Root-finding recursions

Given a function f(x) and an approximation for one of its roots x_n, we can find a better approximation for the root by linearizing the function around x_n

$$f(x) \approx f(x_n)+(x_{n+1}-x_n) f'(x_n)$$

and by setting f(x) to be zero for x=x_n+1. We find that  

$$\fbox {$ \displaystyle x_{n+1}=x_{n}-\frac{f(x_n)}{f'(x_n)} $}$$ (1)

1.

Newton-Raphson's method for the square root

A common choice of the function f is f(x)=x²-s. This function has the advantage that it is easily differentiable, with f'(x)=2x. The recursion relation thus becomes

$$x_{n+1}=x_{n}-\frac{x_n^2-s}{2x_n} =\frac{x_n}{2}+\frac{s}{2x_n}$$

or

$$x_{n+1}=\frac{1}{2}\left(x_n+\frac{s}{x_n}\right)$$

or, after rearrangement,  

$$\fbox {$ \displaystyle x_{n+1}=\frac{s+x_n^2}{2 x_n} $}$$ (2)

The recursion (2) converges to $\pm \sqrt{s}$ depending on the sign of the starting guess $x_0\ne 0$.

2.

Secant method for the square root

A variation of the Newton-Raphson method is to use a finite approximation of the derivative instead of the differential form. In this case, the approximate value of the derivative at step n is

$$f'(x_n)=\frac{f(x_n)-f(x_{n-1})}{x_{n}-x_{n-1}}$$

For the same choice of the function f, f(x)=x²-s, we obtain

$$x_{n+1}=x_{n}-\frac{x_n^2-s}{x_n+x_{n-1}}$$

and  

$$\fbox {$ \displaystyle x_{n+1}=\frac{s+x_{n}x_{n-1}}{x_n+x_{n-1}} $}$$ (3)

In this case, recursion (3) also converges to $\pm \sqrt{s}$ depending on the sign of the starting guesses x₀ and x₁.

3.

Muir's method for the square root

Another possible iterative relation for the square root is Francis Muir's, described by Jon Claerbout 1995:  

$$\fbox {$ \displaystyle x_{n+1}=\frac{s+x_{n}}{x_n+1} $}$$ (4)

This relation belongs to the same family of iterative schemes as Newton and Secant, if we make the following special choice of the function f(x) in (1):  

$$f(x)= \vert x+\sqrt{s}\vert^{\frac{\sqrt{s}-1}{2\sqrt{s}}} \vert x-\sqrt{s}\vert^{\frac{\sqrt{s}+1}{2\sqrt{s}}}$$ (5)

Figure 1 is a graphical representation of the function f(x).

 

muf Figure 1 The graph of the function defined in Equation (1) used to generate Muir's iteration for the square root (solid line). The dashed lines are the plot of the two factors in the equation  .

4.

A general formula for the square root

From the analysis of equations (2), (3), and (4), we can derive the following general form for the square root iteration:  

$$\fbox {$ \displaystyle x_{n+1}=\frac{s+x_{n}\gamma}{x_n+\gamma} $}$$ (6)

where $\gamma$ can be either a fixed parameter, or the value of the iteration at the preceding step, as shown in Table 1.

   

$$\gamma$$ Recursion

$$x_{n+1}=\frac{s+x_n }{x_n+1 }$$ Muir 1

$$x_{n+1}=\frac{s+x_n x_{n-1}}{x_n+x_{n-1} }$$ Secant x_n-1

$$x_{n+1}=\frac{s+x_n^2 }{2 x_n }$$ Newton x_n

$$\sqrt{s} x_{n+1}=\frac{s+x_n\sqrt{s}}{x_n+\sqrt{s}}$$ Ideal

The parameter $\gamma$ is the estimate of the square root at the given step (Newton), the estimate of the square root at the preceding step (Secant), or a constant value (Muir). Ideally, this value should be as close as possible to $\sqrt{s}$.