In this note, we described two strategies for converting a Chebyshev polynomial to an ordinary polynomial with series of powers form. For this work we used of recursive sequences and …
In this note, we described two strategies for converting a Chebyshev polynomial to an ordinary polynomial with series of powers form. For this work we used of recursive sequences and matrix form for coefficients. These strategies help us for finding zeros of some functions. Numerical examples are given to illustrate the effectiveness of different processes.
It is well known that the matrix exponential function has practical applications in engineering and applied sciences. In this paper, we present some new explicit identities to the exponential functions …
It is well known that the matrix exponential function has practical applications in engineering and applied sciences. In this paper, we present some new explicit identities to the exponential functions of a special class of matrices that are known as central-symmetric $X$-form. For instance, $e^{mathbf{A}t}$, $t^{mathbf{A}}$ and $a^{mathbf{A}t}$ will be evaluated by the new formulas in this particular structure. Moreover, upper bounds for the explicit relations will be given via subordinate matrix norms. Eventually, some numerical illustrations and applications are also adapted.
We try to factor symmetric Toeplitz matrix A by A = LPLT for finding inertia. We know A and P have the same inertia, namely they have eigenvalues of the …
We try to factor symmetric Toeplitz matrix A by A = LPLT for finding inertia. We know A and P have the same inertia, namely they have eigenvalues of the same sign. Orthogonal or invertible matrix gives us special properties of this factorization. Thus this process gives us a partible way for finding sign of eigenvalues for symmetric Toeplitz matrices.
We investigate the eigenvalue distribution of banded Hankel matrices with non-zero skew diagonals. This work uses push-forward of an arcsine density, block structures and generating functions. Our analysis is done …
We investigate the eigenvalue distribution of banded Hankel matrices with non-zero skew diagonals. This work uses push-forward of an arcsine density, block structures and generating functions. Our analysis is done by a combination of Chebyshev polynomials, Laplacian determinant expansion and mathematical induction.
Abstract Let L be the infinite lower triangular Toeplitz matrix with first column ( µ , a 1 , a 2 , ..., a p , a 1 , ..., …
Abstract Let L be the infinite lower triangular Toeplitz matrix with first column ( µ , a 1 , a 2 , ..., a p , a 1 , ..., a p , ...) T and let D be the infinite diagonal matrix whose entries are 1, 2, 3, . . . Let A := L + D be the sum of these two matrices. Bünger and Rump have shown that if p = 2 and certain linear inequalities between the parameters µ , a 1 , a 2 , are satisfied, then the singular values of any finite left upper square submatrix of A can be bounded from below by an expression depending only on those parameters, but not on the matrix size. By extending parts of their reasoning, we show that a similar behaviour should be expected for arbitrary p and a much larger range of values for µ , a 1 , ..., a p . It depends on the asymptotics in µ of the l 2 -norm of certain sequences defined by linear recurrences, in which these parameters enter. We also consider the relevance of the results in a numerical analysis setting and moreover a few selected numerical experiments are presented in order to show that our bounds are accurate in practical computations.
Abstract In this note, we consider real nonsymmetric tridiagonal 2-Toeplitz matrices $\bold{B}_n$. First we give the asymptotic spectral and singular value distribution of the whole matrix-sequence $\{\bold{B}_n\}_n$, which is described …
Abstract In this note, we consider real nonsymmetric tridiagonal 2-Toeplitz matrices $\bold{B}_n$. First we give the asymptotic spectral and singular value distribution of the whole matrix-sequence $\{\bold{B}_n\}_n$, which is described via two eigenvalue functions of a $2\times 2$ matrix-valued symbol. In connection with the above findings, we provide a characterization of the eigenvalues and eigenvectors of real tridiagonal 2-Toeplitz matrices $\bold{B}_n$ of even order, thatcan be turned into a numerical effective scheme for the computation of all the entries of $\bold{B}_n^l$, $n$ even and $l$ positive and small compared to $n$. We recall that a corresponding eigenvalue decomposition for odd order tridiagonal 2-Toeplitz matrices was found previously, while, for even orders, an implicit formula for all the eigenvalues is obtained. AMS SC 15B05; 15A18; 65F15.
