Question 49

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?

Secant
  Initialize: x_a, x_b, eps, N   // eps = convergence indicator
                                 // N = maximum no. of iterations
  f_b = f(x_b)
  i = 0
  while (i < N and |f_b| > eps) 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| > eps then            // loop is terminated with i=N
    write "Non-convergence"
  else
    write "return x_b"
  end if