User:Elvek

This is my new user page on Wikipedia. It contains some exercises of writing in Latex and some proposed changes/additions to Wikipedia pages, particularly about numerical methods.

”Five interesting facts about Wikipedia/Wikiversity websites are:


 * 1) It is an open source platform where everybody can contribute an appropriate study material
 * 2) A user can create an account relatively easily, but even without an account a user can edit existing pages
 * 3) Beginners can exercise editting pages using the Sandbox. This is the place where one can write related or unrelated material. Since the pages in the "Sandbox" are deleted regularly, it is not important to be precise here (unless the user wants to transfer the page as a regular page from the Sandbox).
 * 4) A file can be uploaded to the website and a link created using  [[Media:Filename]] syntax.
 * 5) The symbols/formulae can be added by typing textual commands (unlike with the MS Word equation editor).

An example of a formula follows:


 * $$Pressure=\chi(\omega)\left(\mathcal{P}\int\limits_{-\infty}^{\infty}{F_x(\alpha,x,y,z)\over\alpha-\alpha_1}d\alpha\right)^{\nabla\left({\sum\limits_{k=0}^{k=\infty}{\frac{\partial G_k(x,y,z)}{\partial x}}}\right)}$$

Newton's method convergence rate

(a) By reading the page on the Newton's method, one can notice that there is no proof of the convergence rate. Although there are many ways in which this method can fail, the main advantage of this method over many other methods is that it's convergence rate is quadratic. It would be convenient to have a proof directly displayed or attached as a file via a link. This would also help students dealing with this or similar subjects to see the general approach how a convergence rate of a numerical method can be analyzed, such that similar analysis can be used to analyze other numerical methods.


 * For equation numbering, see w:Help:Displaying_a_formula.
 * Only number equations that you actually refer to.
 * Put in cross links to topics you mention, like Taylor's Theorem

(b)

Proof of quadratic convergence for the Newton's iterative method
The function $$\text{f(x)}$$ can be represented by a Taylor series expansion about a point that is considered to be close to a root of f(x). Let us denote this root as $${\alpha}$$. The Taylor series expansion of f(x) about an $${x_n}\,$$ is:

For $$x=\alpha\,$$; $$f(x)=f(\alpha)=0\,$$, since $$x=\alpha\,$$ is the root. Then, ($$) becomes:

Using the Lagrange form of the Taylor series expansion remainder
 * $$ R_m=\frac 1 {(m+1)!}f^{(m+1)}(\xi)(\alpha-x_0)^{(m+1)}, \text{ where m=1 and } \xi\in[x_0,\alpha] $$

the equation ($$) becomes
 * $$ 0=f(x_0)+{f^\prime(x_0)}\left(\alpha-x_0\right)+\frac 1 {2!}{f^{\prime\prime} (\xi)}\left(\alpha-x_0\right)^2 $$

It should be noted that the point $$x_0$$ stands for the initial guess, but the same form of expansion is valid for Taylor series expansion about an arbitrary point $$x_n\,$$ obtained after n iterations

By dividing equation ($$) by $$ f^\prime(x_n)\,$$ and rearranging the expression, following can be obtained:

Now, the goal is to bring in the new iteration n+1 and relate it to the old one n. At this point, the Newton's formula can be used to transform the expression ($$). The Newton's formula is given by

The intention is to eliminate the (generaly unknown) root $$\alpha$$ from ($$) via using formula ($$) and get the error terms involved in the relation. Equation ($$) can be adjusted for expression ($$):
 * $$ \underbrace{\underbrace{\frac {f(x_n)}{f^\prime(x_n)}-x_n}_{-x_{n+1}}+\alpha}_{\epsilon_{n+1}}=-\frac 1 {2!}\frac {f^{\prime\prime} (\xi)}{f^\prime(x_n)}\underbrace{\left(\alpha-x_n\right)^2}_{{\epsilon^2}_{n}} $$

That is,

After taking absolute value (scalar function of a scalar variable is considered here) of both sides of ($$), following is obtained:

Equation ($$) shows that the convergence rate is quadratic if following conditions are satisfied:


 * 1) $$f'(x)\ne0; \forall x\in I \text{, where }I \text{ is the interval }[\alpha-r,\alpha+r] \text{ for some } r \ge \left\vert(\alpha-x_0)\right\vert;\,$$
 * 2) $$f''(x) \text{ is finite },\forall x\in I; \,$$
 * 3) $$ x_0 \,$$ sufficiently close to the root  $$ \alpha \,$$

The term sufficiently close in this context means the following:

(a) Taylor approximation is accurate enough such that we can ignore higher order terms,

(b) $$\frac 1 {2}\left |{\frac {f^{\prime\prime} (x_n)}{f^\prime(x_n)}}\right |<C\left |{\frac {f^{\prime\prime} (\alpha)}{f^\prime(\alpha)}}\right |, \text{ for some } C<\infty,\,$$

(c) $$C \left |{\frac {f^{\prime\prime} (\alpha)}{f^\prime(\alpha)}}\right |\epsilon_n<1, \text{ for }n\in \Zeta ^+\cup\{0\} \text{ and }C \text{ satisfying condition (b) }.\, $$

Finally, ($$) can be expressed in the following way:
 * $$ \left | {\epsilon_{n+1}}\right | \le M{{\epsilon}^2}_n \, $$

where M is the supremum of the variable coefficient of $${\epsilon^2}_n \, $$ on the interval $$I\,$$ defined in the condition 1, that is:

$$M=\underset{x \in I }{sup} \frac 1 {2}\left |{\frac {f^{\prime\prime} (x)}{f^\prime(x)}}\right |. \,$$

The initial point $$x_0 \,$$ has to be chosen such that conditions 1 through 3 are satisfied, where the third condition requires that $$M\left |\epsilon_0 \right |<1.\,$$