Spectral factorization revisited |
A common choice of the function is . This
function has the advantage that it is easily differentiable, with
. The recursion relation thus becomes
The recursion (2) converges to depending on the sign of the starting guess .
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 is
For the same choice of the function , , we
obtain
In this case, recursion (3) also converges to depending on the sign of the starting guesses and .
Another possible iterative relation for the square root is Francis
Muir's, described by Claerbout (1995):
This relation belongs to the same family of iterative schemes as
Newton and Secant, if we make the following special choice of the
function in (1):
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 . | |
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From the analysis of equations (2), (3),
and (4), we can derive the following general form for
the square root iteration:
Spectral factorization revisited |