Spectral and computational properties of band symmetric toeplitz matrices

Бини, Дарио Андреа; Capovani, Milvio

doi:10.1016/0024-3795(83)80009-3

Cited by 134 publications

(80 citation statements)

References 8 publications

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“…In this case it is known (see [4,5,29]) that all the matrices A n (f ) are hermitian, banded (in the way induced from the considered structure) and semipositive definite if f ≥ 0. Moreover A n (f ) is ill-conditioned whenever f takes the zero value; it is singular if A ∈ {C, τ , H} and f vanishes at a grid point w [n] i .…”

Section: Multilevel Algebra and Toeplitz Matricesmentioning

confidence: 99%

“…Therefore in the Circulant case we use the Strang correction and we replace f and g with their positive versions f + and g + defined as in (2). Furthermore, the conditioning number of A n (f ) is K(A n (f )) K(A n (g)) 2 [min i (n i )] 4 . We solve the linear system A n (z)x = b where n = (n 1 , n 2 ), z ∈ {f , g}, A ∈ {τ , C} and the data vector b is obtained from the exact solution x taking four different Table 4 Tau case: number of iterations increasing dimension n = (n 1 , n 2 ) for τ n (f ) and τ n (g), where f (x, y) = (4 − 2 cos(x) − 2 cos(y)) 2 and g(x, y) = (4 + 2 cos(x) + 2 cos(y))(8 − cos(x) − cos(y)) Table 5 Circulant case: number of iterations increasing dimension n = (n 1 , n 2 ) for C n (f + ) and C n (g + ), where f + and g + are f (x, y) = (4 − 2 cos(x) − 2 cos(y)) 2 and g(x, y) = (4 + 2 cos(x) + 2 cos(y))(8 − cos(x) − cos(y)) plus them Strang corrections respectively types of solutions (constant, periodic, …).…”

Section: Tau and Circulant Algebrasmentioning

confidence: 99%

“…, m − 1. Define A 0 = A, b 0 = b and choose two classes of iterative methods S i , S i as in(4). If there exist two real positive numbers δ pre , δ post satisfying…”

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confidence: 99%

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A V-cycle Multigrid for multilevel matrix algebras: proof of optimality

2007

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We analyze the convergence rate of a multigrid method for multilevel\ud linear systems whose coefficient matrices are generated by a real\ud and nonnegative multivariate polynomial $f$ and belong to multilevel\ud matrix algebras like circulant, tau, Hartley, or are of Toeplitz type.\ud \ud In the case of matrix algebra linear systems, we prove that the\ud convergence rate is independent of the system dimension even in\ud presence of asymptotical ill-conditioning (this happens iff $f$\ud takes the zero value). More precisely, if the $d$-level coefficient\ud matrix has partial dimension $n_r$ at level $r$, with $r=1,\dots,d$,\ud then the size of the system is $N(\mi{n})=\prod_{r=1}^d n_r$,\ud $\mi{n}=(n_1, \dots, n_d)$, and $O(N(\mi{n}))$ operations are\ud required by the considered $V$-cycle Multigrid in order to compute the solution\ud within a fixed accuracy. Since the total arithmetic cost is\ud asymptotically equivalent to the one of a matrix-vector product, the\ud proposed method is optimal. Some numerical experiments concerning\ud linear systems arising in 2D and 3D applications are considered\ud and discussed

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Section: Multilevel Algebra and Toeplitz Matricesmentioning

confidence: 99%

Section: Tau and Circulant Algebrasmentioning

confidence: 99%

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A V-cycle Multigrid for multilevel matrix algebras: proof of optimality

2007

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“…However, while the preconditioning systems with τ matrices [5] or circulant matrices [14] can be easily solved in a parallel PRAM model of computation (where at each step each processor may perform at most one operation) in O(log n) parallel steps, our preconditioning system involves band Toeplitz matrices. In the literature we find good band-solvers [24], but, unfortunately, these algorithms are inherently sequential.…”

Section: Introductionmentioning

confidence: 99%

“…This problem can be overcome, for instance, by applying three different strategies (see Sect. 2.1) which, making use of matrix algebras [5,14,19,20], allow one to solve the preconditioning system in O(log n) parallel steps (see [4,23]). …”

Section: Introductionmentioning

confidence: 99%

The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

Serra

1999

Numerische Mathematik

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In previous works [21][22][23] we proposed the use of τ [5] and band Toeplitz based preconditioners for the solution of 1D and 2D boundary value problems (BVP) by means of the preconditioned conjugate gradient (PCG) methods. As τ and band Toeplitz linear systems can be solved [4] by using fast sine transforms [8], these methods become especially attractive in a parallel environment of computation. In this paper we extend this technique to the nonlinear, nonsymmetric case and, in addition, we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear. Therefore these methods work much finer than those based on separable preconditioners [18,45], on incomplete LU factorizations [36,13,27], and on circulant preconditioners [9,30,35] since the latter two techniques do not assure a linear rate of convergence. On the other hand, the proposed technique has a wider range of application since it can be naturally used for nonlinear, nonsymmetric problems and for BVP in which the coefficients of the differential operator are not strictly positive and only piecewise smooth. Finally the several numerical experiments performed here and in [22,23] confirm the effectiveness of the theoretical analysis.

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Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols

Ekström

Serra‐Capizzano

2018

Numerical Linear Algebra App

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Summary It is known that for a tridiagonal Toeplitz matrix, having on the main diagonal the constant a0 and on the two first off‐diagonals the constants a1(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 (a0,a1,a−1)=(2,−1,−1). In the first part of this article, we consider a tridiagonal Toeplitz matrix of the same form (a0,aω,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 a0, aω=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.

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Spectral and computational properties of band symmetric toeplitz matrices

Cited by 134 publications

References 8 publications

A V-cycle Multigrid for multilevel matrix algebras: proof of optimality

A V-cycle Multigrid for multilevel matrix algebras: proof of optimality

The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols

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