Tridiagonal matrix algorithm

Abstract: Received 17 June 1977 and 8 November 1977. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for chrect commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machmery. To copy otherwise, or to repubhsh, requires a fee and/or specific permission. This work was supported m part by the National Science Foundation under Grant NSF DCR 7307998, in part by the Energy Research and Development Administration under Grant US ERDA E(ll-1) 2383, and in part by Yale University under a grant subcontracted from the United States Air Force Office of Scmntific Research Author's address. Department of Computer Sciences, Painter 328, The Umverslty of Texas at Austin, Austin, TX 78712. @) 1978 ACM 0098-3500/78/1200-0391 $00.75

Abstract: Purpose The purpose of this paper is to provide a robust second-order numerical scheme for singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Design/methodology/approach For the parabolic convection-diffusion initial boundary value problem, the authors solve the problem numerically by discretizing the domain in the spatial direction using the Shishkin-type meshes (standard Shishkin mesh, Bakhvalov–Shishkin mesh) and in temporal direction using the uniform mesh. The time derivative is discretized by the implicit-trapezoidal scheme, and the spatial derivatives are discretized by the hybrid scheme, which is a combination of the midpoint upwind scheme and central difference scheme. Findings The authors find a parameter-uniform convergent scheme which is of second-order accurate globally with respect to space and time for the singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Also, the Thomas algorithm is used which takes much less computational time. Originality/value A singularly perturbed delay parabolic convection–diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a hybrid scheme. The method is parameter-uniform convergent and is of second order accurate globally with respect to space and time. Numerical results are carried out to verify the theoretical estimates.

Abstract: It is known that for several classes of matrices, including quadratic and certain tridiagonal matrices, the numerical range is an ellipse. We prove this result for a larger class of matrices, encompassing these published results as well as providing other sufficient conditions for ellipticity. The proof gives an explanation for this phenomenon and, in certain cases, provides an explicit description of the numerical range.

Abstract: An algorithm for reducing a nonsymmetric matrix to tridiagonal form as a first step toward finding its eigenvalues is described. The algorithm uses a variation of threshold pivoting, where at each step, the pivot is chosen to minimize the maximum entry in the transformation matrix that reduces the next column and row of the matrix. Situations are given where the tridiagonalization process breaks down, and two recovery methods are presented for these situations. Although no existing tridiagonalization algorithm is guaranteed to succeed, this algorithm is found to be very robust and fast in practice. A gradual loss of similarity is also observed as the order of the matrix increases.