Let g x 3x4 64x3 a find the positive fixed point of g b fi

Let g (x) = 3x/4 + 64/x^3. a. find the positive fixed point of g. b. find an interval [a, 6], containing the fixed point found in part (a), in which g(x) elementof [a, 6] whenever x elementof [a, 6] and g is a contraction mapping on this interval. The endpoints of the interval are not allowed to coincide with the fixed point. c. What Ls the maximum number of fixed point iterations needed in order to find the fixed point of g with 10\"10 accuracy? The starting point should be one of the endpoints of the interval found in part (b).

Solution

In numerical analysis, fixed-point iteration is a method of computing fixed points of iterated functions.

If a function f defined on the real numbers with real values and given a point x₀ in the domain of f, the fixed point iteration is

x_n+1 = f(x_n), n=0,1,2,...

which gives rise to the sequence x₀,x₁,x₂,... which is hoped to converge to a point x. If f is continuous, then we can prove that the obtained x is a fixed point of f, i.e., f(x) = x

In the given case :-

g(x)= 3(x)/4 + 64/(x)³, n=0,1,2,...

Hence, we can calculate the fixed point of g.

(b) The point that we will get from point a will be the interval in point b i.e. (a,b)