Abstract In this note, we consider real nonsymmetric tridiagonal 2-Toeplitz matrices $\bold{B}_n$. First we give the asymptotic spectral and singular value distribution of the whole matrix-sequence $\{\bold{B}_n\}_n$, which is described …
Abstract In this note, we consider real nonsymmetric tridiagonal 2-Toeplitz matrices $\bold{B}_n$. First we give the asymptotic spectral and singular value distribution of the whole matrix-sequence $\{\bold{B}_n\}_n$, which is described via two eigenvalue functions of a $2\times 2$ matrix-valued symbol. In connection with the above findings, we provide a characterization of the eigenvalues and eigenvectors of real tridiagonal 2-Toeplitz matrices $\bold{B}_n$ of even order, thatcan be turned into a numerical effective scheme for the computation of all the entries of $\bold{B}_n^l$, $n$ even and $l$ positive and small compared to $n$. We recall that a corresponding eigenvalue decomposition for odd order tridiagonal 2-Toeplitz matrices was found previously, while, for even orders, an implicit formula for all the eigenvalues is obtained. AMS SC 15B05; 15A18; 65F15.
We investigate the eigenvalue distribution of banded Hankel matrices with non-zero skew diagonals. This work uses push-forward of an arcsine density, block structures and generating functions. Our analysis is done …
We investigate the eigenvalue distribution of banded Hankel matrices with non-zero skew diagonals. This work uses push-forward of an arcsine density, block structures and generating functions. Our analysis is done by a combination of Chebyshev polynomials, Laplacian determinant expansion and mathematical induction.
Abstract Let L be the infinite lower triangular Toeplitz matrix with first column ( µ , a 1 , a 2 , ..., a p , a 1 , ..., …
Abstract Let L be the infinite lower triangular Toeplitz matrix with first column ( µ , a 1 , a 2 , ..., a p , a 1 , ..., a p , ...) T and let D be the infinite diagonal matrix whose entries are 1, 2, 3, . . . Let A := L + D be the sum of these two matrices. Bünger and Rump have shown that if p = 2 and certain linear inequalities between the parameters µ , a 1 , a 2 , are satisfied, then the singular values of any finite left upper square submatrix of A can be bounded from below by an expression depending only on those parameters, but not on the matrix size. By extending parts of their reasoning, we show that a similar behaviour should be expected for arbitrary p and a much larger range of values for µ , a 1 , ..., a p . It depends on the asymptotics in µ of the l 2 -norm of certain sequences defined by linear recurrences, in which these parameters enter. We also consider the relevance of the results in a numerical analysis setting and moreover a few selected numerical experiments are presented in order to show that our bounds are accurate in practical computations.
In this note, we described two strategies for converting a Chebyshev polynomial to an ordinary polynomial with series of powers form. For this work we used of recursive sequences and …
In this note, we described two strategies for converting a Chebyshev polynomial to an ordinary polynomial with series of powers form. For this work we used of recursive sequences and matrix form for coefficients. These strategies help us for finding zeros of some functions. Numerical examples are given to illustrate the effectiveness of different processes.
We try to factor symmetric Toeplitz matrix A by A = LPLT for finding inertia. We know A and P have the same inertia, namely they have eigenvalues of the …
We try to factor symmetric Toeplitz matrix A by A = LPLT for finding inertia. We know A and P have the same inertia, namely they have eigenvalues of the same sign. Orthogonal or invertible matrix gives us special properties of this factorization. Thus this process gives us a partible way for finding sign of eigenvalues for symmetric Toeplitz matrices.
It is well known that the matrix exponential function has practical applications in engineering and applied sciences. In this paper, we present some new explicit identities to the exponential functions …
It is well known that the matrix exponential function has practical applications in engineering and applied sciences. In this paper, we present some new explicit identities to the exponential functions of a special class of matrices that are known as central-symmetric $X$-form. For instance, $e^{mathbf{A}t}$, $t^{mathbf{A}}$ and $a^{mathbf{A}t}$ will be evaluated by the new formulas in this particular structure. Moreover, upper bounds for the explicit relations will be given via subordinate matrix norms. Eventually, some numerical illustrations and applications are also adapted.
