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2.5.4 Truncated Newton     continued...

The PCG process can be terminated when either one of the following conditions is satisfied: (1) The residual is sufficiently small, (2) the quadratic model of eq. (34) is sufficiently reduced, or (3) a direction of negative curvature is encountered (i.e., ), possible since may not be positive-definite. Two effective truncation tests monitor the relative residual norm (RT) [18] and the decrease of the quadratical model (QT) [47].

The inner loop of a TN algorithm at Newton step k (step 2 of algorithm 2.4) can then be sketched as follows. For clarity, we omit the subscript k from , , , , and q. The sequence of vectors denotes the PCG iterates for , and a small positive number for the negative curvature test, such as , is chosen, along with appropriate values of or (for truncation), around 0.5.