# Matrix multiplication on an upper/lower triangular matrix

I am a statistician working on DNA datasets. I am writing an R package, and I will routinely need to perform matrix multiplication on upper-triangular (sparse) matrices that have hundreds-of-millions of columns/rows.

I will be multiplying these matrices by low-pass and high-pass filters. For example:

Note that when multiplying the right and center, only the first two rows of the 'a' matrix need to be multiplied by the first column of the down-sampler, as the rest all have zeros in the first two entries.

Similarly, when premultiplying the down-sampler with the 'A' matrix, only the first row needs to be multiplied by the first column, as subsequent rows will be multiplying by zeros only.

Clearly when dealing with massive matrices, these unnecessary operations will add time to the calculation that could otherwise be avoided.

My question is: Is there a 'Forward multiplication' function for upper triangular matrices in either the R packages Matrix or c++ (which I can implement in Rcpp) that will avoid the unnecessary row/column operations that are not needed?