Knowledge base dedicated to Linux and applied mathematics.
All the versions of this article: [English] [français]
Fixed point method allows us to solve non linear equations. We build an iterative method, using a sequence wich converges to a fixed point of g, this fixed point is the exact solution of f(x)=0.
The aim of this method is to solve equations of type:
We introduce a convergent sequence to the fixed point of , which is the solution of equation (E).
If and , then g has a fixed point in [a, b].
If and if exists a constant in ]0,1[ such that
Proof of the existence.
We define over [a,b], like follows:
Proof of the unicity.
We suppose now that there exists two fixed points with . We apply the mean value theorem : there exists with:
The sequence is well defined because
and consequently provided that . As previously , we apply the mean value theorem, we show the existence of
for all with
Proof of the Corollary.
The first inequality is obvious. To prove the second inequality,we use
We suppose that converges to :
If exists two constants and such that:
If p=1 and C<1 we say that the sequence converges linearly .
If p=2, convergence is called quadratic convergence.
If p=3, convergence is called cubic convergence.
Obviously several cases are possible, we can build several functions and that also depends on the nature of .
If , we must introduce a Taylor Developement of around . Don’t forget that converges to 0. For example,
a Taylor developement of order 3 gives:
Taylor developement of order k around of gives: