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, ε , N
// ε = convergence indicator
// N = maximum number of iterations
f_b = f(x_b)
i = 0
while (i < N and |f_b | > ε ) do
i = i + 1// update counter
x_t =? // missing expression for
x_a = x_b
x_b = x_t // intermediate value
f_b = f(x_b) // function value at new x_b
end while
if |f_b | > ε then // loop is terminated with i = N
write "Non-convergence"
else
write "return x_b "
end if