# Truncated Newton Methods

The line search and trust-region techniques are suitable if the number of variables $$n$$ is not too large, since the cost per iteration is of order $$n^3$$. Implemented algorithms for problems with a large number of variables tend to use iterative techniques for obtaining a direction $$d_k$$ in a line-search method or a step $$s_k$$ in a trust-region method. These techniques are usually called truncated Newton methods because the iterative technique is stopped (truncated) as soon as a termination criterion is satisfied.
For example, some algorithms use a line search method in which the direction $$d_k$$ satisfies
$\parallel \nabla^2 f(x_k) d_k + \nabla f(x_k) \parallel \leq \eta_k \parallel \nabla f(x_k) \parallel$ for some $$\eta_k \in (0,1)$$.