Synopsis About this title This volume provides a down-to-earth, easily understandable guide to techniques of matrix theory, which are widely used throughout engineering and the physical, life, and social sciences.
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In the block KSS method described in [ 3 , 4 ] and reviewed in the previous section, these interpolation points are the eigenvalues of the block tridiagonal matrix T K from 12 that is produced by block Lanczos iteration. Although each such matrix is small, computing the eigenvalues and eigenvectors for each frequency is computationally expensive. Therefore, in this section, we describe a much faster approach to obtaining estimates of these nodes, at least for high frequencies.
This approach was first presented in [ 11 ] and generalized in [ 6 , 7 ]. We start the first iteration of the block Lanczos algorithm by finding the QR -factorization of R 0 :. Substituting the value of X 1 from 18 into 19 yields.
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Continuing this process, it can be shown that given sufficient regularity of the solution and coefficients of L , each block M j or B j of T K from 12 becomes approximately diagonal at high frequencies. In [ 11 ], it is shown that this decoupling also takes place if the leading coefficient p x of L is not constant.
In [ 6 , 7 ], similar formulas for the nodes are derived for a PDE with homogeneous Neumann boundary conditions, and for a 2-D PDE with periodic boundary conditions. When the matrix A is a finite-difference representation of the underlying differential operator, the block Gaussian quadrature nodes can be represented more accurately if formulas for the eigenvalues of symmetric Toeplitz matrices are used for the leading-order terms in the nodes.
The theory developed in [ 12 ] applies to symmetric positive definite matrices, but this property is not essential [ 17 , 18 ]. The algorithm for Arnoldi iteration, applied to a matrix A and initial vector z 0 , is as follows:. The output of Arnoldi iteration is an upper Hessenberg matrix H m , and a matrix V m with orthonormal columns, such that. Additional details can be found in [ 6 , 7 ]. In this section, we give an overview of EPI methods, as developed by Tokman et al. First, the time derivative F y is expressed in terms of its Taylor expansion around y t n , which yields.
The integral on the right side is then approximated using a quadrature rule. Any such matrix function-vector product is computed using Krylov projection. Arnoldi iteration is applied to A or Lanczos iteration, if A is symmetric with initial vector b. After m iterations, we obtain 21 , from which we obtain an approximation.
Matrices: Methods and Applications : Stephen Barnett :
The accuracy of this approximation is discussed in [ 9 ]. In the case where A is ill-conditioned, the number of iterations m needed for convergence of 25 can be quite large, and this is exacerbated by increasing the spatial resolution in the discretization of the underlying PDE from which 1 arises. When the number of iterations is large, an additional issue, particularly for advection-dominated problems, is the appearance of spurious high-frequency oscillations in the columns of V m , even if the initial vector b represents a smooth function. This can be alleviated by filtering out high-frequency components of the columns of V m after each matrix-vector multiplication.
Future work will include the automatic, adaptive selection of an appropriate threshold for filtering high-frequency components. The behavior of the unfiltered Krylov vectors is not surprising, as similar behavior is displayed by the unsmoothed Fourier method applied to hyperbolic PDEs [ 22 ]. In that work, the proposed remedies were to either use filtering, or increase the number of grid points; the former remedy serves as the motivation for denoising in this context. The main steps in the new approach are as follows: First, it is necessary to determine a cutoff frequency N c.
In the numerical experiments presented in this paper, the value of N c has been determined by experimentation; it is demonstrated in [ 7 ] that the performance is not unduly sensitive to the choice of N c. In future work, an adaptive approach to choosing N c will be developed. We now provide details on how step 4 can be performed efficiently, by minimizing the number of FFTs. To simplify the exposition, we consider the 1-D case, with periodic boundary conditions.
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Expressing this interpolant in Newton form, we have. Arranging the interpolation points in the order indicated above allows us to reduce the number of FFTs needed. Using the relation from Lanczos iteration,. In this section, we compare several versions of EPI methods, as applied to two test problems; additional test problems are featured in [ 6 , 7 ].
All of these approaches are used in the context of two EPI methods. The first is a third-order, two-stage EPI method [ 1 ]. The second is a fifth-order, three-stage EPI method [ 25 ]. Throughout this section, for the purpose of discussing performance as a function of spatial resolution, N refers to the number of grid points per dimension. Relative error plotted against execution time for solving the Allen-Cahn equation 31 using the third-order EPI method We impose homogeneous Neumann boundary conditions, and the initial condition. The Laplacian is discretized using a centered finite difference.
That is, for this problem, the value of N c , as defined in the previous section, is 7. Matrices 4. Multiplication of Matrices 4a. Matrix Multiplication examples 4b. Finding the Inverse of a Matrix 5a. Simple Matrix Calculator 5b.
sitechpharma.com/wp-includes/copiare-rubrica-da-iphone-a-samsung-note-4.php Inverse of a Matrix using Gauss-Jordan Elimination 6. Eigenvalues and Eigenvectors 8. Applications of Eigenvalues and Eigenvectors Eigenvalues and eigenvectors calculator Feedback? On this page Coming up Large Determinants. Multiplication of Matrices. Related, useful or interesting IntMath articles Where did matrices and determinants come from? This article points to 2 interactives that show how to multiply matrices. Here's a method for finding inverses of matrices which reduces the chances of getting lost.
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