[alogo] Iterative

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).

[0_0] [0_1] [0_2] [0_3]

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).

See Also



Gisela-Jordan Engeln und Fritz Reutter Numerische Mathematik fuer Ingenieure Mannheim B.I.-Wissenschaftsverlag 1978, p. 33

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