How can I find all solutions using an iteration formula?

Question

Take the equation $x^2+7x+10=0$, which has roots $-2$ and $-5$.

When I use the iteration formula $x_{n+1}=\dfrac{-10}{x_n+7}$, I always converge on $-2$ but not on $-5$.

I have tried starting values of $-6, -4, -1, 1, 2$ and many others, but I only converge on $-2$.

Is there something wrong with my iteration formula or starting values?

Answer 1

An iterative formula is not a 100% sure method of finding a solution. This is a method based on approximation.

The kind of iteration you are using is a fixed point iteration. It will only work with attractive points, not repulsive points.

Answer 2

There is nothing wrong, when considering sequences defined by $x_{n+1}=f(x_n)$ to study if $l$ such that $f(l)=l$ is attractive or repulsive you have to consider $|f'(l)|$:

• If $|f'(l)|<1$ this is an attractive point.
• If $|f'(l)|>1$ this is a repulsive point (and there is no hope to converge to $l$ if $x_0 \neq l$).
• If $|f'(l)| =1$ you have to study with more precision the function.

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Here $f'(x)=\frac{10}{(x+7)^2}$ so:

• $f'(-2)=\frac{10}{25}\in (-1,1)$
• $f'(-5)=\frac{10}{4} >1$

so $-2$ is attractive but $-5$ is repulsive.

Answer 3

One way of looking at this is to compare the curves $y=x$ (the orange line below) and $y=\dfrac{-10}{x+7}$ (the blue line). The solutions are where the two lines intersect, at $x=-2$ and $x=-5$

In effect, an iteration can be considered as moving vertically from a point on the orange line to a point on the blue line and then horizontally to a point on the orange line.

Two such sequences of iterations are illustrated to demonstrate moving towards the solution at $-2$ and away from the solution at $-5$. As Delta-u says, this is because the slope of the blue curve at $-5$ is steeper than that of the orange curve, with the position reversed at $-2$. (If the slopes were of opposite signs then the iterations would spiral around the solution, and whether they were attracted or repelled would depend on the relative magnitudes of the slopes)