Showing papers on "Tridiagonal matrix algorithm published in 1979"
TL;DR: Roundoff error m the solution of near algebraic systems is studied using a more reahstsc notion of what st means to perturb a problem, namely, that each datum is subject to a relatwely small change.
Abstract: Roundoff error m the solution of hnear algebraic systems is stud,ed using a more reahstsc notion of what st means to perturb a problem, namely, that each datum :s subject to a relatwely small change Th:s ,s particularly appropriate for sparse linear systems The condition number :s determined for th:s approach The effect of scahng on the stabdlty of Gaussmn ellmmat,on is stud:ed, and st is d:scovered that the proper way to scale a system depends on the right-hand s:de However, ff only the norm of the error is of concern, then there ~s a good way to scale that does not depend on the right-hand stde
208 citations
TL;DR: In this paper, the exact solution for the resolvent matrix of a generalized tridiagonal Hamiltonian whose elements are themselves block matrices is obtained, and the capability of the method is demonstrated by applying it to study the electronic structure on the surface of semiconductors.
Abstract: The exact solution for the resolvent matrix of a generalized tridiagonal Hamiltonian whose elements are themselves block matrices is obtained. The capability of the method is demonstrated by applying it to study the electronic structure on the surface of semiconductors. Some interesting insights regarding the difference between the Shockley states and Tamm states are also discussed.
90 citations
TL;DR: In this article, a formula for inverting general band matrices is established, taking a simple form when the matrices are tridiagonal, and as a special case it includes the Bukhberger-Emel'yanenko algorithm for symmetric tridimensional matrices.
73 citations
TL;DR: In this paper, the authors characterize those matrices with nonzero diagonal elements whose inverses are tridiagonal and show that such matrices have an interesting structure, which is similar to the structure of Jacobi matrices.
72 citations
TL;DR: A parallel algorithm for the solution of the general tridiagonal system is presented, based on an efficient implementation of Cramer's rule, in which the only divisions are by the determinant of the matrix.
Abstract: . A parallel algorithm for the solution of the general tridiagonal system is presented. The method is based on an efficient implementation of Cramer's rule, in which the only divisions are by the determinant of the matrix. Therefore, the algorithm is defined without pivoting for any nonsingular system. 0(n) storage is required for n equations and 0(log n) operations are required on a parallel computer with n processors. 0(n) operations are required on a sequential computer. Experimental results are presented from both the CDC 7600 and CRAY-1 computers.
38 citations
TL;DR: Lower bounds for the number of different real eigenvalues as well as for real simple eigen values of a class of real irreducible tridiagonal matrices are given in this article.
17 citations
TL;DR: The decomposition of a block tridiagonal matrix into the product of block lowe and upper matrices is described and an expression of the cost is developed and is shown to compare favourably with decomposition in many cases.
Abstract: The decomposition of a block tridiagonal matrix into the product of block lowe and upper matrices is described. The cost of solving a block tridiagonal system of equations is given and compared to profile gaussian elimination. The desirability of a less expensive method is coupled to physical intuition about a common problem of solving a slowly varying sequence of such systems to motivate an iterative method based on residual correction. The method is described and convergence criteria are derived. An expression of the cost is developed and is shown to compare favourably with decomposition in many cases. Problems and advantages in computer implementation of the method are discussed and results of tests of a particular implementation on a well-known problem are given.