By Gerald L. Alexanderson
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Then xk > xk+1 , lim ∀ k ≥ 0, and k→∞ lim xk = θ. k→∞ (xk+1 − θ) 1 = . (xk − θ)m (2θ)m−1 (i) (ii) Proof. 1 for each real input x0 > 0, the sequence {gm (x0 )}∞ m=2 converges to θ. But if x0 > θ, then 0 < R(x0 ) < 1. From this it follows: (1 + Rm (x0 ))(1 − Rm+1 (x0 )) > (1 − Rm (x0 ))(1 + Rm+1 (x0 )). 18), it is easy to see that for all m ≥ 2, gm (x0 ) > gm+1 (x0 ). 22) implies of (i). 14), and since γm (θ) = −1 1 = . P1m (θ) (2θ)m−1 √ Although we have considered the approximation θ = α where α is a natural number, clearly all the convergence properties extend to the case where α is any positive real number.
Viewing the approximation of polynomials roots over the complex plane is a more challenging task, even for such a simple complex polynomial such as p(z) = z 2 − 2. To begin with, we would have to remember that the polynomial p(z) maps the complex number z = x + iy, corresponding to the point (x, y) in the complex plane, to z 2 − 2 = (x2 − y 2 − 2) + i2xy, another point in the complex plane. Thus we need four dimensions to view the graph of this mapping, which is not possible, given that our physical visualization capabilities are limited to three dimensions only.
0 .. . . . 4 1 0 ... 0 2 4 0 1 ... . . 0 .. −2 + 2i 2 + 2i 0 .. . 2 + 2i 1 ... 1. √ 2 = 1 + lim m→∞ = 2 − 2 lim m→∞ Dm−2 (1) Dm−1 (1) Dm−2 (2) Dm−2 (1 + i) = 1 + i − (−2 + 2i) lim . 0. 1. 00101i Visualizations in Approximation of Square-Roots One way to visualize the process of approximating square-root of a number α > 0 via Newton’s method, is through ordinary graphing. We draw the function p(x) = x2 −α and the iterates on the x-axis. There is a well-known geometric interpretation of the location of the iterates for an arbitrary real polynomial: given xk , xk+1 is the intersection of the tangent line to p(x) at xk , and the x-axis.