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 May 13th, 2012, 01:39 PM #1 Member   Joined: May 2012 Posts: 34 Thanks: 0 newton's method? can someone show me what newton method is? thank you Apply Newton’s method to the equation 1/x?a = 0 to derive the following reciprocal algorithm: xn+1 = 2xn ? ax2n http://www.texify.com/img/%5CLARGE%5...%5E%7B2%7D.gif (which enables a computer to find reciprocals without actually dividing). Then use this algorithm to compute 1/1.6984 correct to five decimal places.
 May 13th, 2012, 03:26 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: newton's method? We are given: $f(x)=\frac{1}{x}-a=0$ We compute: $f'(x)=-\frac{1}{x^2}$ Newtons method gives us the recursion: $x_{n+1}=x_n-\frac{f$$x_n$$}{f#39;$$x_n$$}=x_n-\frac{\frac{1}{x_n}-a}{-\frac{1}{x_n^2}}=x_n+x_n-ax_n^2=2x_n-ax_n^2$ Can you proceed to get the required approximation?
 May 13th, 2012, 03:39 PM #3 Member   Joined: May 2012 Posts: 34 Thanks: 0 Re: newton's method? not quiet, first time i learn about this, what to do next?
 May 13th, 2012, 04:54 PM #4 Member   Joined: May 2012 Posts: 34 Thanks: 0 Re: newton's method? ???
 May 13th, 2012, 05:53 PM #5 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: newton's method? What you do is make an initial guess, which is $x_0$. Then you use the formula to get $x_1$ which you then use to get $x_2$ and so on. As you progress you find $x_n$ converges to the value you are seeking. So, with: $a=1.6984$ and $x_0=0.5$ (you could begin with any positive initial guess) we have: $x_1\approx0.5754$ $x_2\approx0.588484964256$ $x_3\approx0.588789291433$ $x_4\approx0.588789448893$ Since the last two approximations agree to six places, the approximation rounded to five places is: $x\approx0.58879$
 May 13th, 2012, 07:16 PM #6 Member   Joined: May 2012 Posts: 34 Thanks: 0 Re: newton's method? Ok .. the question asking for Then use this algorithm to compute 1/1.6984 correct to five decimal places. why a=1.6984
 May 13th, 2012, 07:22 PM #7 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: newton's method? From $\frac{1}{x}-a=0$ we find: $a=\frac{1}{x}$ where $x=\frac{1}{1.6984}$ hence: $a=1.6984$
 May 13th, 2012, 11:02 PM #8 Member   Joined: May 2012 Posts: 34 Thanks: 0 Re: newton's method? yes! this is good

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### newtons method to find reciprocal

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