Tridiagonal matrix algorithm

Abstract: Publisher Summary This chapter presents the solution of single tridiagonal linear systems and vectorization of the ICCG algorithm on the Cray-1. When the numerical algorithms used to solve the physics equations in codes which model laser fusion are examined, it is found that a large number of subroutines require the solution of tridiagonal linear systems of equations. Radiation transport, thermal- and suprathermal-electron transport, ion thermal conduction, charged-particle, and neutron transport require the solution of tridiagonal systems of equations. The standard algorithm that has been used in the past on CDC 7600s will not vectorize and hence, cannot take advantage of the large speed increases possible on the Cray-1 through vectorization. There is an alternative algorithm for solving tridiagonal systems called cyclic reduction, which allows for vectorization, and is optimal for the Cray-1. Software based on this algorithm is being used in LASNEX to solve tridiagonal linear systems in the subroutines. The new algorithm runs five times faster than the standard algorithm on the Cray-1.

Abstract: Perfect elimination digraphs arise in the study of Gaussian elimination on sparse linear systems. With a view toward numerical computational complexity we show four conditions (C1–C4) to be necessa...

Abstract: Publisher Summary This chapter discusses the applications of an element model for Gaussian elimination. The system of linear equations is considered A x = b, where A is an N × N sparse symmetric positive definite matrix such as those that arise in finite difference and finite element approximations to elliptic boundary value problems in two and three dimensions. A classic method for solving such systems is Gaussian elimination. The chapter presents a method by which kth equation is used to eliminate the kth variable from the remaining N − k equations for k = 1, 2, …, N − 1 and then back-solve the resulting upper triangular system for the unknown vector x. The chapter reviews the graph-theoretic elimination model and highlights a new element model of the elimination process. The model is used to give simple proofs of inherent lower bounds for the work and storage associated with Gaussian elimination. It also explains the way by which element model, combined with the unusual idea of recomputing rather than saving the factorization, leads to a minimal storage sparse elimination algorithm that requires significantly less storage than regular sparse elimination for the five- or nine-point.

Abstract: The applications of infinite systems of linear first order differential equations with 2L+1-term recursion formulas are discussed. It is shown that such systems can be written as a system of linear tridiagonal vector equations of dimensionL. A general method is presented by which the initial value problem can be solved by iteration. For special but physically important initial conditions the solution is given by a matrix continued fraction. The eigenvalues of the tridiagonal vector recurrence relations are obtained as the roots of aL×L determinant the elements of which are determined by a matrix continued fraction. The applicability of the method is demonstrated by calculating the eigenvalues of the laser Fokker-Planck operator.