How can we calculate derivatives of discrete (tabulated) functions? Write down its algorithm. [5]
2.
Find the following integral using composite trapezoidal rule for using 2 segments (k=2) and 4 segments (k=4).
$\int_{2}^{4} (x^3 + 2) dx$
[5]
Solution of Nonlinear Equations
1.
How secant methods differs from Newton Raphson method? Derive the formula for Secant Method. Solve the equation using Secant method. Assume error precision as 0.01. Discuss the drawbacks of the Newton Raphson method.
$\cos x + 2\sin x - x^2 = 0$
[10]
2.
Define the terms approximate error and relative approximate error? Discuss the working of Half Interval method for finding the roots of non-linear equation. [5]
Solution of Ordinary Differential Equations
1.
Approximate the solution of $y' = 3x^2$, $y(1) = 1$ using Taylor's series method using first four terms. Approximate the value of $y(2)$.
$y' = 3x^2, \quad y(1) = 1$
[5]
2.
Solve the Poisson's equation $\nabla^2 f = xy$ and $f=2$ on boundary by assuming square domain $0 \leq x \leq 3$ and $0 \leq y \leq 3$ and $h=1$. [5]
3.
Write down the program for solving ordinary differential equation using Heun's method. [5]
Solving System of Linear Equations
1.
How Gauss Jordan method differs from Gauss Elimination method? Solve the following system of equations using Gauss Jordan method. How can we use Gauss Jordan method to find the inverse of a matrix? Discuss.
$2x - y + 4z = 15$
$2x + 3y - 2z = 4$
$3x + 2y - 4z = -4$
[10]
2.
Factorise the following matrix using Cholesky method.