Hessenberg Form Research Articles

On the FOM Algorithm for the Resolution of the Linear Systems Ax = b

In this paper, we propose another version of the full orthogonalization method (FOM) for the resolution of linear system Ax = b, based on an extended definition of Sturm sequence in the calculation of the determinant of an upper hessenberg matrix in o(n2). We will also give a new version of Givens method based on using a tensor product and matrix addition. This version can be used in parallel calculation.

A generalized unitary Hessenberg matrix

This paper describes the use of a generalized isometric Arnoldi algorithm to reduce a unitary matrix, via unitary similarity, to a product of elementary reflectors and permutations. The computation is analogous to the reduction of a unitary matrix to a unitary Hessenberg matrix using the isometric Arnoldi algorithm. In the case in which A is a shift matrix, the reduction provides a novel recurrence for the factor R in the QR factorization of a Toeplitz-like matrix.

Fast computation of eigenvalues of companion, comrade, and related matrices

The class of eigenvalue problems for upper Hessenberg matrices of banded-plus-spike form includes companion and comrade matrices as special cases. For this class of matrices a factored form is developed in which the matrix is represented as a product of essentially 2×2 matrices and a banded upper-triangular matrix. A non-unitary analogue of Francis’s implicitly-shifted QR algorithm that preserves the factored form and consequently computes the eigenvalues in O(n 2) time and O(n) space is developed. Inexpensive a posteriori tests for stability and accuracy are performed as part of the algorithm. The results of numerical experiments are mixed but promising in certain areas. The single-shift version of the code applied to companion matrices is much faster than the nearest competitor.

Ratio Asymptotics, Hessenberg Matrices, and Weak Asymptotic Measures

We discuss the relationship between ratio asymptotics for general orthogonal polynomials and the asymptotics of the associated Bergman shift operator. More specifically, we consider the case in which a measure is supported on an infinite compact subset of the complex plane. We show that there is a straightforward connection between the corresponding orthonormal polynomials exhibiting ratio asymptotics and the corresponding Bergman shift operator being asymptotically Toeplitz. We also discuss a connection to the weak asymptotics of the measures derived from the orthonormal polynomials.

On the Number of 1-factors of Bipartite Graphs

Abstract: In this paper, we investigated relationships between the Fibonacci, Lucas, Padovan numbers and 1-factors of some bipartite graphs with upper Hessenberg adjacency matrix. We calculated permanent of these upper Hessenberg matrices by contraction method and show that their permanents are equal to elements of the Fibonacci, Lucas and Padovan numbers. At the end of the paper, we give some Maple 13 procedure in order to calculate numbers of 1-factors of above-mentioned bipartite graphs.

Necessary and sufficient conditions for GMRES complete and partial stagnation

In this paper we give necessary and sufficient conditions for the complete or partial stagnation of the GMRES iterative method for solving real linear systems. Our results rely on a paper by Arioli, Pták and Strakoš (1998), characterizing the matrices having a prescribed convergence curve for the residual norms. We show that we have complete stagnation if and only if the matrix A is orthonormally similar to an upper or lower Hessenberg matrix having a particular first row or column or a particular last row or column. Partial stagnation is characterized by a particular pattern of the matrix Q in the QR factorization of the upper Hessenberg matrix generated by the Arnoldi process.

A new method to compute the terms of generalized order- k Fibonacci numbers

In this paper, we give a new method to determine the terms of generalized order- k Fibonacci numbers and Fibonacci type sequences by using the inverse of various Hessenberg and triangular matrices. In addition, instead of obtaining the n -th term only, we are able to determine successive ( n + 1 ) terms of these sequences with this method simultaneously.

Eigenvalue computations based on IDR

The induced dimension reduction (IDR) method, which has been introduced as a transpose-free Krylov space method for solving nonsymmetric linear systems, can also be used to determine approximate eigenvalues of a matrix or operator. The IDR residual polynomials are the products of a residual polynomial constructed by successively appending linear smoothing factors and the residual polynomials of a two-sided (block) Lanczos process with one right-hand side and several left-hand sides. The Hessenberg matrix of the OrthoRes version of this Lanczos process is explicitly obtained in terms of the scalars defining IDR by deflating the smoothing factors. The eigenvalues of this Hessenberg matrix are approximations of eigenvalues of the given matrix or operator.

Efficient generalized Hessenberg form and applications

This article proposes an efficient algorithm for reducing matrices to generalized Hessenberg form by unitary similarity, and recommends using it as a preprocessor in a variety of applications. To illustrate its usefulness, two cases from control theory are analyzed in detail: a solution procedure for a sequence of shifted linear systems with multiple right hand sides (e.g. evaluating the transfer function of a MIMO LTI dynamical system at many points) and computation of the staircase form. The proposed algorithm for the generalized Hessenberg reduction uses two levels of aggregation of Householder reflectors, thus allowing efficient BLAS 3-based computation. Another level of aggregation is introduced when solving many shifted systems by processing the shifts in batches. Numerical experiments confirm that the proposed methods have superior efficiency.

