Implementation and convergence evaluation of multiple-precision iterative CG and PCG solvers for GPUS

To solve systems of linear equations with sparse coefficient matrices, iterative Krylov subspace methods such as conjugate gradients (CG) and preconditioned conjugate gradients (PCG) are widely used. The convergence rate and numerical reliability of these methods may suffer from rounding errors. One way to reduce the impact of rounding errors is to use arithmetic over numbers with increased word length, which is called multiple-precision arithmetic. In the paper, we consider multiple-precision iterative CG and PCG solvers for graphics processing units (GPUs) and evaluate their convergence rate. The preconditioned implementation uses a diagonal preconditioner, which is well suited for parallel computing. The residue number system is employed to support multiple-precision floating-point arithmetic. The matrix-vector product is implemented as a hybrid kernel, in which a double-precision matrix, represented in the CSR storage format, is multiplied by a dense multiple-precision vector. A two-stage algorithm is used to compute the parallel multiple-precision dot product. Experimental results with sparse matrices of different sizes show that higher arithmetic precision improves the convergence rate of iterative methods.