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Notes of 4.8 Newton's Method

Notes of 4.8 Newton's Method - Iteration Fonnula Let f be a...

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Veryfewequations ofthe form f(x) = 0 canbe solvedexactly, sowe must often settle for approximate solutions. Newton's Method isamethod for findingapproximate solutions for f(x) = 0, one atatime. relyingon the following idea: They-coordinates 011 thetangent line 10 a curve closely approximate y-coordinates on thecurve itself, near thepoint of tangency. Suppose we have afunctionfand we want to find the roots of f. That is,we wishto solve f(x) == 0, or equivalently, findthex.•intercepts ofthe graph of f To use Newton's Method, you must have aninitial guess, .Yo, for aparticular root. If Xo isnot given, look at thegraph off andchoose aconvenient value nearthe desired root. The nextguess, xl, willoccur atthe intersection ofthex-axiswith the tangent line to Y == lex) at (xo,f(xo». Once we have xI \ we repeat the process to get x2 from Xl, repeat again to get X3 from x2, and soon. This iterative process is generalized inthe"fact" below: Fact: (Newton's Iteration Fonnula) Let
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Unformatted text preview: Iteration Fonnula) Let f be a function and let Xo bean initial guess for a root. For /l ~ 0, Horizontal Tangent Lines: Although it is unlikely, it is possible that at some guess X n , f'(x n ) == O. Of course, at this point Newton's Iteration Formula becomes undefined and the process fails. In graphical terms, the tangent line at x n is horizontal, and so it never intersects the x~axisto produce the next guess! Near-Horizontal Tangent Lines: If the tangent line at x n is near-horizontal, the x-intercept of the tangent line could take us far away from the desired root. At the very least, it will take several iterations to get back to the desired root. What's worse, fmay have a second root out near this far-away x-intercept. If this is the case, Newton's method will work beautifully, but will find tlus second root instead of the desired root!...
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