  We propose a practical implementation of high-order fully implicit
Runge-Kutta(IRK) methods in a multiple precision floating-point environment.
Although implementations based on IRK methods in an IEEE754 double precision
environment have been reported as RADAU5 developed by Hairer and SPARK3
developed by Jay, they support only 3-stage IRK families. More stages and
higher-order IRK formulas must be adopted in order to decrease truncation
errors, which become relatively larger than round-off errors in a multiple
precision environment. We show that SPARK3 type reduction based on the
so-called W-transformation is more effective than the RADAU5 type one for
reduction in computational time of inner iteration of a high-order IRK process,
and that the mixed precision iterative refinement method is very efficient in a
multiple precision floating-point environment. Finally, we show that our
implementation based on high-order IRK methods with embedded formulas can
derive precise numerical solutions of some ordinary differential equations.

Numerical Analysis