(1) |

- 1.
**Newton-Raphson's method for the square root**A common choice of the function

or or, after rearrangement,*f*is*f*(*x*)=*x*-^{2}*s*. This function has the advantage that it is easily differentiable, with*f*'(*x*)=2*x*. The recursion relation thus becomes(2) The recursion (2) converges to depending on the sign of the starting guess .

- 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*isFor the same choice of the function

and*f*,*f*(*x*)=*x*-^{2}*s*, we obtain(3) In this case, recursion (3) also converges to depending on the sign of the starting guesses

*x*and_{0}*x*._{1}- 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:

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

**muf**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 .

Figure 1- 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:

(6)

The parameter 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 .

4/20/1999