Direct and Iterative Methods for Block Tridiagonal Linear Systems

Block tridiagonal systems of linear equations occur frequently in scientific computations, often forming the core of more complicated problems. Numerical methods for solution of such systems are studied with emphasis on efficient methods for a vector computer. A convergence theory for direct methods under conditions of block diagonal dominance is developed, demonstrating stability, convergence and approximation properties of direct methods. Block elimination (LU factorization) is linear, cyclic odd-even reduction is quadratic, and higher-order methods exist. The odd-even methods are variations of the quadratic Newton iteration for the inverse matrix, and are the only quadratic methods within a certain reasonable class of algorithms. Semi-direct methods based on the quadratic convergence of odd-even reduction prove useful in combination with linear iterations for an approximate solution. An execution time analysis for a pipeline computer is given, with attention to storage requirements and the effect of machine constraints on vector operations.