  Singly-TASE operators for the numerical solution of stiff differential
equations were proposed by Calvo et al. in J.Sci. Comput. 2023 to reduce the
computational cost of Runge-Kutta-TASE (RKTASE) methods when the involved
linear systems are solved by some $LU$ factorization. In this paper we propose
a modification of these methods to improve the efficiency by considering
different TASE operators for each stage of the Runge-Kutta. We prove that the
resulting RKTASE methods are equivalent to $W$-methods (Steihaug and
Wolfbrandt, Mathematics of Computation,1979) and this allows us to obtain the
order conditions of the proposed Modified Singly-RKTASE methods (MSRKTASE)
through the theory developed for the $W$-methods. We construct new MSRKTASE
methods of order two and three and demonstrate their effectiveness through
numerical experiments on both linear and nonlinear stiff systems. The results
show that the MSRKTASE schemes significantly enhance efficiency and accuracy
compared to previous Singly-RKTASE schemes.
