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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.