Soft error resilient QR factorization for hybrid system with GPGPU

The general purpose graphics processing units (GPGPUs) are increasingly deployed for scientific computing due to their performance advantages over CPUs. What followed is the fact that fault tolerance has become a more serious concern compared to the period when GPGPUs were used exclusively for graphics applications. Using GPUs and CPUs together in a hybrid computing system increases flexibility and performance but also increases the possibility of the computations being affected by soft errors, for example, in the form of bit flips. In this work, we propose a soft error resilient algorithm for QR factorization on such hybrid systems. Our contributions include: (1) a checkpointing and recovery mechanism for the left-factor Q whose performance is scalable on hybrid systems; (2) optimized Givens rotation utilities on GPGPUs to efficiently reduce an upper Hessenberg matrix to an upper triangular form for the protection of the right factor R; and (3) a recovery algorithm based on QR update on GPGPUs. Experimental results show that our fault tolerant QR factorization can successfully detect and recover from soft errors in the entire matrix with little overhead on hybrid systems with GPGPUs.

High-Order Three-Term Recursions, Riemann–Hilbert Minors and Nikishin Systems on Star-Like Sets

We study monic polynomials Q n (x) generated by a high-order three-term recursion xQ n (x)=Q n+1(x)+a n−p Q n−p (x) with arbitrary p≥1 and a n >0 for all n. The recursion is encoded by a two-diagonal Hessenberg operator H. One of our main results is that, for periodic coefficients a n and under certain conditions, the Q n are multiple orthogonal polynomials with respect to a Nikishin system of orthogonality measures supported on star-like sets in the complex plane. This improves a recent result of Aptekarev–Kalyagin–Saff, where a formal connection with Nikishin systems was obtained in the case when $\sum_{n=0}^{\infty}|a_{n}-a|<\infty$ for some a>0. An important tool in this paper is the study of ‘Riemann–Hilbert minors’, or equivalently, the ‘generalized eigenvalues’ of the Hessenberg matrix H. We prove interlacing relations for the generalized eigenvalues by using totally positive matrices. In the case of asymptotically periodic coefficients a n , we find weak and ratio asymptotics for the Riemann–Hilbert minors and we obtain a connection with a vector equilibrium problem. We anticipate that in the future, the study of Riemann–Hilbert minors may prove useful for more general classes of multiple orthogonal polynomials.

Rank-1 preservers on Hessenberg matrices

Let be the space of all upper Hessenberg matrices over a field where is a positive integer greater than . In this paper, linear maps preserving rank- on are characterized, i.e. is a linear rank- preserver on if and only if (i) is an -dimensional rank- subspace or (ii) there exist nonsingular upper Hessenberg matrices and such that for all or for all where with .

The Hessenberg matrix and the Riemann mapping function

We consider a Jordan arc Γ in the complex plane \({\mathbb C}\) and a regular measure μ whose support is Γ. We denote by D the upper Hessenberg matrix of the multiplication by z operator with respect to the orthonormal polynomial basis associated with μ. We show in this work that, if the Hessenberg matrix D is uniformly asymptotically Toeplitz, then the symbol of the limit operator is the restriction to the unit circle of the Riemann mapping function ϕ(z) which maps conformally the exterior of the unit disk onto the exterior of the support of the measure μ. We use this result to show how to approximate the Riemann mapping function for the support of μ from the entries of the Hessenberg matrix D.

Eigenvalue Computations Based on IDR

The induced dimension reduction (IDR) method, which has been introduced as a transpose-free Krylov space method for solving nonsymmetric linear systems, can also be used to determine approximate eigenvalues of a matrix or operator. The IDR residual polynomials are the products of a residual polynomial constructed by successively appending linear smoothing factors and the residual polynomials of a two-sided (block) Lanczos process with one right-hand side and several left-hand sides. The Hessenberg matrix of the OrthoRes version of this Lanczos process is explicitly obtained in terms of the scalars defining IDR by deflating the smoothing factors. The eigenvalues of this Hessenberg matrix are approximations of eigenvalues of the given matrix or operator.

On new algorithms for inverting Hessenberg matrices

A modification of the Ikebe algorithm for computing the lower half of the inverse of an (unreduced) upper Hessenberg matrix, extended to compute the entries of the superdiagonal, is considered in this paper. It enables us to compute the inverse of a quasiseparable Hessenberg matrix in O(n2) times. A new factorization expressing the inverse of a nonsingular Hessenberg matrix as a product of two suitable matrices is obtained. Because this allows us the use of back substitution for the inversion of triangular matrices, the inverse is computed with complexity O(n3). Some comparisons with results obtained using other recent inversion algorithms are also provided.

