Algorithms for Special Tridiagonal Systems

Abstract

Algorithms for the solution of symmetric diagonally dominant tridiagonal systems of linear equations with constant diagonals are considered. Such systems occur, for example, when solving certain constant-coefficient elliptic partial differential equations by the Fourier method. In particular, the specialized $LU$ factorization method of Malcolm and Palmer (SpLU), the cyclic reduction method of Hockney (CR), and the reversed triangular factorization method of Evans (RTF) are considered. An interesting property of the first two algorithms is that they may be terminated early for highly diagonally dominant systems. A new implementation of RTF that also has this property is presented, significantly reducing its operation count in many cases. The slightly perturbed systems that arise from problems with Neumann or periodic boundary conditions are also considered, with extensions given for SpLU to the periodic case and RTF to the Neumann case. Floating-point operation counts are given for each method, and the results of experiments on a single scalar processor are reported.