Ridders' method: Difference between revisions

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==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, the method begins by evaluating the function at the midpoint <math>x_1 = (x_0 +x_2)/2 </math>. One then finds the unique exponential function <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>. Specifically, parameter <math>a</math> is determined by

:<math>e^{a(x_1 - x_0)} = \frac{f(x_1)-\operatorname{sign}[f(x_0)]\sqrt{f(x_1)^2 - f(x_0)f(x_2)}}{f(x_2)} .</math>