where is the hessian of at xd :diag ( /( :[ -iis an ( nmatrixand )is an diagonal matrix#- : - furtherwe define hn :( be- )- ( )- using this notation and applying the matrix blockwise inversion formula ( )we obtain the inverse of the matrix of jacobihn #- - - hn -hn be - - bhn - - bhn be- be -dhn dhn dhn thereforethe search direction in newton' root-finding method is given byf (xb dx - / dx - - - / (dx swhere dx :-( )- ( / dx and we have assumed that is positive-definite matrix if at any step of the iteration the matrix fails to be positive-definitethen newton' root-finding algorithm may fail to converge thusany practical implementation will have to include fail-safe feature to guard against this possibility in summaryfor given penalty parameter we can locate the approximate nonnegative minimizer of usingfor examplethe version of the newton-raphson rootfinding method given in algorithm interior-point method in practiceone needs to choose sufficiently small value for nso that the output xn of algorithm is good approximation to xargmin > (xalternativelyone can create decreasing sequence of penalty parameters and compute the corresponding solutions xn xn of the penalized problems in the so-called interiorpoint methoda given xni is used as an initial guess for computing xni+ and so on until the approximation to the minimizer xargmin > (xis deemed accurate here is an matrix of zeros and is the identity matrix