[imtringued]

There is a mild lack of context here. If you have a single vector and want to solve LUx=b, you actually have matrix vector multiplication. It's the batched LUX=B case, where X and B are matrices where you need matrix matrix multiplication.

For those who don't know. One of the most useful properties of triangular matrices is that the block matrices in the diagonal blocks are triangular matrices themselves. This means you an solve a subset of the x using the first triangular block. Since the sub-x vector is now known, you can now do a forward multiplication against the non-triangular blocks that take your sub-x vector as input and subtract them from the b vector. This is the same as if you removed one of the columns or rows in the triangular matrix. The remaining matrix stays triangular, which means you can just keep repeating this until the entire matrix is solved.