Robust reduced-step Newton method

In the study of the performance of Newton s method for a simple 2-D system, we have seen that far away from the solution, the update steps are large and lie in erratic directions. The efficiency and robustness of Newton s method (and quasi-Newton variations such as that of Broyden) are improved dramatically through use of a reduced-step algorithm, in which only a fraction of the update vector is accepted. The full update vector is generated by solving the linear system 

This fractional step is chosen such that the norm of the function vector at the new estimate is smaller than at the old one, yielding the descent criterion 

Is there always some 0 such tliat die descent criterion using the 2-norm, v 2 = vv, 

Under what conditions will this method find a limiting point that is not a solution  

To answer the first question, we use the path integral relation 

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