Note that the formula in the middle of page 58 of the text book should have the derivative of g, as g'(x) determines the rate of convergence (or divergence).
A general method to accelerate iterative processes is Aitken's acceleration. The book provides no explanation for the derivation of the formula, therefore, see below:
We want to accelerate the convergence of a sequence of points x(k), where k=0,1,..,infinity, defined by x(k+1) = g(x(k)). We assume the sequence converges to the fixed point x(infinity) = g(x(infinity)).
Denote e(k) = x(k+1) - x(k) = g(x(k)) - x(k). In practice, e(k) is used to measure the error of the iteration process, if |e(k)| is small enough we terminate the sequence. By the assumption of convergence we have that e(k) goes to 0 as k goes to infinity, and e(infinity) = 0.
Consider now e(k) as a function of x: e(k) = e(x(k)). To accelerate the convergence, we wish to find the value for x for which e(x) = 0. Suppose we know e(x(k)) and e(x(k+1)). To find a root of e(x) = 0, we apply the idea of the secant method: construct a line through the points (x(k),e(x(k))) and (x(k+1),e(x(k+1))). An approximation for the root of e(x) = 0 is where the line intersects the x-axis.
The formula for the line through the points (x(k),e(x(k)) and (x(k+1),e(x(k+1))) is
e(x(k+1)) - e(x(k))
y - e(x(k)) = --------------------- (x - x(k))
x(k+1) - x(k)
To compute the intersection with the x-axis, we set y = 0 in the equation above and solve for x:
x(k+1) - x(k)
0 - e(x(k)) --------------------- + x(k) = x
e(x(k+1)) - e(x(k))
To simplify, we compute the following:
e(x(k+1)) = g(x(k+1)) - x(k+1) = x(k+2) - x(k+1)
- ( e(x(k)) = g(x(k)) - x(k) = x(k+1) - x(k) )
--------------------------------------------------------
e(x(k+1)) - e(x(k)) = x(k+2) - 2*x(k+1) + x(k)
Thus, we obtain
( x(k+1) - x(k) )^2
x = x(k) - -------------------------
x(k+2) - 2*x(k+1) + x(k)