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TRSM - theoretical stumbling block

A possible answer to this question is the following:

if M - the inverse of the triangular matrix L, is relatively insensitive to 
perturbations in L, then multiplying by M is safe.

for more details see for example

Reliable Solution of Tridiagonal Systems of Linear Equations -  Ilan Bar-On, 
Mauro Leoncini , SINUM 38, 4

lan Bar-On, Checking nonsingularity of tridiagonal matrices, ELA Volume 6 
(1999/2000), pp. 11-19. 

Ilan Bar-On