Spectral and computational properties of band symmetric toeplitz matrices
TL;DR: Two algorithms, based on the bisection technique and Newton's method, are shown to be very fast for computing the eigenvalues of a 7− or 5-diagonal BST-matrix.
About: This article is published in Linear Algebra and its Applications.The article was published on 1983-07-01 and is currently open access. It has received 172 citations till now. The article focuses on the topics: Matrix (mathematics) & Toeplitz matrix.
Citations
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TL;DR: In this article, the authors proposed antireflective boundary conditions (BCs) for deblurring and detecting the regularization parameters in the presence of noise, which can be related to the algebra of the matrices that can be simultaneously diagonalized by the (fast) sine transform DST I.
Abstract: In a recent work Ng, Chan, and Tang introduced reflecting (Neumann) boundary conditions (BCs) for blurring models and proved that the resulting choice leads to fast algorithms for both deblurring and detecting the regularization parameters in the presence of noise. The key point is that Neumann BC matrices can be simultaneously diagonalized by the (fast) cosine transform DCT III. Here we propose antireflective BCs that can be related to $\tau$ structures, i.e., to the algebra of the matrices that can be simultaneously diagonalized by the (fast) sine transform DST I. We show that, in the generic case, this is a more natural modeling whose features are (a) a reduced analytical error since the zero (Dirichlet) BCs lead to discontinuity at the boundaries, the reflecting (Neumann) BCs lead to C0 continuity at the boundaries, while our proposal leads to C1 continuity at the boundaries; (b) fast numerical algorithms in real arithmetic for both deblurring and estimating regularization parameters. Finally, simple yet significant 1D and 2D numerical evidence is presented and discussed.
138 citations
Cites background from "Spectral and computational properti..."
...Let Q be the n-dimensional discrete sine of type I (see [3]) with entries...
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...2), then every matrix of the class can be written as (see Bini and Capovani [3]) T (v)−H(σ2(v), Jσ(v)), where v = (v0, ....
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..., [3] or (3....
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TL;DR: An adaptive algorithm which has a input the coefficients of A and return an iterative Multigrid method with convergence speed independent of the mesh spacing h and with an asymptotical cost of O(n).
Abstract: We introduce a class of Multigrid methods for solving banded, symmetric Toeplitz systems Ax=b. We use a, special choice of the projection operator whose coefficients simply depend on some spectral properties of A. This choice leads to an iterative Multigrid method with convergence rate smaller than 1 independent of the condition number K2(A) and of the dimension of the matrix. In the second part the B0 class is introduced: this class, of Toeplitz matrices contains the linear space generated by the matrices arising from the finite differences discretization of the differential operators
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, m∈N +. To sum up we present an adaptive algorithm which has a input the coefficients of A and return an iterative Multigrid method with convergence speed independent of the mesh spacing h and with an asymptotical cost of O(n).
119 citations
TL;DR: It is proved the V-cycle multigrid iteration to have a convergence rate independent of the dimension even in presence of ill-conditioning, and the corresponding method is optimal.
Abstract: In this paper we are interested in the solution by multigrid strategies of multilevel linear systems whose coefficient matrices belong to the circulant, Hartley, or $\tau$ algebras or to the Toeplitz class and are generated by (the Fourier expansion of) a nonnegative multivariate polynomial f. It is well known that these matrices are banded and have eigenvalues equally distributed as f, so they are ill-conditioned whenever f takes the zero value; they can even be singular and need a low-rank correction.
We prove the V-cycle multigrid iteration to have a convergence rate independent of the dimension even in presence of ill-conditioning. If the (multilevel) coefficient matrix has partial dimension nr at level r, r=1,...,d, then the size of the algebraic system is $N(n)=\prod_{r=1}^d n_r$, O(N(n)) operations are required by our technique, and therefore the corresponding method is optimal.
Some numerical experiments concerning linear systems arising in applications, such as elliptic PDEs with mixed boundary conditions and image restoration problems, are considered and discussed.
111 citations
Cites methods from "Spectral and computational properti..."
...In that case an interesting matrix algebra approximation is provided by the τ algebra [3]....
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TL;DR: A particular class of preconditioners for the conjugate gradient method and other iterative methods is proposed for the solution of linear systems An,mx=b, whereAn,m is ann×n positive definite block Toeplitz matrix withm×m Toe Plitz blocks.
