ABSTRACT
Equation (3.125) is in the general iteration form, xi+] = g(xJ Differentiating g(x) and evaluating the result at X = ex yields g'(ex) = O. Substituting this result into Eq. (3.50) shows that Eq. (3.125) is convergent. Further analysis yields
g"m 2 ei+1 = -2-ei (3.126)
where ~ is between Xi and ex, which shows that Eq. (3.125) converges quadratically. Next consider the variation where Newton's basic method is applied to the function
u(x) = j'(x)
If f(x) has m repeated roots,j(x) can be expressed as f(x) = (x - ex)lIlh(x) (3.128)
where the deflated function hex) does not have a root at x = ex, that is, h(ex) i= O. Substituting Eq. (3.128) into Eq. (3.127) gives
u(x) __ (x - r)1Il hex) (3.129) m(x - ex)IIl-l h(x) + (x - ex)lIlg'(X)
which yields
u(x) = (x - ex)h(x) mh(x) + (x - ex)g'(x) (3.130)
method, with second-order convergence, can be applied to u(x) to give
Differentiating Eq. (3.127) gives
(3.131 )
(3.132)
Substituting Eqs. (3.127) and (3.132) into Eq. (3.131) yields an alternate form of Eq. (3.131):
The advantage ofEq. (3.133) over Newton's basic method for repeated roots is that Eq. (3.133) has second-order convergence. There are several disadvantages. There is an additional calculation for f"(x;). Equation (3.133) requires additional effort to evaluate. Round-off errors may be introduced due to the difference appearing in the denominator of Eq. (3.133). This method can also be used for simple roots, but it is less efficient than Newton's basic method in that case.
