Acceleration of a preconditioning method for ill-conditioned dense linear systems by use of a BLAS-based method

We are interested in accurate numerical solutions of ill-conditioned linear systems using floating-point arithmetic. Recently, we proposed a preconditioning method to reduce the condition numbers of coefficient matrices. The method utilizes an LU factorization obtained in working precision arithmetic and requires matrix multiplication in quadruple precision arithmetic. In this note, we aim to accelerate the preconditioning method from a practical point of view. For this purpose, we apply a more efficient method of accurate matrix multiplication based on BLAS in the preconditioning method. We demonstrate excellent performance of the BLAS-based preconditioning method by numerical experiments.