When one refers to the CG method, one often means the * linear*
Conjugate Gradient; that is, the implementation for the convex quadratic
form. In this case, minimizing
is equivalent to solving the * linear* system .
Consequently, the conjugate directions , as well as the step
lengths , can be computed in closed form. Below we sketch such
an algorithm from a given . We define the residual vector and use the vectors below
to denote the CG search vectors.

Note here that only a few vectors are stored, the product
is required but not knowledge (or storage) of * per se*, and the
cost only involves several scalar and vector operations. The value of
the step length can be derived in the closed form above
by minimizing the quadratic function
as a function of (producing
) and
then using
the conjugacy relation: for all **j<k** [45].

See the linear algebra chapter for further details and examples.