Method
Newton's Method attempts to construct a sequence from an initial guess that converges towards such that . This is called a stationary point of .
The second order Taylor expansion of function around (where ) is:, attains its extremum when its derivative with respect to is equal to zero, i.e. when solves the linear equation:
(Considering the right-hand side of the above equation as a quadratic in, with constant coefficients.)
Thus, provided that is a twice-differentiable function well approximated by its second order Taylor expansion and the initial guess is chosen close enough to, the sequence defined by:
will converge towards a root of, i.e. for which .
Read more about this topic: Newton's Method In Optimization
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