# Fixed point theorem

### From CFD-Wiki

The fixed point theorem is a useful tool for proving existence of solutions and also for convergence of iterative schemes. In fact the latter is used to prove the former. If the problem is to find the solution of

then we convert it to a problem of finding a fixed point

*Statement*

Let be a complete metric space with distance and let be contractive, i.e., for some ,

Then there exists a unique fixed point such that

*Proof*

Choose any and define the sequence by the following iterative scheme

If we can show that the sequence has a unique limit independent of the starting value then we would have proved the existence of the fixed point. The first step is to show that is a Cauchy sequence.

Now using the triangle inequality we get, for

where the last inequality follows because . Now since as we see that

which proves that is a Cauchy sequence. Since is complete, this sequence converges to a unique limit . It is now left to show that the limit is a fixed point.

and since is the limit of the sequence we see that the right hand side goes to zero so that

The uniqueness of the fixed point follows easily. Let and be two fixed points. Then

and since we conclude that

## Application to solution of linear system of equations

We will apply the fixed point theorem to show the convergence of Jacobi iterations for the numerical solution of the linear algebraic system

under the condition that the matrix is diagonally dominant. Assuming that the diagonal elements are non-zero, define the diagonal matrix

and rewriting we get

which is now in the form of a fixed point problem with

For measuring the distance between two vectors let us choose the maximum norm

We must show that is a contraction mapping in this norm. Now

so that the j'th component is given by

From this we get

and

Hence the mapping is contractive if

which is just the condition of diagonal dominance of matrix . Hence if the matrix is diagonally dominant then the fixed point theorem assures us that the Jacobi iterations will converge.

## Application to Newton-Raphson method

Consider the problem of finding the roots of an equation

If an approximation to the root is already available then the next approximation

is calculated by requiring that

Using Taylors formula and assuming that is differentiable we can approximate the left hand side,

so that the new approximation is given by

This defines the iterative scheme for Newton-Raphson method and the solution if it exists, is the fixed point of

If we assume that

in some neighbourhood of the solution then

Thus condition (*) guarantees that is contractive and the Newton iterations will converge.