I was reading about the convergence analysis of the regula falsi method in the book A friendly introduction to numerical analysis by B. Bradie.
I got stuck in the last step where $e_n=\lambda e_{n-1}$ and $$\lambda= \frac{l f''(p)}{2f'(p)+l f''(p)}$$ which comes from equation (4). However, I think we should have one more term in the denominator $a_n-p$ or $b_n-p$. Why did the author not consider it? If we consider it, then we won't get $e_n=\lambda e_{n-1}$ as now we also have one $e_{n-1}$ in the denominator.
Convergence Analysis
Does the sequence of approximations generated by the method of false
position, $\{p_n\}$ converge to a root $p$? If so, what is the order
of convergence? To answer these questions, we need to examine the
associated sequence of errors, $\{e_n\}$, where $e_n = p_n – p$. The
sequence $\{p_n\}$ converges if and only if $|e_n| \to 0$ as $n \to
\infty$ and the order of convergencee is ddetermined by the asymptotic
relationship between $|e_n|$ and $|e_{n-1}|$.The error sequence $\{e_n\}$ is governed by what is known as the
error evolution equation. To construct the error evolution equation, take the equation for $p_n$ and subtract $p$ from both sides. This
yields: $$p_n – p = b_n – p – f(b_n) \frac{b_n – a_n}{f(b_n) –
f(a_n)}$$ Next, approximate the function values $f(a_n)$ and $f(b_n)$
by the second degree Taylor polynomials $$f(a_n) \approx
f'(p)(a_n-p)+\frac{f''(p)}{2}(a_n-p)^2,$$ $$f(b_n) \approx
f'(p)(b_n-p)+\frac{f''(p)}{2}(b_n-p)^2$$
Best Answer
The removal of one term is due to the fact that one of the bounds will converge to the root while the other will not. Suppose $a\to p$. Then $l=b-p$ and $(b+a-2p)\to(b+p-2p)=(b-p)=l$. On the other hand, $a-p$ is written as $e_{n-1}$, the term $\lambda$ is being multiplied to.