Steffensen's method: Difference between revisions

Revision as of 01:25, 12 July 2008

Revision as of 01:31, 12 July 2008

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Steffensen's method is then very similar to the Newton's method, except that it uses the divided difference<math>L(f(x),x)\ </math> instead of the derivative <math>Df(x)\ </math>. It is thus defined by

: <math>x_{kn+1} = x_kx_n + [I - L(f(x_kx_n), x_kx_n)]^{-1}(f(x_kx_n) - x_kx_n), \, </math>

for ''k'' <math>n= 1,\ 2,\ 3,\ ...&nbsp;</math>, and where <math>I\ </math> is the identity operator. If the operator <math>L\ </math> satisfies

: <math>\|L(u,v)-L(x,y)\| \le K \big( \|u-x\| + \|v-y\| \big) </math>