Analysis of the GCR method with mixed precision arithmetic using QuPAT

To verify computation results of double precision arithmetic, a high precision arithmetic environment is needed. However, it is difficult to use high precision arithmetic in ordinary computing environments without any special hardware or libraries. Hence, we designed the quadruple precision arithmetic environment QuPAT on Scilab to satisfy the following requirements: (i) to enable programs to be written simply using quadruple precision arithmetic; (ii) to enable the use of both double and quadruple precision arithmetic at the same time; (iii) to be independent of any hardware and operating systems.To confirm the effectiveness of QuPAT, we applied the GCR method for ill-conditioned matrices and focused on the scalar parameters α and β in GCR, partially using DD arithmetic. We found that the use of DD arithmetic only for β leads to almost the same results as when DD arithmetic is used for all computations. We conclude that QuPAT is an excellent interactive tool for using double precision and DD arithmetic at the same time.

► We confirm the effectiveness of QuPAT that enables to use DD arithmetic. ► We examined whether convergence of the Generalized Conjugate Residual (GCR) method could be improved with partial use of DD arithmetic. ► We found that convergence is improved by using DD arithmetic for computing both the scalar beta and the vectors p and q.