So lets say you have the following two polynomials:
$$f_1(x)=(x-1)(x-2)^2$$
and
$$f_2(x)=(x-1)^2(x-2)$$
And we want know whether Newton-Raphon's method will converge quadraticly or linearly for finding the root $x=2$. How would I go by doing this?
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You could conduct numerical experiments (if that is allowed in your course and you have access to computational tools which makes it attractive to do). – mathreadler Aug 08 '17 at 20:07
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I do know how you can perform the newton raphson method and see how the error changes after testing some starting point. However I was thinking there is some other easier and less time consuming way to check this since the solution to this problem states that $f_1(x)$ is linear because 2 is a double root and $f_2(x)$ is quadratic because 2 is a simple root, which I don't understand. – osk Aug 08 '17 at 20:24
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I don't remember all the details of Newton Rhapsons method, I mostly remember the practical methods to find stuff out, but if you are reading it as part of a course maybe they want a theoretical explanation instead of a practical method. Maybe you can ask your professor what approach would be acceptable. – mathreadler Aug 08 '17 at 20:29