- Give me a fixed point, and I will move the Earth.(Archimedes)
To solve an equation of equations , in unknowns , with thus , it is natural to connect to solution of the IVP: Find such that
with some given initial value . If it turns out that as increases, the function tends to some value , then could be expected to become small, and if so, we would have
and we would be led to set for some large and consider to be an approximate solution of with small residual .
If has several different solutions, which is often the case, then we could expect to capture different solutions by chosing different initial values .
Computing by Forward Euler with time step , we would have
If would become small for increasing , then would become small and thus would be an approximate solution of with small residual .
Solving u=g(u) by Fixed Point Iteration
With , we are thus led to study the convergence of the iteration, referred to as Fixed Point Iteration:
To this end we take the difference of (2) for two consecutive steps to get
If is Lipschitz continuous with Lipschitz constant , then
We see that if , then becomes vanishingly small as increases, which by (1) means that becomes vanishingly small and thus may be viewed as an approximate solution of in the sense that the residual is small. In the next chapter we will consider the error in the approximate root compared to an exact root with vanishingly small residual .
We see that if then the convergence is fast, and if then the convergence is slow. If then the residual is reduced with a factor 2 in each iteration step, that is with a binary digit per step.