Stability of parallel triangular system solvers

Several parallel algorithms have been proposed for the solution of triangular systems. The stability of four of them is analysed here: a fan-in algorithm, a block elimination method, a method based on a factorized power series expansion of the matrix inverse, and a method based on a divide and conquer matrix inversion technique. New forward error and residual bounds are derived, including an improvement on the bounds of Sameh and Brent for the fan-in algorithm. A forward error bound is identified that holds not only for all the methods described here, but for any triangular equation solver that does not rely on algebraic cancellation; among the implications of the bound is that any such method is extremely accurate for certain special types of triangular systems