List of top Numerical Analysis Questions

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

If \( P(x) \) is a polynomial of degree 5 and \[ \alpha = \sum_{i=0}^{6} P(x_i) \left( \prod_{\substack{j=0
j\neq i}}^{6} (x_i - x_j)^{-1} \right), \] where \( x_0, x_1, \ldots, x_6 \) are distinct points in the interval \([2,3]\), then the value of \( \alpha^2 - \alpha + 1 \) is __________.

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).

The first derivative of a function \( f \in C^\infty(-3, 3) \) is approximated by an interpolating polynomial of degree 2, using the data \[ (-1, f(-1)), (0, f(0)) \text{ and } (2, f(2)). \] It is found that \[ f'(0) \approx -\frac{2}{3} f(-1) + \alpha f(0) + \beta f(2). \] Then, the value of \( \frac{1}{\alpha \beta} \) is

The first derivative of a function \( f \in C^\infty(-3, 3) \) is approximated by an interpolating polynomial of degree 2, using the data \[ (-1, f(-1)), (0, f(0)) \text{ and } (2, f(2)). \] It is found that \[ f'(0) \approx -\frac{2}{3} f(-1) + \alpha f(0) + \beta f(2). \] Then, the value of \( \frac{1}{\alpha \beta} \) is