The Experts below are selected from a list of 309 Experts worldwide ranked by ideXlab platform
Emilio Torrano - One of the best experts on this subject based on the ideXlab platform.
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The Hessenberg Matrix and the Riemann mapping function
Advances in Computational Mathematics, 2013Co-Authors: Carmen Escribano, Antonio Giraldo, M. Asunción Sastre, Emilio TorranoAbstract: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 ?(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.
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two applications of the subnormality of the Hessenberg Matrix related to general orthogonal polynomials
Linear Algebra and its Applications, 2011Co-Authors: V. Tomeo, Emilio TorranoAbstract:Abstract In this paper we prove two consequences of the subnormal character of the Hessenberg Matrix D when the hermitian Matrix M of an inner product is a moment Matrix. If this inner product is defined by a measure supported on an algebraic curve in the complex plane, then D satisfies the equation of the curve in a noncommutative sense. We also prove an extension of the Krein theorem for discrete measures on the complex plane based on properties of subnormal operators.
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The Hessenberg Matrix and the Riemann mapping
arXiv: Spectral Theory, 2011Co-Authors: Carmen Escribano, Antonio Giraldo, M. Asunción Sastre, Emilio TorranoAbstract:We consider a Jordan arc \Gamma in the complex plane \mathbb{C} and a regular measure \mu whose support is \Gamma . We denote by D the upper Hessenberg Matrix of the multiplication by z operator with respect to the orthonormal polynomial basis associated with \mu . 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 \phi(z) which maps conformally the exterior of the unit disk onto the exterior of the support of the measure \mu . We use this result to show how to approximate the Riemann mapping function for the support of \mu from the entries of the Hessenberg Matrix D.
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Hessenberg Matrix for sums of Hermitian positive definite matrices and weighted shifts
Journal of Computational and Applied Mathematics, 2011Co-Authors: Carmen Escribano, Antonio Giraldo, María Asunción Sastre, Emilio TorranoAbstract:Abstract 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.
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Computing the Hessenberg Matrix associated with a self-similar measure
Journal of Approximation Theory, 2011Co-Authors: Carmen Escribano, Antonio Giraldo, María Asunción Sastre, Emilio TorranoAbstract:We introduce in this paper a method to calculate the Hessenberg Matrix of a sum of measures from the Hessenberg matrices of the component measures. Our method extends the spectral techniques used by G. Mantica to calculate the Jacobi Matrix associated with a sum of measures from the Jacobi matrices of each of the measures. We apply this method to approximate the Hessenberg Matrix associated with a self-similar measure and compare it with the result obtained by a former method for self-similar measures which uses a fixed point theorem for moment matrices. Results are given for a series of classical examples of self-similar measures. Finally, we also apply the method introduced in this paper to some examples of sums of (not self-similar) measures obtaining the exact value of the sections of the Hessenberg Matrix.
William B. Gragg - One of the best experts on this subject based on the ideXlab platform.
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Convergence of the unitary QR algorithm with a unimodular Wilkinson shift
Mathematics of Computation, 2002Co-Authors: Tai-lin Wang, William B. GraggAbstract:In applying the QR algorithm to compute the eigenvalues of a unitary Hessenberg Matrix, a projected Wilkinson shift of unit modulus is proposed and proved to give global convergence with (at least) a quadratic asymptotic rate for the QR iteration. Experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence.
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Constructing a Unitary Hessenberg Matrix from Spectral Data
Numerical Linear Algebra Digital Signal Processing and Parallel Algorithms, 1991Co-Authors: Gregory S. Ammar, William B. Gragg, L. ReichelAbstract:We consider the numerical construction of a unitary Hessenberg Matrix from spectral data using an inverse QR algorithm. Any upper unitary Hessenberg Matrix H with nonnegative subdiagonal elements can be represented by 2n − 1 real parameters. This representation, which we refer to as the Schur parametrization of H, facilitates the development of efficient algorithms for this class of matrices. We show that a unitary upper Hessenberg Matrix H with positive subdiagonal elements is determined by its eigenvalues and the eigenvalues of a rank-one unitary perturbation of H. The eigenvalues of the perturbation strictly interlace the eigenvalues of H on the unit circle.
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Inverse problems for orthogonal matrices, Toda flows, and signal processing
[1992] Proceedings of the 31st IEEE Conference on Decision and Control, 1Co-Authors: L. Faybusovich, Gregory S. Ammar, William B. GraggAbstract:The authors consider Toda flows induced on the set of orthogonal upper Hessenberg matrices. The explicit formulas for the evolution of Schur parameters are given. Since Schur parameters determine orthogonal Hessenberg matrices uniquely, an explicit description is obtained of the evolution of a given orthogonal Hessenberg Matrix under the Toda flow. >
Raf Vandebril - One of the best experts on this subject based on the ideXlab platform.
