Theorem Let the function F(x) be defined in the closed interval D = [a,b], where it satisfies: (i) a <= F(x) <= b, for every x in [a,b], and also one (or both) of the following conditions: (ii1) |F(x) - F(x')| <= L*|x - x'|, with 0 <= L < 1 (i.e. it is Lipshitz continuous). (ii2) F is differentiable and |F'(x)| <= L < 1. Then the equation x = F(x) has exactly one solution (z) in D. z is the limit of the sequence, starting with an arbitrary x0 in [a,b] and defined iteratively through xn+1 = F(xn).
The above theorem is the basic one for iterative procedures in calculating the roots of equations. The picture shows an application of it. It is applied to the function F(x) = p-q*sin(x), which for the actual values of the constants selected satisfies the conditions of the theorem. In particular the interval [a,b] is mapped by F on [a',b'] which is inside [a,b]. One can modify the values of a, b and see that the resulting sequence may diverge (some of the conditions of the theorem are then broken).
![[alogo]](alogo.png){kind=small}
Iterative ![[0_0]](Iterative_files/Iterative_0_0_0.png)
![[0_1]](Iterative_files/Iterative_0_0_1.png)
![[0_2]](Iterative_files/Iterative_0_0_2.png)
![[0_3]](Iterative_files/Iterative_0_0_3.png)