Ridders' method: Difference between revisions

Revision as of 11:33, 31 May 2017

Revision as of 11:33, 31 May 2017

Line 4:

==Method==

Given two values of the independent variable, <math>x_0</math> and <math>x_2</math>, which are on two different sides of the root being sought, i.e.,<math>f(x_0)f(x_2) < 0</math>.The method begins by evaluating the function at the midpoint <math>x_1 = (x_0 +x_2)/2 </math> between the two points. One then finds the unique exponential function of the form <math>e^{ax}</math> such that function <math>h(x)=f(x)e^{ax}</math> satisfies <math>h(x_1)=(h(x_0) +h(x_2))/2 </math>. The false position method is then applied to the transformed values, leading to a new value ''x''₃, between ''x''₀ and ''x''₂, which can be used as one of the two bracketing values in the next step of the iteration.

The other bracketing value is taken to be ''x''₃ if f(''x''₃) has the opposite sign to f(''x''₄), or otherwise whichever of ''x''₁ and ''x''₂ has f(x) of opposite sign to f(''x''₄).