Order of matrix multiplication

[Bernardo Sulzbach]

I noticed today that Octave always performs multiplication from left to 
right.

This is, as one could expect, properly documented:


    -- Built-in Function: mtimes (X1, X2, ...)
        Return the matrix multiplication product of inputs.

        This function and x * y are equivalent.  If more arguments are
        given, the multiplication is applied cumulatively from left to
        right (...)


This is evidently not the best solution from a performance point of view 
if a multiplication substantially reduces the size of the product.

Locally, profiling
    m * m * m * m * x
against
    m * (m * (m * (m * x)));
when the size of m is [2048 2048] and the size of x is [2048 1] produces 
the following results

    Took 9.752126 seconds when using the default order.
    Took 0.035902 seconds when going from right to left.
    Maximum difference is 0.000732422.

I may be oversimplifying the analysis of numerical error here, but I 
would say that the second solution introduces less error as the number 
of floating-point operations is much smaller.

---

Understanding that trying to determine the "best" order of matrix 
multiplication during execution may not be a trivial task, I still 
wonder why Octave approach is so naive.

Additionally, would there be any function out there which I could use 
instead of mtimes which performed some analysis on the dimensions of the 
input matrices and optimized the order of multiplication without 
requiring me to do it manually?