In the past few years, Bogoya, Böttcher, Grudsky, and Maximenko obtained the precise asymptotic expansion for the eigenvalues of a Toeplitz matrix Tn(f), under suitable assumptions on the generating function …
In the past few years, Bogoya, Böttcher, Grudsky, and Maximenko obtained the precise asymptotic expansion for the eigenvalues of a Toeplitz matrix Tn(f), under suitable assumptions on the generating function f, as the matrix size n goes to infinity. On the basis of several numerical experiments, it was conjectured by Serra-Capizzano that a completely analogous expansion also holds for the eigenvalues of the preconditioned Toeplitz matrix Tn(u)− 1Tn(v), provided f = v/u is monotone and further conditions on u and v are satisfied. Based on this expansion, we here propose and analyze an interpolation–extrapolation algorithm for computing the eigenvalues of Tn(u)− 1Tn(v). The algorithm is suited for parallel implementation and it may be called "matrix-less" as it does not need to store the entries of the matrix. We illustrate the performance of the algorithm through numerical experiments and we also present its generalization to the case where f = v/u is non-monotone.
In this note, we described two strategies for converting a Chebyshev polynomial to an ordinary polynomial with series of powers form. For this work we used of recursive sequences and …
In this note, we described two strategies for converting a Chebyshev polynomial to an ordinary polynomial with series of powers form. For this work we used of recursive sequences and matrix form for coefficients. These strategies help us for finding zeros of some functions. Numerical examples are given to illustrate the effectiveness of different processes.
Summary It is known that for a tridiagonal Toeplitz matrix, having on the main diagonal the constant a 0 and on the two first off‐diagonals the constants a 1 (lower) …
Summary It is known that for a tridiagonal Toeplitz matrix, having on the main diagonal the constant a 0 and on the two first off‐diagonals the constants a 1 (lower) and a −1 (upper), which are all complex values, there exist closed form formulas, giving the eigenvalues of the matrix and a set of associated eigenvectors. For example, for the 1D discrete Laplacian, this triple is ( a 0 , a 1 , a −1 )=(2,−1,−1). In the first part of this article, we consider a tridiagonal Toeplitz matrix of the same form ( a 0 , a ω , a − ω ), but where the two off‐diagonals are positioned ω steps from the main diagonal instead of only one. We show that its eigenvalues and eigenvectors can also be identified in closed form and that interesting connections with the standard Toeplitz symbol are identified. Furthermore, as numerical evidences clearly suggest, it turns out that the eigenvalue behavior of a general banded symmetric Toeplitz matrix with real entries can be described qualitatively in terms of the symmetrically sparse tridiagonal case with real a 0 , a ω = a − ω , ω =2,3,…, and also quantitatively in terms of those having monotone symbols. A discussion on the use of such results and on possible extensions complements the paper.
In this paper we consider a family of tetradiagonal (= four non-zero diagonals) Toeplitz matrices with a limiting set consisting in one analytic arc only and obtain individual asymptotic expansions …
In this paper we consider a family of tetradiagonal (= four non-zero diagonals) Toeplitz matrices with a limiting set consisting in one analytic arc only and obtain individual asymptotic expansions for all the eigenvalues, as the matrix size goes to infinity. Additionally, we provide specific expansions for the extreme eigenvalues which are the eigenvalues approaching the extreme points of the limiting set. In contrast to previous related works, we study non-Hermitian Toeplitz matrices having non-canonical distribution and a real limiting set. The considered family does not belong to the so-called simple-loop class, nevertheless we manage to extend the theory to this case. The achieved formulas reveal the fine details of the eigenvalue structure and allow us to directly calculate high accuracy eigenvalues, even for matrices of relatively small size.
