Understanding the False Position Method and Its Iterations

[ziad1985]

I'm tying to use the false position method to find the root r of f(x)=0, between 2 points a0 and b0.
I have [a(i),b(i)] enclose in it f(x)=0.
There is something I don't get, if the number of iteration i goes to infinity, why the length of [a(i),b(i)] doesn't unnecessary goes to 0 ?
I understand that this have to do with the f(a0) et f(b0) the first 2 points chosen.
can somebody explain this ?

 

[HallsofIvy]

I'm tying to use the false position method to find the root r of f(x)=0, between 2 points a0 and b0.
I have [a(i),b(i)] enclose in it f(x)=0.
There is something I don't get, if the number of iteration i goes to infinity, why the length of [a(i),b(i)] doesn't unnecessary goes to 0 ?
I understand that this have to do with the f(a0) et f(b0) the first 2 points chosen.
can somebody explain this ?

You seem to be confusing two different methods- in false position the length of the interval [ai, bi] does NOT necessairily go to 0!

That does happened with "bisection": if f(a0)< 0 and f(b0)> 0 (and f is continuous) then you know there is a root somewhere between a0 and b0. Take your next point to be the midpoint of the interval: c= (a0+ b0)/2. if f(c)> 0 then there is a root between a0 and c: let a1= a0, b1= c. If f(c)< 0 then there is a root between b0 and c: let a1= c, b1= b0.

In this case, because we are always dividing the interval in half the length of (ai, bi) is the (b0- a0)/2ⁱ which obviously goes to 0.

False position, however, starts with two points, a0 and b0, such that f(a0)< 0 and f(b0)> 0 (or vice-versa) and calculates the slope of the line between them (f(b0)- f(a0))/(b0- a0) and uses that to determine a1. In true "false position", it is always the "a" point that is replace while the b point remains equal to b0. ai approaches a root while bi= b0 for all i so the length of [ai, bi] does NOT go to 0.

You might want to look at Wikipedia's article on it:
http://en.wikipedia.org/wiki/False_position_method

 

[ziad1985]

"necessarily" "unnecessary", that was a typo, I just noticed it.
I know the difference between the 2 methods.

If the initial end-points a0 and b0 are chosen such that f(a0) and f(b0) are of opposite signs, then one of the end-points will converge to a root of f. Asymptotically, the other end-point will remain fixed for all subsequent iterations while the one end-point always being updated. As a result, unlike the bisection method, the width of the bracket does not tend to zero. As a consequence, the linear approximation to f(x), which is used to pick the false position, does not improve in its quality.

So I can say that of if at the start f(a0) and f(b0) are of opposite signs...
I would get a non zero length of [ai, bi]...
I think I got it, I'm going to try the example given on wiki and several others and try them myself.