I have a question about Newton's Method

[gungun8799]

Hi, guys, I have a questions to ask you about Newton's Method.
I knew that Newton's Method was used to estimate the true value of the root of the real number by analysis the graph of that particular function, but I just wonder that do I have to plot the graph of the function everytime I want to use Newton's method, my understanding was 'yes', I think that ploting the graph help me to know "what value" will be used to be x1 or the approximate answer I was looking for, I'm I right or wrong ?

suppose the problem wants me to find x intersect of f(x) = x^7-1000 by using Newton's Method

Do I have to plot this graph first in order to approximate the value of x1
Thank you so much, sir !

 

[Ray Vickson]

Hi, guys, I have a questions to ask you about Newton's Method.
I knew that Newton's Method was used to estimate the true value of the root of the real number by analysis the graph of that particular function, but I just wonder that do I have to plot the graph of the function everytime I want to use Newton's method, my understanding was 'yes', I think that ploting the graph help me to know "what value" will be used to be x1 or the approximate answer I was looking for, I'm I right or wrong ?

suppose the problem wants me to find x intersect of f(x) = x^7-1000 by using Newton's Method

Do I have to plot this graph first in order to approximate the value of x1
Thank you so much, sir !

No, usually plotting is out of the question. For example, some optimization codes may apply Newton's Method as subroutines, but cannot look at plots because they are inanimate computer programs. For certain classes of functions, we know that Newton will always work, starting from any point; for other types of functions, we know that Newton can fail spectacularly if the wrong initial point is chosen. Usually, computer codes do not apply "pure" Newton, but rather, a form of "safeguarded Newton", that tries to guard against failure. These are safer, but slower than pure Newton methods.

However, when solving problems one-by-one manually (although using software to assist you) a plot is usually advisable.

Your example function f(x) = x^7-1000 is "strictly convex", at least in the interval x >= 0, so as long as roundoff errors do not give you negative x values, you should get rapid convergence from *any* positive starting point (try it and see).

RGV

 

[gungun8799]

Thank you very very much Mr.Ray, your post helps me a lot...^_^