(0) = b Ax (0) , p (0) = r (0) r (k) = hrE( x (k)) , d (k)i h Ad (k) , d (k)i d (k) = rE( x (k)) + || v (k)|| || v (k 1)||d (k 1) v (k) = rE( x (k)) = Ax (k) b x (k+1) = x (k) r (k) d (k) Iterations: x (k+1) = argmin E( x ) s.t. x x (k) 2 span(rE( x (0)) , . . . , rE( x (k))) Intuition: Conjugate Gradient Ax = b () min x 2Rn E( x ) = 1 2 h Ax, x i h x, b i Proposition: 8 ` < k, hrE( x k) , rE( x `)i = 0