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Short recurrences for computing extended Krylov bases for Hermitian and unitary matrices
Numerische Mathematik, 2015Co-Authors: Clara Mertens, Raf VandebrilAbstract:It is well known that the projection of a Matrix $$A$$ A onto a Krylov subspace span $$\left\{ \mathbf {h}, A\mathbf {h}, A^2\mathbf {h}, \ldots , A^{k-1}\mathbf {h}\right\} $$ h , A h , A 2 h , … , A k - 1 h , with $$A \in \mathbb {C}^{n \times n}$$ A ∈ C n × n and $$\mathbf {h} \in \mathbb {C}^n$$ h ∈ C n , results in a Hessenberg Matrix. We show that the projection of the Matrix $$A$$ A onto an extended Krylov subspace, which is of the form span $$\left\{ A^{-k_r}\mathbf {h}, \ldots , A^{-2}\mathbf {h},A^{-1}\mathbf {h}, \mathbf {h}, A\mathbf {h}, A^2 \mathbf {h}, \ldots , A^{k_\ell } \mathbf {h} \right\} $$ A - k r h , … , A - 2 h , A - 1 h , h , A h , A 2 h , … , A k ℓ h , is a Matrix of so-called extended Hessenberg form which can be characterized uniquely by its $$QR$$ Q R -factorization. This $$QR$$ Q R -factorization will be presented by means of a pattern of $$2 \times 2$$ 2 × 2 unitary rotations. We will show how this rotation pattern leads to new insights and allows to elegantly predict the structure of the Matrix. In case $$A$$ A is Hermitian or unitary, this extended Hessenberg Matrix is banded and structured, allowing the design of short recurrence relations. For the unitary case, coupled two term recurrence relations are derived of which the coefficients capture all information necessary for a sparse factorization of the corresponding extended Hessenberg Matrix.
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A numerical example showing that deflations based on rotation leads to higher relative accuracy in QR algorithm
PAMM, 2014Co-Authors: Thomas Mach, Raf VandebrilAbstract:We present a numerical example illustrating that the deflation procedure in Francis's implicitly shifted QR algorithm can be improved by a deflation criteria based on the QR decomposition of the upper Hessenberg Matrix. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
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inverse eigenvalue problems for extended Hessenberg and extended tridiagonal matrices
Journal of Computational and Applied Mathematics, 2014Co-Authors: Thomas Mach, Marc Van Barel, Raf VandebrilAbstract:In inverse eigenvalue problems one tries to reconstruct a Matrix, satisfying some constraints, given some spectral information. Here, two inverse eigenvalue problems are solved. First, given the eigenvalues and the first components of the associated eigenvectors (called the weight vector) an extended Hessenberg Matrix with prescribed poles is computed possessing these eigenvalues and satisfying the eigenvector constraints. The extended Hessenberg Matrix is retrieved by executing particularly designed unitary similarity transformations on the diagonal Matrix containing the eigenvalues. This inverse problem closely links to orthogonal rational functions: the extended Hessenberg Matrix contains the recurrence coefficients given the nodes (eigenvalues), poles (poles of the extended Hessenberg Matrix), and a weight vector (first eigenvector components) determining the discrete inner product. Moreover, it is also sort of the inverse of the (rational) Arnoldi algorithm: instead of using the (rational) Arnoldi method to compute a Krylov basis to approximate the spectrum, we will reconstruct the orthogonal Krylov basis given the spectral info. In the second inverse eigenvalue problem, we do the same, but refrain from unitarity. As a result we execute possibly non-unitary similarity transformations on the diagonal Matrix of eigenvalues to retrieve a (non)-symmetric extended tridiagonal Matrix. The algorithm will be less stable, but it will be faster, as the extended tridiagonal Matrix admits a low cost factorization of O(n) (n equals the number of eigenvalues), whereas the extended Hessenberg Matrix does not. Again there is a close link with orthogonal function theory, the extended tridiagonal Matrix captures the recurrence coefficients of bi-orthogonal rational functions. Moreover, it is again sort of inverse of the nonsymmetric Lanczos algorithm: given spectral properties, we reconstruct the two basis Krylov matrices linked to the nonsymmetric Lanczos algorithm.
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MULTIPLE RECURRENCES AND THE ASSOCIATED Matrix STRUCTURES STEMMING FROM NORMAL MATRICES
SIAM Journal on Numerical Analysis, 2014Co-Authors: Clara Mertens, Raf VandebrilAbstract:There are many classical results in which orthogonal vectors stemming from Krylov subspaces are linked to short recurrence relations, e.g., three term recurrences for Hermitian and short rational recurrences for unitary matrices. These recurrence coefficients can be captured in a Hessenberg Matrix, whose structure reflects the relation between the spectrum of the original Matrix and the recurrences. The easier the recurrences, the faster the orthogonal vectors can be computed possibly resulting in computational savings in the design of, e.g., iterative solvers. In this article we focus on multiple recurrence relations, i.e., the $(j+1)$st orthogonal vector satisfies $\mathbf{q}_{j+1} = \sum_{i=j-m}^j \rho_{j,i}A\mathbf{q}_i - \sum_{i=j-\ell}^j \gamma_{j,i}\mathbf{q}_i$ with $\rho_{j,i}$, $\gamma_{j,i}$ scalars and $A$ the Matrix defining the Krylov space. Though many compelling results are around, the structure of the corresponding Hessenberg Matrix is mostly deduced by analyzing the inner product relatio...
