Abstract. The notion of Compact Fourier Analysis (CFA) is discussed. The CFA allows description of multigrid (MG) in a nutshell and offers a clear overview on all MG components. The principal idea of CFA is to model the MG mechanisms by means of scalar generating functions and matrix functions (block symbols). The formalism of the CFA approach is presented by describing the symbols of the fine and coarse grid problems, the prolongation and restriction, the smoother, and the coarse grid correction, resp. smoothing corrections. CFA uses matrix functions and their features (e.g. product, inverse, adjoint, norm, spectral radius, eigenvectors, eigenvalues), and scalar functions and their roots. This leads to an elementary description and allows for an easy analysis of MG algorithms. A first application is to utilize the CFA for deriving MG as a direct solver, i.e. an MG cycle that will converge in just one iteration step. Necessary and sufficient conditions that have to be fulfilled by the MG components are given for obtaining MG functioning as a direct solver. Furthermore, new general and practical smoothers and transfer operators that lead to efficient MG methods are introduced. In addition, we study sparse approximations of the Galerkin coarse grid operator yielding efficient and practicable MG algorithms (approximately direct solvers). Numerical experiments demonstrate the theoretical results.Key words. multigrid, Fourier analysis, generating function, block symbol, Toeplitz matrices AMS subject classifications. 65N55, 65F10, 65F15, 65N12, 15A121. Introduction. A crucial point for the efficiency of a multigrid (MG) method is the appropriate choice of its components, which allows for an efficient interplay between smoother and coarse grid correction. In many cases, this coordination can be made by use of Local Fourier Analysis (LFA), which is an important quantitative tool for the development of powerful MG methods [3,24,23,13,4]. This approach has been generalized for structured matrices by employing generating functions expressing, e.g. the symbol of the smoother, of the projection or of the standard discrete Laplace operator in terms of trigonometric polynomials. It is based on the connection between Toeplitz, resp. circulant matrices, and trigonometric functions. It was analyzed by S. Serra Capizzano, R. Chan, T. Huckle, and coauthors in [10,11,5,14,18,15,20,21]. In this paper we complete this formal approach in order to represent a full two-grid step in terms of the block symbol, called also block generating (matrix) function. Furthermore, we show how to use the block symbol formalism for a multigrid Fourier analysis.We consider the notion of Compact Fourier Analysis (CFA), which can be seen as a reformulation and generalization of LFA based on matrix functions. Instead of discrete operators on a grid we consider analytic matrix functions (block symbols) of small order that capture the behavior of the full matrices [14]. This allows the use of matrix features, such as product and eigendecomposition, for descr...
In this paper we discuss classical sufficient conditions to be satisfied from the grid transfer operators in order to obtain optimal two-grid and V-cycle multigrid methods utilizing the theory for Toeplitz matrices. We derive relaxed conditions that allow the construction of special grid transfer operators that are computationally less expensive while preserving optimality. This is particularly useful when the generating symbol of the system matrix has a zero of higher order, like in the case of higher order PDEs. These newly derived conditions allow the use of rank deficient grid transfer operators. In this case the use of a pre-relaxation iteration that is lacking the smoothing property is proposed. Combining these pre-relaxations with the new rank deficient grid transfer operators yields a substantial reduction of the convergence rate and of the computational cost at each iteration compared with the classical choice for Toeplitz matrices.\ud The proposed strategy, i.e. a rank deficient grid transfer operator plus a specific pre-relaxation, is applied to linear systems whose system matrix is a Toeplitz matrix where the generating symbol is a high-order polynomial. The necessity of using high-order polynomials as generating symbols for the grid transfer operators usually destroys the Toeplitz structure on the coarser levels. Therefore, we discuss some effective and computational cheap coarsening strategies found in the literature. In particular, we present numerical results showing near-optimal behavior while keeping the Toeplitz structure on the coarser levels
In 1968 Cryer conjectured that the growth factor of an n ×n Hadamard matrix is n. In 1988 Day and Peterson proved this only for the Hadamard-Sylvester class. In 1995 Edelman and Mascarenhas proved that the growth factor of a Hadamard matrix of order 12 is 12. In the present paper we demonstrate the pivot structures of a Hadamard matrix of order 16 and prove for the first time that its growth factor is 16. The study is divided in two parts: we calculate pivots from the beginning and pivots from the end of the pivot pattern. For the first part we develop counting techniques based on symbolic manipulation for specifying the existence or non-existence of specific submatrices inside the first rows of a Hadamard matrix, and so we can calculate values of principal minors. For the second part we exploit sophisticated numerical techniques that facilitate the computations of all possible (n − j)×(n − j) minors of Hadamard matrices for various values of j. The pivot patterns are obtained by utilizing appropriately the fact that the pivots appearing after the application of Gaussian elimination on a completely pivoted matrix are given as quotients of principal minors of the matrix.
C. Koukouvinos, M. Mitrouli and Jennifer Seberry, in "Growth in Gaussian elimination for weighing matrices, W (n, n-1)", Linear Algebra and its Appl., 306 (2000), 189-202, conjectured that the growth factor for Gaussian elimination of any completely pivoted weighing matrix of order n and weight n-1 is n-1 and that the first and last few pivots are (1,2,2,3 or 4, ..., n-1 or (n-1)/2, , (n-1)/2, n-1) for n > 14. In the present paper we study the growth problem for skew and symmetric conference matrices. An algorithm for extending a k × k matrix with elements 0, ±1 to a skew and symmetric conference matrix of order n is described. By using this algorithm we show the unique W(8, 7) has two pivot structures. We also prove that unique W(10,9) has three pivot patterns.
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