Gate CS-2015-2 Question Paper With Solutions

Q. 63 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: xa, xb, ε, N     // ε = convergence indicator
fb = f(xb) i = 0
while (i < N and |fb| > ε) do
   i = i + 1                 // update counter
   xt = ?                    // missing expression for
                             // intermediate value
   xa = xb                   // reset xa
   xb = xt                   // reset xb
   fb = f(xb)                // function value at new xb
end while
if |fb| > ε
  then // loop is terminated with i = N
  write “Non-convergence”
else
  write “return xb”
end if

(A) x_b – (f_b– f(x_a)) f_b/ (x_b – x_a)

(B) x_a– (f_a– f(x_a)) f_a/ (x_b – x_a)

(C) x_b – (f_b – x_a) f_b/ (x_b – f_b(x_a)

(D) x_a – (x_b – x_a) f_a/ (f_b – f(x_a))

Answer: (C)

Explanation: