List of top Numerical Analysis Questions on Numerical solutions of nonlinear equations asked in GATE MA

The real sequence generated by the iterative scheme
\[ x_n = \frac{x_{n-1}}{2} + \frac{1}{x_{n-1}}, \quad n \geq 1 \]

The initial value problem \[ \frac{dy}{dx} = f(x, y), \quad y(x_0) = y_0 \] is solved by using the following second-order Runge-Kutta method: \[ K_1 = h f(x_i, y_i) \] \[ K_2 = h f(x_i + \alpha h, y_i + \beta K_1) \] \[ y_{i+1} = y_i + \frac{1}{4} (K_1 + 3 K_2), \quad i \geq 0 \] where \( h \) is the uniform step length between the points \( x_0, x_1, \dots, x_n \) and \( y_i = y(x_i) \). The value of the product \( \alpha \beta \) is __________ (round off to TWO decimal places).

Consider the fixed-point iteration \[ x_{n+1} = \varphi(x_n), n \geq 0, \] with \[ \varphi(x) = 3 + (x - 3)^3, x \in (2.5, 3.5), \] and the initial approximation \( x_0 = 3.25 \). Then, the order of convergence of the fixed-point iteration method is