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I am really stuck on problems $9$ and $10$.

Here is what I tried to do:

I can use Lagrange's interpolation formula to get the approximating polynomial, however, in order to apply Newton Raphson's method to find roots, I must get exactly the function $f(x)$.Is it possible to obtain such function $f(x)$ or do I have to consider the approximating polynomial itself as $f(x)$?

Any help would be appreciated...

Thanks in advance...

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  • $\begingroup$ I think they are asking you to do it graphically. For example, they provide $x_0$ and want you to use the graph and the Newton-Raphson iteration to find $x_1, x_2, ...$. For example, $$x_1 = x_0 - \dfrac{f(x_0)}{f'(x_0)}$$ $\endgroup$ – Moo Aug 8 '17 at 17:09
  • $\begingroup$ @Moo:i can understand that.but,in order to find out the iterations ,i would need f(x) .please elaborate a bit more... $\endgroup$ – Abhishek Shrivastava Aug 8 '17 at 17:11
  • $\begingroup$ What is the function value at $x_0$? Don't you know that value from the grapgh of the function they provide? $\endgroup$ – Moo Aug 8 '17 at 17:12
  • $\begingroup$ To do it graphically, just sketch a tangent line at each of the iteration points, follow it to its root, then make that your new "$x_0$" and repeat. $\endgroup$ – Ian Aug 8 '17 at 17:13
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Question 9 asks you to to find the root graphically using the Newton-Raphson method. I would physically draw the tangents to the curve, to show how the roots are (approximately) determined in a few iterations. The answer is, of course, ~0.2 and 0.6, depending on where you start the algorithm, but the start is given to you.

Since you don't actually know the function plotted, there is little use in interpolating or approximating the plot. This question seems to me to be primarily about showing that you have understood the way the methods works, rather than about calculating a highly accurate solution.

The same goes for question 10.

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  • $\begingroup$ :in problem no 9,i can clearly see the roots are .2 and .6.but how does drawing tangents give info regarding roots.I know it might be something really basic to ask,but,it will really be nice if you can explain that $\endgroup$ – Abhishek Shrivastava Aug 8 '17 at 17:26
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    $\begingroup$ Your teacher expects you to produce a plot similar to the following: link The ability to construct this plot demonstrates that you have understood the basis of the method. If you cannot create this plot, which is a graphical representation of Moo's answer, then I would suggest you read up further on the Newton-Raphson method. $\endgroup$ – Kobs Aug 8 '17 at 17:31
  • $\begingroup$ :but by the given initial condition,first iteration is point is at infinity...how to proceed now $\endgroup$ – Abhishek Shrivastava Aug 8 '17 at 17:51
  • $\begingroup$ At $x_0=0$ we have $f(x_0)=-0.5$ and $f'(x_0)\approx \Delta y/\Delta x =0.5/0.6=5/6$, thus we get $$x_1 = 0 - (-0.5/(5/6))=0.6$$ Next, $f(x_1)=0$, $f'(x_1) \approx -1/0.2=-5$, leads to $$ x_2 = x_1 - f(x_1) /f'(x_1) = 0.6 - 0 = 0.6\ \text{ (again)}$$ and we see that $x_3 =x_2$... $\endgroup$ – AD. Jan 5 '18 at 8:50
  • $\begingroup$ Also, in the above $\Delta y/\Delta x$ are estimated using a ruler... $\endgroup$ – AD. Jan 5 '18 at 8:51

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