GATE CS 2015 Set 2 Q39 — Numerical Methods (Historical)

The secant method is used to find the root of an equation $f(x)=0$. It is started from two distinct estimates $x_a$ and $x_b$ for the root. It is an iterative procedure involving linear interpolation to a root. The iteration stops if $f(x_b)$ is very small and then $x_b$ is the solution. The procedure is given below. Observe that there is an expression which is missing and is marked by ?. Which is the suitable expression that is to be put in place of ? so that it follows all steps of the secant method?

$\textbf{Secant}$

Initialize $x_a, x_b, \varepsilon, N$ // $\varepsilon$ = convergence integer
// N = maximum no. iterations
$f_b=f(x_b)$
i=0
while ( i < N and $|f_b| > \varepsilon)$ do
     i = i+1 // update counter
     $x_t$ =? //missing expression for intermediate value
     $x_a =x_b$ //reset $x_a$
     $x_b = x_t$  // reset $x_b$
     $f_b = f(x_b)$  //function value at new $x_b$
end while
if $|f_b| > \varepsilon$ then // loop is terminated with i=N
     write "Non-convergence"
else
     write "return $x_b$"
end if