Another method for solving a system Ax = b when A is nonsingular and symmetric is the conjugate...

Another method for solving a system Ax = b
when A is nonsingular and symmetric is the conjugate residual method. In this
method the direction vectors are constructed to be an A2-orthogonalized version
of the residuals rk = b − Axk. The error function E(x) = |Ax − b| 2
decreases monotonically in this process. Since the directions are based on rk
rather than the gradient of E, which is 2Ark, the method extends the simplicity
of the conjugate gradient method by implicit use of the fact that A2 is
positive definite. The method is