i was trying to work with few problems for my computer oriented numerical method ...which i had in college ...

i am not in college anymore ... but i failed that subject ... i am trying to improve this maths field as much as i possibly can , because its a difficult subject for me .....

let met start with a few things i am familiar with ...

then we have simulaneous equations ...which looks like these ...

i think these two numerical methods can be applied to it if you have to deal with equations like these ...

These methods are for determining solutions of (systems of) nonlinear equations. The equations may involve transcendental functions, but for example also higher order polynomials. (Note that the system of three equations that you wrote down above consists of linear equations only.) Newton's method is a good default choice, but there are caveats. Nonlinear root finders may also occur as part of the numerical solution of other problems, for example involving differential equations.

polynomials with degree greater than one , which is not linear ...and when the usual factorisation methods wont work ??? for my understandings sake ...

for some reason .. i thought i could narrow it down to few types of equations and its numerical methods ...
but this subject again .. even though its just the three methods ... is still confusing ....

so this

will work for ...

non linear higher order polynomials ...??
transcendental functions
differentiation
intergration
differential equation ...

works for ...

non linear higher order polynomials ...??
transcendental functions
differentiation
intergration
differential equation ...

works for ...

non linear higher order polynomials ...??
transcendental functions
differentiation
intergration
differential equation ...

These are all polynomial expressions, for example we write ## x^2 - 3x + 2 = 0 ## and call it a second degree polynomial equation, or a quadratic equation. First degree polynomial equations are also called linear equations.

Note that these are all linear equations, and together we call them a system of linear equations, or a linear system.

Yes, as the titles imply these methods are used to find numerical solutions to linear systems.

Again as the titles imply these methods are used to find numerical solutions to non-linear systems. Quadratic systems e.g. ## x = 3t; y = 10t - \frac {9.8}2 t^2; y = 0 ## are non-linear, and so is any system that contains terms other than a variable multiplied by a constant e.g. ## x = \cos t; y = \sin t; t = 2 ##.

No, they are all for solving systems of equations i.e. working out what values of the variables solve all the constraints. (Numerical) methods for differentiation and integration solve different classes of problems, although if you look at Newton's method in your last link, this can also be used for solving differential equations.

There are many numerical methods for many different problems. Even when we restrict ourselves to systems of linear equations, there are various different methods that have different benefits and drawbacks, depending on the structure of the particular problem at hand.

I think you may benefit from a well-structured introductory book with many examples. Would you like that? I have my own favorites on numerical analysis, but maybe in this case it is better when others make some recommendations first. Or did your course already come with prescribed literature?

I enjoy https://www.amazon.com/Introduction-Numerical-Analysis-Kendall-Atkinson/dp/0471624896/ by Atkinson, but I'm not sure whether the level is appropriate for you. It does not give codes, but presents the basic algorithms clearly enough so you can do your own coding in the language of your choice. Atkinson, together with Han, has also written Elementary Numerical Analysis, but I don't know this book.

There is also the book by Cleve Moler, Numerical Computing with MATLAB, which is available for free. It is much more applied than Atkinson's book and is tied to the MATLAB environment, which is a good but non-free software package suitable for the exploration of numerical methods. Octave is a free alternative, but I don't know how well it goes together with Moler's book.

Many other textbooks exist, so I also encourage you to browse an online bookstore and have a look at the reviews. It depends a lot on your personal style.