This paper is continuation of previous work by the present author, where explicitformulas for the eigenvalues associated with several tridiagonal matrices were given. In this paperthe associated eigenvectors are calculated …
This paper is continuation of previous work by the present author, where explicitformulas for the eigenvalues associated with several tridiagonal matrices were given. In this paperthe associated eigenvectors are calculated explicitly. As a consequence, a result obtained by WenChyuan Yueh and independently by S. Kouachi, concerning the eigenvalues and in particular thecorresponding eigenvectors of tridiagonal matrices, is generalized. Expressions for the eigenvectorsare obtained that differ completely from those obtained by Yueh. The techniques used herein arebased on theory of recurrent sequences. The entries situated on each of the secondary diagonals arenot necessary equal as was the case considered by Yueh.
Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = …
Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ 1 , . . ., λn} . If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A , and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ = {λ 1 , . . ., λn} of complex numbers in the left-half plane, that is, with Re λ i ≤ 0, i = 2, . . ., n , is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.
Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained for the eigenvalues of a Toeplitz matrix, under suitable assumptions on the generating function, the precise asymptotic expansion as the matrix size …
Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained for the eigenvalues of a Toeplitz matrix, under suitable assumptions on the generating function, the precise asymptotic expansion as the matrix size goes to infinity. In this article we provide numerical evidence that some of these assumptions can be relaxed. Moreover, based on the eigenvalue asymptotics, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric Toeplitz matrices with a high level of accuracy and a relatively low computational cost.
We establish the asymptotic zero distribution for polynomials generated by a four-term recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics …
We establish the asymptotic zero distribution for polynomials generated by a four-term recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these polynomials. We can apply this result to three examples of multiple orthogonal polynomials, in particular Jacobi-Piñeiro, Laguerre I and the example associated with modified Bessel functions. We also discuss an application to Toeplitz matrices.
An elementary and direct proof of the Szegö formula is given, for both eigen and singular values. This proof, which is based on tools from linear algebra and does not …
An elementary and direct proof of the Szegö formula is given, for both eigen and singular values. This proof, which is based on tools from linear algebra and does not rely on the theory of Fourier series, simultaneously embraces multilevel Toeplitz matrices, block Toeplitz matrices and combinations of them. The assumptions on the generating function f are as weak as possible; indeedf is a matrix-valued function of p variables, and it is only supposed to be integrable. In the case of singular values f(x), and hence the block p-level Toeplitz matrices it generates, are not even supposed to be square matrices. Moreover, in the asymptotic formulas for eigen and singular values the test functions involved are not required to have compact support.
A study is made of the numerical condition of the coordinate map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{M_n}</mml:annotation> …
A study is made of the numerical condition of the coordinate map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{M_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which associates to each polynomial of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-slanted-equals n minus 1"> <mml:semantics> <mml:mrow> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\leqslant n - 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the compact interval [<italic>a, b</italic>] the <italic>n</italic>-vector of its coefficients with respect to the power basis. It is shown that the condition number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper M Subscript n Baseline double-vertical-bar Subscript normal infinity Baseline double-vertical-bar upper M Subscript n Superscript negative 1 Baseline double-vertical-bar Subscript normal infinity"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\left \| {{M_n}} \right \|_\infty }{\left \| {M_n^{ - 1}} \right \|_\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> increases at an exponential rate if the interval [<italic>a, b</italic>] is symmetric or on one side of the origin, the rate of growth being at least equal to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 plus StartRoot 2 EndRoot"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mrow> <mml:annotation encoding="application/x-tex">1 + \sqrt 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the more difficult case of an asymmetric interval around the origin we obtain upper bounds for the condition number which also grow exponentially.
