Notes of 4.8 Newton's Method

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

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Very few equations of the form f(x) = 0 can be solved exactly, so we must often settle for approximate solutions. Newton's Method is a method for finding approximate solutions for f(x) = 0, one at a time. relying on the following idea: They-coordinates 011 the tangent line 10 a curve closely approximate y-coordinates on the curve itself, near the point of tangency. Suppose we have a functionfand we want to find the roots of f. That is, we wish to solve f(x) == 0, or equivalently, find the x.•intercepts ofthe graph of f To use Newton's Method, you must have an initial guess, .Yo, for a particular root. If Xo is not given, look at the graph off and choose a convenient value near the desired root. The next guess, xl, will occur at the intersection of the x-axis with 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 so on. This iterative process is generalized in the "fact" below: Fact: (Newton's
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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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This note was uploaded on 12/12/2011 for the course MAC 2311 taught by Professor Teague during the Spring '11 term at Santa Fe College.

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