Tridiagonal matrix algorithm

Abstract: Tridiagonal or Jacobi matrices arise in several problems in electrical engineering as well as in different areas of mathematics. — However, there is little known on the inverses of such matrices. The explicit inverses of general tridiagonal matrices are presented in this paper. The method is based on an efficient implementation of Cramer's rule or Coate's flow graph technique and the theory of continuants. The technique is well suited to computer programming and is extremely efficient in symbolical analysis of electrical networks. It could also be efficiently implemented on a parallel computer. — Sensitivity studies or modern optimization techniques used in iterative network synthesis often require the computation of partial derivatives. The proposed technique permits these to be calculated analytically with great case.

Abstract: In this paper we given an algorithm of low computational complexity which determines the eigenvalues of a symmetric tridiagonal matrix.

Abstract: We discuss the exact solvability of a lattice model with a flow within the matrix product ansatz to interacting many-body systems from the point of view of a tridiagonal boundary algebra. It reveals deep symmetry properties of the multiparticle process which allow for the exact solution in the stationary state and puts the description of the dynamics into perspective.

Abstract: The tridiagonal Toeplitz linear systems occur repeatedly in the solution of the implicit finite difference equations derived from linear first order hyperbolic equations, i.e. the Transport equation, under a variety of boundary conditions. Interest in fast direct methods for solving these kind of linear systems has long been a hot spot of research. A parallel algorithm for certain tridiagonal Toeplitz linear systems on distributed memory multicomputers is presented. Derivation of the algorithm is introduced. The algorithm is based on the factorization of the coefficient matrix and the principle of ‘divide and conquer’ in designing parallel algorithms. Authors make full use of the special structure of the coefficient matrix. By using the customary nesting procedure, Horner's formula, authors avoid the necessary of quantities α i, (-α) i(i=2,3,…,m ) and (α m) i, (-α m) i(i=2,3,…,p-1 ). This reduces the algorithm's redundancy computation caused by parallelization. The complexity of the algorithm is analyzed using Log P model. Its communication mechanism is very simple. The communication complexity is only related to p , the number of processors, and not related to n , the size of the matrix. The algorithm's parallel efficiency is high. The analysis of complexity and numerical experiments show that the algorithm's speedup satisfy: S p(n)→p(n→+∞) . This is the best a parallel algorithm can reach. The algorithm has been implemented on parallel computers. The results of numerical experiments about the algorithm on a distributed memory multicomputer are presented in this paper.