Eliminating ϕ10 does not interact with eliminating ϕ14 and can be carried out with SIMD style parallelism. Novel placement of data in RAM is probably required. After the first three groups have been eliminated, however, the last group presents the same recurrence dilemma as we saw initially. At least we have solved 3/4 of the problem. We can declare victory or notice that the remaining problem is exactly equivalent to the original problem, except for being only 1/4 as big. We can recurse.
After the 25 unknowns have been computed, recursively or recurrently, we must then compute the other 75 values, which is again appropriate for normal SIMD.