Families of Algorithms for Reducing a Matrix to Condensed Form

In a recent paper it was shown how memory traffic can be diminished by reformulating the classic algorithm for reducing a matrix to bidiagonal form, a preprocess when computing the singular values of a dense matrix. The key is a reordering of the computation so that the most memory-intensive operations can be “fused.” In this article, we show that other operations that reduce matrices to condensed form (reduction to upper Hessenberg form and reduction to tridiagonal form) can be similarly reorganized, yielding different sets of operations that can be fused. By developing the algorithms with a common framework and notation, we facilitate the comparing and contrasting of the different algorithms and opportunities for optimization on sequential architectures. We discuss the algorithms, develop a simple model to estimate the speedup potential from fusing, and showcase performance improvements consistent with the what the model predicts.

A probabilistic evolution approach trilogy, part 1: quantum expectation value evolutions, block triangularity and conicality, truncation approximants and their convergence

This is the first one of three companion papers focusing on the “probabilistic evolution approach (PEA)” which has been developed for the solution of the explicit ODE involving problems under certain consistent impositions. The main purpose here is the determination of the expectation value of a given operator in quantum mechanics by solving only ODEs, not directly using the wave function. To this end we first define a basis operator set over the Kronecker powers of an appropriately defined “system operator vector”. We assume that the target operator’s commutator with the system’s Hamiltonian can be expressed in terms of the above-mentioned basis operators. This assumption leads us to an infinite set of linear homogeneous ODEs over the expectation values of the basis operators. Its coefficient matrix is in block Hessenberg form when the target operator has no singularity, and beyond that, it may become block triangular when certain conditions over the system’s potential function are satisfied. The initial conditions are the basic determining agents giving the probabilistic nature to the solutions of the obtained infinite set of ODEs. They may or may not have fluctuations depending on the nature of the probability density. All these issues are investigated in a phenomenological and constructive theoretical manner in this paper. The remaining two papers are devoted to further details of PEA in quantum mechanics, and, the application of PEA to systems defined by Liouville equation.

Numerical evaluation of the stability of stationary points of index-2 differential-algebraic equations: Applications to reactive flash and reactive distillation systems

The dynamic behavior of many chemical processes can be represented by an index-2 system of differential-algebraic equations. This index can be reduced by differentiation, but unfortunately the index reduced systems are not guaranteed to possess the same stability characteristics as that of the original system. When the set of differential-algebraic equations can be written in Hessenberg form, the matrix pencil of the linearized system can be used to directly evaluate the stability of a steady state without the need for index reduction. Direct evaluations of stability of reactive flash and reactive distillation are presented. It is also shown that a commonly used index reduction will always result in null eigenvalues at steady state. Stabilization methods were successfully applied to this reduced system. An alternative index reduction method for a reactive flash is generalized and shown to be highly sensitive to minor changes in the jacobian.

Asymptotics for Hessenberg Matrices for the Bergman Shift Operator on Jordan Regions

Let \(G\) be a bounded Jordan domain in the complex plane. The Bergman polynomials \(\{p_n\}_{n=0}^\infty \) of \(G\) are the orthonormal polynomials with respect to the area measure over \(G\). They are uniquely defined by the entries of an infinite upper Hessenberg matrix \(M\). This matrix represents the Bergman shift operator of \(G\). The main purpose of the paper is to describe and analyze a close relation between \(M\) and the Toeplitz matrix with symbol the normalized conformal map of the exterior of the unit circle onto the complement of \(\overline{G}\). Our results are based on the strong asymptotics of \(p_n\). As an application, we describe and analyze an algorithm for recovering the shape of \(G\) from its area moments.

Equilibrium problem for the eigenvalues of banded block Toeplitz matrices

AbstractWe consider banded block Toeplitz matrices Tn with n block rows and columns. We show that under certain technical assumptions, the normalized eigenvalue counting measure of Tn for n → ∞ weakly converges to one component of the unique vector of measures that minimizes a certain energy functional. In this way we generalize a recent result of Duits and Kuijlaars for the scalar case. Along the way we also obtain an equilibrium problem associated to an arbitrary algebraic curve, not necessarily related to a block Toeplitz matrix.For banded block Toeplitz matrices, there are several new phenomena that do not occur in the scalar case: (i) The total masses of the equilibrium measures do not necessarily form a simple arithmetic series but in general are obtained through a combinatorial rule; (ii) The limiting eigenvalue distribution may contain point masses, and there may be attracting point sources in the equilibrium problem; (iii) More seriously, there are examples where the connection between the limiting eigenvalue distribution of Tn and the solution to the equilibrium problem breaks down. We provide sufficient conditions guaranteeing that no such breakdown occurs; in particular we show this if Tn is a Hessenberg matrix.