Tridiagonal matrix algorithm

Abstract: Complete pivoting is known to be numerically preferable to partial pivoting for solving systems of linear algebraic equations by Gaussian elimination. However, partial pivoting requires less computational work. Hence we should like to use partial pivoting provided we can easily recognize numerical difficulties. We propose an effective and inexpensive test for this purpose.

Abstract: A powerful new simulation procedure is developed for multicomponent distillation columns processing nonideal solutions or reactive solutions. Although the tridiagonal matrix algorithm is incorporated, the main calculational loop is the Newton-Raphson method in which liquid mole fractions are chosen for the independent variables and the functions to be zeroed are originally defined. The procedure does not vitiate at all such great advantages of the conventional tridiagonal matrix method as the ready incorporation of heat balances, nonideality of the solutions and chemical reactions, and still the algorithm and computer programming are rather simple. Nevertheless, the procedure presents much greater stability in finding converged solutions. A total of eight numerical experiments are made under a variety of conditions, and in all cases rigorous solutions are obtained achieving convergence in three to nine iterations.

Abstract: A parallel solver for general tridiagonal irreducible systems is described.Solver based on Spike framework and Givens-QR with occasional low-rank modification.Modifications handle singularities exposed by QR in blocks of the parallel partition.The GPU implementation has similar performance to existing methods.Method returns accurate results when current GPU tridiagonal solvers fail. g-Spike, a parallel algorithm for solving general nonsymmetric tridiagonal systems for the GPU, and its CUDA implementation are described. The solver is based on the Spike framework, applying Givens rotations and QR factorization without pivoting. It also implements a low-rank modification strategy to compute the Spike DS decomposition even when the partitioning defines singular submatrices along the diagonal. The method is also used to solve the reduced system resulting from the Spike partitioning. Numerical experiments with problems of high order indicate that g-Spike is competitive in runtime with existing GPU methods, and can provide acceptable results when other methods cannot be applied or fail.