(i) a <= F(x) <= b, for every x in [a,b],

and also one (or both) of the following conditions:

(ii

(ii

Then the equation x = F(x) has exactly one solution (z) in D.

z is the limit of the sequence, starting with an arbitrary x

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).

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