You calculate the factorization (QR, LU, or similar), but you never calculate the inverse of the factorization: A x = b A => QR # QR decomposition

You calculate the factorization (QR, LU, or similar), but you never calculate the inverse of the factorization: A x = b A => QR # QR decomposition Q R x = b R x = Q' b # Q is inverted by transposing x = backsubstitute(R, Q' b) Q was the transpose of it's inverse, and so you don't get any rounding error when you transpose it. R didn't need to be inverted because it was in a form where back substitution lets you apply the inverse of R to be without needing the inverse of R.