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An Implicit Multishift $QR$-Algorithm for Hermitian Plus Low Rank Matrices
SIAM Journal on Scientific Computing, 2010Co-Authors: Raf Vandebril, Gianna M. Del CorsoAbstract:Hermitian plus possibly non-Hermitian low rank matrices can be efficiently reduced into Hessenberg form. The resulting Hessenberg Matrix can still be written as the sum of a Hermitian plus low rank Matrix. In this paper we develop a new implicit multishift $QR$-algorithm for Hessenberg matrices, which are the sum of a Hermitian plus a possibly non-Hermitian low rank correction. The proposed algorithm exploits both the symmetry and low rank structure to obtain a $QR$-step involving only $\mathcal{O}(n)$ floating point operations instead of the standard $\mathcal{O}(n^2)$ operations needed for performing a $QR$-step on a Hessenberg Matrix. The algorithm is based on a suitable $\mathcal{O}(n)$ representation of the Hessenberg Matrix. The low rank parts present in both the Hermitian and low rank part of the sum are compactly stored by a sequence of Givens transformations and a few vectors. Due to the new representation, we cannot apply classical deflation techniques for Hessenberg matrices. A new, efficient technique is developed to overcome this problem. Some numerical experiments based on matrices arising in applications are performed. The experiments illustrate effectiveness and accuracy of both the $QR$-algorithm and the newly developed deflation technique.
Carmen Escribano - One of the best experts on this subject based on the ideXlab platform.
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The Hessenberg Matrix and the Riemann mapping function
Advances in Computational Mathematics, 2013Co-Authors: Carmen Escribano, Antonio Giraldo, M. Asunción Sastre, Emilio TorranoAbstract: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 ?(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.
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The Hessenberg Matrix and the Riemann mapping
arXiv: Spectral Theory, 2011Co-Authors: Carmen Escribano, Antonio Giraldo, M. Asunción Sastre, Emilio TorranoAbstract:We consider a Jordan arc \Gamma in the complex plane \mathbb{C} and a regular measure \mu whose support is \Gamma . We denote by D the upper Hessenberg Matrix of the multiplication by z operator with respect to the orthonormal polynomial basis associated with \mu . 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 \phi(z) which maps conformally the exterior of the unit disk onto the exterior of the support of the measure \mu . We use this result to show how to approximate the Riemann mapping function for the support of \mu from the entries of the Hessenberg Matrix D.
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Hessenberg Matrix for sums of Hermitian positive definite matrices and weighted shifts
Journal of Computational and Applied Mathematics, 2011Co-Authors: Carmen Escribano, Antonio Giraldo, María Asunción Sastre, Emilio TorranoAbstract:Abstract 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.
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Computing the Hessenberg Matrix associated with a self-similar measure
Journal of Approximation Theory, 2011Co-Authors: Carmen Escribano, Antonio Giraldo, María Asunción Sastre, Emilio TorranoAbstract:We introduce in this paper a method to calculate the Hessenberg Matrix of a sum of measures from the Hessenberg matrices of the component measures. Our method extends the spectral techniques used by G. Mantica to calculate the Jacobi Matrix associated with a sum of measures from the Jacobi matrices of each of the measures. We apply this method to approximate the Hessenberg Matrix associated with a self-similar measure and compare it with the result obtained by a former method for self-similar measures which uses a fixed point theorem for moment matrices. Results are given for a series of classical examples of self-similar measures. Finally, we also apply the method introduced in this paper to some examples of sums of (not self-similar) measures obtaining the exact value of the sections of the Hessenberg Matrix.
Mongi Benhamadou - One of the best experts on this subject based on the ideXlab platform.
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On the FOM Algorithm for the Resolution of the Linear Systems Ax = b
Advances in Linear Algebra & Matrix Theory, 2014Co-Authors: Mongi BenhamadouAbstract: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.
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On the calculation of the multiplicity of a real eigenvalue of Hessenberg Matrix
Advances in Engineering Software, 1999Co-Authors: Mongi BenhamadouAbstract:Abstract In this article, we describe three algorithms: the first one is used to compute the multiplicity of a real eigenvalue of a Hessenberg Matrix A ∈ R n × n in o ( n 3 ); the second for the computation of the determinant of a Hessenberg Matrix in o ( n 2 ) and the third for the computation of the multiplicity of a real root of a polynomial in o ( n 2 ). It is a new matricial form and represents generalized Horner scheme.
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On the inverse's computation of a real, upper Hessenberg Matrix
Advances in Engineering Software, 1993Co-Authors: Mongi BenhamadouAbstract:Abstract If A ϵ R n×n is an upper Hessenberg Matrix, regular and irreducible, we propose two methods to calculate the inverse of A. The first is based on QR factorisation, the second on Y. Ikebe's method. We compare the numerical results using a microcomputer.