Transcription of Newton’s Method
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Newton’s Method
Jim LambersMAT 419/519 Summer Session 2011-12 Lecture 9 NotesThese notes correspond to Section in the s MethodFinding the minimum of the functionf(x), wheref:D Rn R, requires finding its criticalpoints, at which f(x) =0. In general, however, solving this system of equations can be quitedifficult. Therefore, it is often necessary to usenumerical methodsthat compute anapproximatesolution. We now present one such Method , known asNewton s Methodor :D Rn Rnbe a function that is differentiable onD. As it is a vector-valued function,it has component functionsgi(x),i= 1,2, .. , n, and thus we haveg(x) = g1(x)g2(x) gn(x) ,x s Method is aniterativemethod that computes an approximate solution to the systemof equationsg(x) =0. The Method requires an initial guessx(0)as input. It then computessubsequent iteratesx(1),x(2),..that, hopefully, will converge to a solutionx ofg(x) = idea behind newton s Method is to approximateg(x) near the current iteratex(k)by afunctiongk(x) for which the system of equationsgk(x) is easy to solve, and then use the solutionas the next iteratex(k+1), after which this process is repeated.
Newton’s Method is an iterative method that computes an approximate solution to the system of equations g(x) = 0. The method requires an initial guess x(0) as input. It then computes subsequent iterates x(1), x(2), ::: that, hopefully, will converge to a solution x of g(x) = 0.
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