We can also compute the residual iteratively as ˆrk+1=ˆb−ˆAˆxk+1, which equals ˆb−ˆA(ˆxk+αkˆpk), or (ˆb−ˆAˆxk)−αkˆAˆpk, which gives ˆrk+1=ˆrk−ˆAˆpk,