Abstract: A particular class of preconditioners for the conjugate gradient method and other iterative methods is proposed for the solution of linear systemsAn,mx=b, whereAn,m is ann×n positive definite block Toeplitz matrix withm×m Toeplitz blocks. In particular we propose a sparse preconditionerPn,m such that the condition number of the preconditioned matrix turns out to be less than a suitable constant independent of bothn andm, even if the condition number ofAn,m tends to ∞. This leads to iterative methods which require a number of steps independent ofm andn in order to reduce the error by a given factor.
104 citations
Cites background or methods from "Spectral and computational properti..."
...In this case in [19, 6, 9], by using a preconditioner in the two-level circulant algebra [8] and in the r..., class [ 2 ], respectively, it is shown that good clustering properties of the preconditioned matrix are achieved....
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...An alternative strategy to solve the former linear system uses the following decomposition [ 2 ]: for each Pn.m with external bandwidth 2p + 1 and internal bandwidth 2q + 1 there exists r(Pn, m) E Z..m such that...
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...The solution of a system with coefficient matrix r(P,,m) costs O(nm log nm) [ 2 ], that is, the cost of a few bidimensional sine transforms....
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...This leads to preconditioned iterative methods where the convergence rate is independent of m and n. Each step of the method requires the solution of a system of the type P,,mX = b, which can be easily obtained by a band matrix solver [14, 2 ] adapted to block matrices [l 3]....
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References
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Book•
01 Jan 1965
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Abstract: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography Index.
7,422 citations
Book•
16 Feb 2013
TL;DR: This well written book is enlarged by the following topics: B-splines and their computation, elimination methods for large sparse systems of linear equations, Lanczos algorithm for eigenvalue problems, implicit shift techniques for theLR and QR algorithm, implicit differential equations, differential algebraic systems, new methods for stiff differential equations and preconditioning techniques.
Abstract: This well written book is enlarged by the following topics:
$B$-splines and their computation, elimination methods for
large sparse systems of linear equations, Lanczos algorithm for
eigenvalue problems, implicit shift techniques for the $LR$ and
$QR$ algorithm, implicit differential equations, differential
algebraic systems, new methods for stiff differential
equations, preconditioning techniques and convergence rate of
the conjugate gradient algorithm and multigrid methods for
boundary value problems. Cf. also the reviews of the German
original editions.
6,270 citations
TL;DR: In this article, Toeplitz forms are used for the trigonometric moment problem and other problems in probability theory, analysis, and statistics, including analytic functions and integral equations.
Abstract: Part I: Toeplitz Forms: Preliminaries Orthogonal polynomials. Algebraic properties Orthogonal polynomials. Limit properties The trigonometric moment problem Eigenvalues of Toeplitz forms Generalizations and analogs of Toeplitz forms Further generalizations Certain matrices and integral equations of the Toeplitz type Part II: Applications of Toeplitz Forms: Applications to analytic functions Applications to probability theory Applications to statistics Appendix: Notes and references Bibliography Index.
2,279 citations
01 Jan 1974
TL;DR: A computer algorithm that computes the discrete Fourier transform much faster than other algorithms, is explained and examples and detailed procedures are provided to assist the reader in learning how to use the algorithm.
1,823 citations
TL;DR: A modified version of the Fast Fourier Transform is developed and described and it is suggested that this form is of general use in the development and classification of various modifications and extensions of the algorithm.
Abstract: A modified version of the Fast Fourier Transform is developed and described. This version is well adapted for use in a special-purpose computer designed for the purpose. It is shown that only three operators are needed. One operator replaces successive pairs of data points by their sums and differences. The second operator performs a fixed permutation which is an ideal shuffle of the data. The third operator permits the multiplication of a selected subset of the data by a common complex multiplier.If, as seems reasonable, the slowest operation is the complex multiplications required, then, for reasonably sized date sets—e.g. 512 complex numbers—parallelization by the method developed should allow an increase of speed over the serial use of the Fast Fourier Transform by about two orders of magnitude.It is suggested that a machine to realize the speed improvement indicated is quite feasible.The analysis is based on the use of the Kronecker product of matrices. It is suggested that this form is of general use in the development and classification of various modifications and extensions of the algorithm.
362 citations