We find the parabola that passes through the points \((x_0, f_0)\text{,}\) \((x_1, f_1)\) and \((x_2, f_2)\text{,}\) where \(f_j = f(x_j)\text{.}\) Although the quadratic can be written in standard form, it is easier if we write it as
\begin{equation*}
q(x) = a_0 + a_1 (x-x_1) + a_2 (x-x_1)^2
\end{equation*}
We also use
\begin{equation*}
\begin{aligned} h_1 \amp = x_1-x_0 \amp h_2 \amp = x_2 - x_1 \amp \delta_1 \amp = \frac{f_1-f_0}{h_1} \amp \delta_2 \amp = \frac{f_2-f_1}{h_2} \end{aligned}
\end{equation*}
substituting the three points into \(q(x)\) results in
\begin{align}
q(x_0) \amp = f_0 = a_0 + a_1 (x_0-x_1) + a_2 (x_0-x_1)^2 = a_0 -a_1 h_1 + a_2 h_1^2 \tag{3.5.2}\\
q(x_1) \amp = f_1 = a_0 \tag{3.5.3}\\
q(x_2) \amp = f_2 = a_0 + a_1 (x_2-x_1) + a_2 (x_2-x_1)^2 = a_0 + a_1 h_2 + a_2 h_2^2 \tag{3.5.4}
\end{align}
Plugging in
\(a_0=f_1\) into
(3.5.2) and divide by
\(h_1\) results in
\begin{equation}
\frac{f_0-f_1}{h_1} = -\delta_1 = - a_1 + a_2 h_1\tag{3.5.5}
\end{equation}
\begin{equation}
\frac{f_2-f_1}{h_2} = \delta_2 = a_1 + a_2h_2\tag{3.5.6}
\end{equation}
\begin{equation*}
\delta_2 - \delta_1 = a_2 (h_1 + h_2)
\end{equation*}
which results in
\begin{equation*}
a_2 = \frac{\delta_2-\delta_1}{h_1+h_2}
\end{equation*}
and then subtracting
\(h_2\) times
(3.5.5) and
\(h_1\) times
(3.5.6) results in
\begin{equation*}
\delta_1 h_2 + \delta_2 h_1 = a_1(h_2+h_1)
\end{equation*}
or
\begin{equation*}
a_1 = \frac{\delta_1 h_2 + \delta_2 h_1 }{h_2+h_1}
\end{equation*}
This shows that the quadratic that passes through the points is
\begin{equation*}
q(x) = a_0 + a_1 (x-x_1) + a_2(x-x_1)^2
\end{equation*}
with
\begin{equation*}
\begin{aligned} a_0 \amp =f_1 \amp a_1 \amp =\frac{\delta_1 h_2 + \delta_2 h_1 }{h_2+h_1} \amp a_2 \amp = \frac{\delta_2-\delta_1}{h_1+h_2} \end{aligned}
\end{equation*}
Next, Muller’s method uses the root of the quadratic to find the root, however, only one of the roots is needed.
\begin{equation*}
x-x_1 = - \frac{2a_0}{a_1 \pm \sqrt{a_1^2-4a_0a_2}} = \frac{-a_1 \pm \sqrt{a_1^2 - 4a_0a_2}}{2 a_2}
\end{equation*}
The root is the one closer to the point \(x_2\) or the one with the larger denominator. Thus we will take the sign that is the same as the sign of \(a_1\text{.}\)
