What are the problems with polynomial interpolation for a large number of data set? How such problems are addressed? Explain with an example. [5]
Numerical Differentiation and Integration
1.
Evaluate the following integration using Romberg integration.
$\int_{0}^{1} \frac{\sin^2 x}{x} dx$
[5]
Solution of Nonlinear Equations
1.
How can Horner’s rule be used to evaluate the f(x) and f(x) of a polynomial at a given point? Explain. Write an algorithm and program to calculate a real root of a polynomial using Horner’s rule. [10]
2.
How the half-interval method can be estimate a root of a non-linear equation? Find a real root of the following equation using the half-interval method to correct up to two decimal places.
$x^2 - e^{-x} - x = 1$
[5]
3.
Calculate the real root of the given equation using fixed point iteration correct up to 3 significant figures.
$2x^3 - 2x = 5$
[5]
Solution of Ordinary Differential Equations
1.
What is a higher-order differential equation? How can you solve the higher-order differential equation? Explain. Solve the following differential equation for 1 ≤ x ≥ 2, taking h = 0.25
$\frac{d^2y}{dx^2} + 3 \frac{dy}{dx} + 5y = 0, \text{ with } y(1) = 1 \text{ and } y'(1) = 2$
[10]
2.
Solve the following differential equation for $1 \leq x \leq 2$, taking $h = 0.25$ using Heun’s method.
$y'(x) + x^2y = 3x, \text{ with } y(1) = 1$
[5]
3.
Consider a metallic plate of size 90cm by 90cm. The two adjacent sides of the plate are maintained at a temperature of $100^\circ C$ and the remaining two adjacent sides are held at $200^\circ C$. Calculate the steady-state temperature at interior points assuming a grid size of 30 cm by 30 cm. [5]
Solving System of Linear Equations
1.
Write matrix factorization? How can be used to solve a system of linear equations? Factorize the given matrix A and solve the system of equations Ax = b for given b using L and U matrices.