Mathematics iteration

Question

Hi I recently was learning the newton raphson method but before I started learning that I came across a method for solving equations that cannot be solved using formulae or other methods called iteration.i don't understand how iteration works to solve for roots of equations for example the equation x^2-x-1 is set equal to 0 and rearranged in some way to make x the subject my question is how does plugging in a certain value of x in this equation converge to a root ?

Answer 1

By IVT, you are sure that the equation $$F(x)=x^2-x-1=0$$ has a root in the intervall $ \alpha $ $ [1,2] $.

There are many many other ways to write your equation in an equivalent form

$$x=x^2-1$$ or $$x=\frac{1}{x-1}$$ ...

or

$$x=x+2(x^2-x-1)$$

many and many

or in a general form $$x=g(x)$$

an iterative method is a repetitive computation, starting by a value $ x_0 $, near the root, and you find successively $ x_1 $, $x_2 $, ....

using the recursive formula $$x_{n+1}=g(x_n)$$

for example $$x_1=g(x_0)$$ $$x_2=g(x_1)$$ and so on.

To be sure that the sequence $ x_1,x_2,x_3,....$ converges to the root $ \alpha $, you must choose the function $ g $ which satisfies $$|g'(x)|\le K<1$$

when $ x $ is close to $\alpha$.

Newton-Raphson corresponds to the case

$$g(x)=x-\frac{f(x)}{f'(x)}$$