Root Finding - Numerical Methods - General Aptitude (GA) Previous Year Questions

The bisection method is applied to compute a zero of the function \(f(x) = x^4 - x^3 - x^2 - 4\) in the interval [1,9]. The method converges to a solution after _____ iterations.

In the LU decomposition of the matrix \(\begin{bmatrix} 2 & 2 \\ 4 & 9 \end{bmatrix}\), if the diagonal elements of \(U\) are both 1, then the lower diagonal entry \(l_{22}\) of \(L\) is ________.

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 no. 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
                // 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| > ε then  // loop is terminated with i=N
  write "Non-convergence"
else
  write "return x_b"
end if

The number of roots of \(e^x + 0.5x^2 - 2 = 0\) in the range \([-5, 5]\) is