Journal Article On the Application of a Generalization of Toeplitz Matrices to the Numerical Solution of Integral Equations with Weakly Singular Convolution Kernels Get access S. N. CHANDLER-WILDE, S. N. …
Journal Article On the Application of a Generalization of Toeplitz Matrices to the Numerical Solution of Integral Equations with Weakly Singular Convolution Kernels Get access S. N. CHANDLER-WILDE, S. N. CHANDLER-WILDE Department of Mathematics, Coventry PolytechnicCoventry CV1 5FB Search for other works by this author on: Oxford Academic Google Scholar M. J. C. GOVER M. J. C. GOVER School of Mathematical Sciences, Bradford UniversityBradford BD7 1DP Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Numerical Analysis, Volume 9, Issue 4, October 1989, Pages 525–544, https://doi.org/10.1093/imanum/9.4.525 Published: 01 October 1989 Article history Received: 31 October 1988 Published: 01 October 1989
Bridging cultures that have often been distant, Julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing. Julia is …
Bridging cultures that have often been distant, Julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing. Julia is designed to be easy and fast and questions notions generally held to be “laws of nature" by practitioners of numerical computing: \beginlist \item High-level dynamic programs have to be slow. \item One must prototype in one language and then rewrite in another language for speed or deployment. \item There are parts of a system appropriate for the programmer, and other parts that are best left untouched as they have been built by the experts. \endlist We introduce the Julia programming language and its design---a dance between specialization and abstraction. Specialization allows for custom treatment. Multiple dispatch, a technique from computer science, picks the right algorithm for the right circumstance. Abstraction, which is what good computation is really about, recognizes what remains the same after differences are stripped away. Abstractions in mathematics are captured as code through another technique from computer science, generic programming. Julia shows that one can achieve machine performance without sacrificing human convenience.
Fiedler pencils are a family of strong linearizations for polynomials expressed in the monomial basis, that include the classical Frobenius companion pencils as special cases. We generalize the definition of …
Fiedler pencils are a family of strong linearizations for polynomials expressed in the monomial basis, that include the classical Frobenius companion pencils as special cases. We generalize the definition of a Fiedler pencil from monomials to a larger class of orthogonal polynomial bases. In particular, we derive Fiedler-comrade pencils for two bases that are extremely important in practical applications: the Chebyshev polynomials of the first and second kind. The new approach allows one to construct linearizations having limited bandwidth: a Chebyshev analogue of the pentadiagonal Fiedler pencils in the monomial basis. Moreover, our theory allows for linearizations of square matrix polynomials expressed in the Chebyshev basis (and in other bases), regardless of whether the matrix polynomial is regular or singular, and for recovery formulas for eigenvectors, and minimal indices and bases.
Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix $$T_n(f)$$ , as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated …
Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix $$T_n(f)$$ , as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated generating function f. A restriction is that f has to be polynomial, monotone, and scalar-valued. In this paper we focus on the case where $$\mathbf {f}$$ is an $$s\times s$$ matrix-valued trigonometric polynomial with $$s\ge 1$$ , and $$T_n(\mathbf {f})$$ is the block Toeplitz matrix generated by $$\mathbf {f}$$ , whose size is $$N(n,s)=sn$$ . The case $$s=1$$ corresponds to that already treated in the literature. We numerically derive conditions which ensure the existence of an asymptotic expansion for the eigenvalues. Such conditions generalize those known for the scalar-valued setting. Furthermore, following a proposal in the scalar-valued case by the first author, Garoni, and the third author, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric block Toeplitz matrices with a high level of accuracy and a low computational cost. The resulting algorithm is an eigensolver that does not need to store the original matrix, does not need to perform matrix-vector products, and for this reason is called matrix-less. We use the asymptotic expansion for the efficient computation of the spectrum of special block Toeplitz structures and we provide exact formulae for the eigenvalues of the matrices coming from the $$\mathbb {Q}_p$$ Lagrangian Finite Element approximation of a second order elliptic differential problem. Numerical results are presented and critically discussed.
Let p1,p2,…,pn be distinct positive real numbers and m be any integer. Every symmetric polynomial f(x,y)∈C[x,y] induces a symmetric matrix f(pi,pj)i,j=1n. We obtain the determinants of such matrices with an …
Let p1,p2,…,pn be distinct positive real numbers and m be any integer. Every symmetric polynomial f(x,y)∈C[x,y] induces a symmetric matrix f(pi,pj)i,j=1n. We obtain the determinants of such matrices with an aim to find the determinants of Pm=(pi+pj)mi,j=1n and B2m=(pi−pj)2mi,j=1n for m∈N (where N is the set of natural numbers) in terms of the Schur polynomials. We also discuss and compute determinant of the matrix Km=pim+pjmpi+pji,j=1n for any integer m in terms of the Schur and skew-Schur polynomials.
Abstract In this short note, we provide a brief proof for a recent determinantal formula involving a particular family of banded matrices.
Abstract In this short note, we provide a brief proof for a recent determinantal formula involving a particular family of banded matrices.
It is known that the generating function $f$ of a sequence of Toeplitz matrices $\{T_n(f)\}_n$ may not describe the asymptotic distribution of the eigenvalues of $T_n(f)$ if $f$ is not …
It is known that the generating function $f$ of a sequence of Toeplitz matrices $\{T_n(f)\}_n$ may not describe the asymptotic distribution of the eigenvalues of $T_n(f)$ if $f$ is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of $T_n(f)$ are real for all $n$, then they admit an asymptotic expansion of the same type as considered in previous works [1,10,12,13], where the first function $g$ appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of $T_n(f)$. After validating this working hypothesis through a number of numerical experiments, drawing inspiration from [12], we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function $g$. The proposed algorithm is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $g$. Future research directions are outlined at the end of the paper.
The gaussoid axioms are conditional independence inference rules which characterize regular Gaussian CI structures over a three-element ground set. It is known that no finite set of inference rules completely …
The gaussoid axioms are conditional independence inference rules which characterize regular Gaussian CI structures over a three-element ground set. It is known that no finite set of inference rules completely describes regular Gaussian CI as the ground set grows. In this article we show that the gaussoid axioms logically imply every inference rule of at most two antecedents which is valid for regular Gaussians over any ground set. The proof is accomplished by exhibiting for each inclusion-minimal gaussoid extension of at most two CI statements a regular Gaussian realization. Moreover we prove that all those gaussoids have rational positive-definite realizations inside every $\varepsilon$-ball around the identity matrix. For the proof we introduce the concept of algebraic Gaussians over arbitrary fields and of positive Gaussians over ordered fields and obtain the same two-antecedental completeness of the gaussoid axioms for algebraic and positive Gaussians over all fields of characteristic zero as a byproduct.
Small- to medium-sized polynomial eigenvalue problems can be solved by linearizing the matrix polynomial and solving the resulting generalized eigenvalue problem using the QZ algorithm. The QZ algorithm, in turn, …
Small- to medium-sized polynomial eigenvalue problems can be solved by linearizing the matrix polynomial and solving the resulting generalized eigenvalue problem using the QZ algorithm. The QZ algorithm, in turn, requires an initial reduction of a matrix pair to Hessenberg-triangular (HT) form. In this paper, we discuss the design and evaluation of high-performance parallel algorithms and software for HT reduction of a specific linearization of matrix polynomials of arbitrary degree. The proposed algorithm exploits the sparsity structure of the linearization to reduce the number of operations and improve the cache reuse compared to existing algorithms for unstructured inputs. Experiments on both a workstation and a high-performance computing system demonstrate that our structure-exploiting parallel implementation can outperform both the general LAPACK routine \tt DGGHRD and the prototype implementation \tt DGGHR3 of a general blocked algorithm.
Positive definite matrices factor into A=LLT (Cholesky). Symmetric indefinite matrices need a symmetric middle factor in A=LPLT. Then A and P have the same inertia (eigenvalues of the same sign). …
Positive definite matrices factor into A=LLT (Cholesky). Symmetric indefinite matrices need a symmetric middle factor in A=LPLT. Then A and P have the same inertia (eigenvalues of the same sign). We construct P through elimination, so the inertias agree for all leading minors of A and P. When restricting P to be a variant of a symmetric permutation in which diagonal 1's can be replaced by 0's or −1's, it is unique.
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 …
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.