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.
1)) is used to approximate f(x), and it crosses the x-axis at x 2 = 1:41 6, which is already very close to the exact solution. Example Newton’s Method can be used to compute the reciprocal of a number a without perform-3
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