormal Hessenberg matrices.     

The global Hessenberg and CMRH methods for linear systems with multiple right-hand sides

In this paper, we introduce two new methods for solving large sparse nonsymmetric linear systems with several right-hand sides. These methods are the global Hessenberg and global CMRH methods. Using the global Hessenberg process, these methods are less expensive than the global FOM and global GMRES methods [9]. Theoretical results about the new methods are given, and experimental results that show good performances of these new methods are presented.     

A dichotomy result about Hessenberg matrices associated with measures in the unit circle

We characterize Hessenberg matrices D associated with measures in the unit circle ν, which are matrix representations of compact and actually Hilbert Schmidt perturbations of the forward shift operator as those with recursion coefficients verifying , ie, associated with measures verifying Szegö condition. As a consequence, we obtain the following dichotomy result for Hessenberg matrices associated with measures in the unit circle: either D = S _R+ K ₂ with K ₂, a Hilbert Schmidt matrix, or there exists an unitary matrix U and a diagonal matrix Λ such that with K ₂, a Hilbert Schmidt matrix. Moreover, we prove that for 1 ≤ p ≤ 2, if , then D = S _R+ K _p with K _p an absolutely p summable matrix inducing an operator in the p Schatten class. Some applications are given to classify measures on the unit circle.     

Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays

A symmetrizer of a nonsymmetric matrix A is the symmetric matrixX that satisfies the equationXA =A ^tX, wheret indicates the transpose. A symmetrizer is useful in converting a nonsymmetric eigenvalue problem into a symmetric one which is relatively easy to solve and finds applications in stability problems in control theory and in the study of general matrices. Three designs based on VLSI parallel processor arrays are presented to compute a symmetrizer of a lower Hessenberg matrix. Their scope is discussed. The first one is the Leiserson systolic design while the remaining two, viz., the double pipe design and the fitted diagonal design are the derived versions of the first design with improved performance.     

On sparse countably infinite Steiner triple systems

We give a general construction for Steiner triple systems on a countably infinite point set and show that it yields 2 ${}^{\text{?}{}_{\text{0}}}$ nonisomorphic systems all of which are uniform and r‐sparse for all finite r?4. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 115–122, 2010     

Updating the QR decomposition of block tridiagonal and block Hessenberg matrices

We present an efficient block-wise update scheme for the QR decomposition of block tridiagonal and block Hessenberg matrices. For example, such matrices come up in generalizations of the Krylov space solvers MinRes, SymmLQ, GMRes, and QMR to block methods for linear systems of equations with multiple right-hand sides. In the non-block case it is very efficient (and, in fact, standard) to use Givens rotations for these QR decompositions. Normally, the same approach is also used with column-wise updates in the block case. However, we show that, even for small block sizes, block-wise updates using (in general, complex) Householder reflections instead of Givens rotations are far more efficient in this case, in particular if the unitary transformations that incorporate the reflections determined by a whole block are computed explicitly. Naturally, the bigger the block size the bigger the savings. We discuss the somewhat complicated algorithmic details of this block-wise update, and present numerical experiments on accuracy and timing for the various options (Givens vs. Householder, block-wise vs. column-wise update, explicit vs. implicit computation of unitary transformations). Our treatment allows variable block sizes and can be adapted to block Hessenberg matrices that do not have the special structure encountered in the above mentioned block Krylov space solvers.