About manipulation of equation, when do you add and when do you move terms? I'm reading a book related to numerical methods (Chapra, Canale) and on the topic of fixed point iteration, (copying from the book) we have to rearrange the function $\operatorname{f}(x) = 0$ so that $x$ is on the left side of the equation:
$x= {g}(x)$
Example:
Manipulation for the following
$x^2-2x+3=0$ will become $x = \frac{x^2+3}{2} $
and adding $x$ to both sides for the following
$sin(x) = 0$ will become $x = sin(x) + x$
--
Looking at them they somehow make sense, but for the first equation I would say that I could just add $x$ on both sides so we'd have an equation equal to $x$.
But now $x^2-x+3=x$ as is seems wrong; I could move the $x$ term on the left side to the right to produce a result same as above, but why should I when I already have $=x$?
I still don't understand when I should do an addition to both sides or just manipulating the placement of the term.
Thank you
 A: Here are some considerations for choosing one way rather than another to rewrite an equation into fixed-point form.
(1) Does the rewriting introduce any extra ("artifact") roots?
(2) Does the resulting function $g(x)$ serve as a "contraction mapping"?
Your book, Chapra and Canale, "Numerical Methods for Engineers"(?), will likely have some information on these ideas, but here are some quick thoughts.
The fixed-point iterations can always be defined from chosen starting point $x_0$:
$$ x_{k+1} = g(x_k), k = 0,1,2,3,\ldots $$
but whether these converge to a root of $f(x) = 0$ very much depends on how $g(x)$ is chosen and (to a lesser extent) on the choice of the starting point.
If $\{x_k\}$ converges to $x_*$ and $g(x)$ is continuous in a neighborhood of $x_*$, then $x_*$ is a solution of $x = g(x)$. Whether it is also a solution of the original equation $f(x)=0$ will depend on if the rewriting steps have introduced any artifact roots, i.e. the roots of $x=g(x)$ not satisfying the original equation $f(x)=0$.  Typical ways this can happen are squaring both sides of equation or multiplying both sides by an expression that introduces a new root.  Adding $x$ to both sides will not introduce an artifact root, because it is a reversible step (one can subtract $x$ from both sides).
The deeper property desired for $g(x)$ is that of being a contraction mapping in a neighborhood of the root $x_*$ we are seeking.  For some historical background, see the excellent Answer there by Willie Wong.  In any case the idea is that if $\{x_k\}$ converges, the terms should (eventually) be close to the limit $x_*$ and hence close to each other.  The contraction mapping property guarantees that this will happen because for some constant $0\lt c \lt 1$:
$$ |x_{k+1} - x_k| = |f(x_k) - f(x_{k-1}| < c |x_k - x_{k-1}| $$
In other words the "gaps" between terms will eventually shrink by at least a factor $c$ with each iteration.
In most cases a good way to check for the contraction mapping property is by showing the function $g(x)$ is differentiable in a neighborhood of $x_*$ and has a derivative less than one in absolute value.
The Mean Value Theorem equates $f(x) - f(y)$ to $f'(z)\cdot (x-y)$, for some $z$ between $x$ and $y$.  Hence if $|f'(z)|\lt 1$, we will have $|f(x)-f(y)|\lt |x-y|$.  In higher dimensions (functions of more than one argument) a similar contraction analysis involving the Jacobian (rather than the ordinary derivative) can be applied.
