The quality of line search in these nonlinear CG algorithms is crucial
to preserve the mutual conjugacy property of the search directions
and to ensure that each generated direction is one
of descent. A technique known as * restarting* is typically used
to preserve a linear convergence rate by resetting to the
steepest descent direction, for example, after a given number of linear
searches (e.g., **n**). Preconditioning may also
be used as in the linear case to accelerate
convergence.

In sum, the greatest virtues of CG methods are their modest storage and
computational requirements (both order **n**), combined
with much better convergence than SD. These properties have made
them popular linear-solvers and minimization choices in many
applications, perhaps the only candidates for
very large problems.
The linear CG is often applied to systems arising from discretizations
of partial differential equations
where the matrices are frequently
positive-definite, sparse, and structured [20,25].