On the other hand, if then one can show that the function has a simple root at .
Derivation
(c) John H. Mathews 2004
![[Graphics:Images/NewtonImprovedMod_gr_43.gif]](../Images/NewtonImprovedMod_gr_43.gif){kind=small}
then
one can show that the function ![[Graphics:Images/NewtonImprovedMod_gr_44.gif]](../Images/NewtonImprovedMod_gr_44.gif){kind=small}
has
a simple root at ![[Graphics:Images/NewtonImprovedMod_gr_45.gif]](../Images/NewtonImprovedMod_gr_45.gif){kind=small}
. then
one can show that the function has
a simple root at . ![[Graphics:../Images/NewtonImprovedMod_gr_46.gif]](../Images/NewtonImprovedMod_gr_46.gif)
![[Graphics:../Images/NewtonImprovedMod_gr_47.gif]](../Images/NewtonImprovedMod_gr_47.gif)