I have a closed set of 4 linear matrix (3x3) transforms. Let's name them (A,B,C,D). Closed set means that $D*C*B*A=E$, where $E=eye(3)$. Their numeric representation is known from experiment and, therefore, is subject to some error. Of course, knowing all 4 elements of the closed set is excessive, and the errors may lead to non self-consistency, when the above equation is not satisfied, so I would like to use this extra information to compensate for the errors.
For my purposes I need to calculate relative transforms from element #0 to all other elements, which are given by these transforms. Because of the errors this is ambiguos, for example:
for (0→0) transform I could use both $E$ and $D*C*B*A$;
for (0→1) transform: $A$ and $B^{-1}*C^{-1}*D^{-1}$;
for (0→2): $B*A$ and $C^{-1}*D^{-1}$, finally
for (0→3): $C*B*A$ and $D^{-1}$
I would like to use some optimization method or other algorithm to ensure that my transforms had possibly smaller error and were self-consistent, i.e. didn't depend on direction of traverse.
Thanks in advance