# newton-raphson method in numerical analysis [closed]

excuse my weak math language and skill, came here to look for help :) any answer will be much appreciated and useful to me.

we have to do a research on the newton raphson method in numerical analysis, but i have no clue what the method do or even how to use it, i looked up some stuff on the internet but they weren't so useful to me(not simple enough for me to understand), cause i'm not that great at math so i came here asking for help.

what i need to do is find a problem, explain it and its solution and a bit explanation of the method itself and to explain if its well posed? and what does also well posed mean ?. and also write about the history of the method.

sorry if i'm asking too much or if i'm asking the wrong question, this place seems full of people who know stuff about math :), any answer will be appreciated and helpful to a fellow student xD, even if you direct me to places with simple information that a normal guy can understand that will be helpful as well.

thanks in advance, and hope u have a nice day :)

First thing i checked was wikipedia, http://en.wikipedia.org/wiki/Newton%27s_method, and i understood its a method to find approximations for the roots of a function. Like $f(x)=0$. If i got a normal function like $ax^2+bx+c = 0$ i think i can find the roots of the function.

1. But i don't understand the concept of approximations. Do i have the function? And i cant solve it so i have to search for approximations. And do i know how many roots the function has? Is it polynomial?

Second website i checked was http://math.ubc.ca/~anstee/math104/104newtonmethod.pdf its a pdf article which made stuff easier and harder at the same time.

1. Like what is a well defined function?
2. And what is the first guess. ? And how do i iterate guesses ? And what is the h ( difference from real root ) !
3. And what kinds of functions is it good for ? Like all kind of functions ?
4. And how can i make a second guess after the first one. And how do i know if its better and closer to the real root ? Or do i just say $x_0$ and $x_1$ is that enough ? Is it theoretical ? Or does provide real numbers ?

Third place i checked was http://jstor.org/stable/2973939?seq=1#page_scan_tab_contents and its an article of the american math journal . They showed the calculation that newton did and how he noticed that its failing because of wrong guess is far from real root.

1. But still i didn't understand how he did the calculations and how the method is actually used . But here i was looking for the methods history and how it was developed and not how to use it.

2. And what is numerical solving ? Is different from algebraic solving ?

## closed as unclear what you're asking by GAVD, JonMark Perry, Claude Leibovici, Shailesh, GNUSupporter 8964民主女神 地下教會Jan 22 '18 at 12:40

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Please cite one or two of the sites that you looked at and comment on what you understand and don't understand. – LutzL Jan 20 '18 at 16:19
• Sure thing man. First thing i checked was wikipedia. en.m.wikipedia.org/wiki/Newton%27s_method and i understood its a method to find approximations for the roots of a function. Like f(x)=0. If i got a normal function like ax^2+bx+c = 0 i think i can find the roots of the function. But i dont understand the concept of approximations. Do i have the function ? And i cant solve it so i have yo search for aproximations. And do i know how many roots the function has ? Is it polynomial ? Second website i checked was math.ubc.ca/~anstee/math104/104newtonmethod.pdf its a pdf articl – Majd Mahajenah Jan 20 '18 at 19:02
• Wich made stuff easier and harder at the same time . Like what is a well defined function ? And what is the first guess. ? And how do i iterate guesses ? And what is the h ( difference from real root ) ! And what kinds of functions is it good for ? Like all kind of functions ? And how can i make a second guess after the first one. And how do i know if its better and closer to the real root ? Or do i just say x0 and x1 is that enough ? Is it theoritical ? Or does provide real numbers ? – Majd Mahajenah Jan 20 '18 at 19:09
• Third place i checked was jstor.org/stable/2973939?seq=1#page_scan_tab_contents and its an article of the american math journal . They showed the calculation that newton did and how he noticed that its failing because of wrong guess is far from real root. But still i didnt understand how he did the calculations and how the method is actually used . But here i was looking for the methods history and how it was developed and not how to use it. – Majd Mahajenah Jan 20 '18 at 19:16
• And what is numerical solving ? Is different from algebric solving ? Im sorry again i know i sound stupid. But english is my 3rd language and math terms are a bit hard for me as well – Majd Mahajenah Jan 20 '18 at 19:18

ad 7. Numerical solving produces approximations to the solution of a problem. Algebraic solving provides a (finite) formula that allows to directly compute the solution (where the computation then again introduces error, but often the result is the best approximation available for the tools used.)

ad 1. The approximation of the function used for the Newton method is the linear Taylor polynomial $f(x)\approx f(x_0)+f'(x_0)(x-x_0)$. The solution of the linear equation is then easy to obtain.

ad 2. & 4. A "well-defined function" is a function that allows the application of the method. Theoretically you need differentiability (twice continuously differentiable for fast convergence). Practically you need the function in some algorithmic form that also allows the easy calculation of the derivative.

ad 3. There is a difference between global and local methods. Newton is a local method, that is, you expect that the first point is miraculously already close to the root. Global methods like random or systematic search of the domain can use the local method to refine their results. This also means that in the application of Newtons method it is recommended to observe the decline in the function value and report in failure if it does not happen, or not fast enough.

ad 3. & 6. $h_k=-f'(x_k)^{-1}f(_k)$ is the Newton step. You iterate by finding the linear approximation in the new point.

ad 6. Newton, Simpson, Raphson originally computed roots of cubic polynomials. Newton started by computing the full shifted polynomial $p_k(h)=p(x_k+h)$, that is, he computed all coefficients of $p_k$ and then disregarded the quadratic and cubic coefficients. Simpson realized that one only needs the value of the first derivative and Raphson found the formulation for general functions (for whatever was understood as function at that time, about 1670-1720).

• cannot thank you enough man. this great you are the best. thanks again :) – Majd Mahajenah Jan 21 '18 at 12:05