Numerical Method Model question question paper

What do you mean by ill condition? Compare Gauss elimination method and Gauss jordan method of solving simultaneous equations.

Using Gauss jordan method solve the given system of equation

6x_1 – 2x_2 + x_3 = 11

x_1 + 2x_2 – 5x_3 = -1

-2x_1 + 7x_2 + 2x_3 = 5

Define the terms interpolation and extrapolation. Find the Langrange interpolation polynomial to fit the following data and find value of y(10).

x
5
6
9
11
y
12
13
14
16

Solve the ordinary differential equation given below by using Eular,s method. And calculate the value of y' = 3x^2 + 1 ; y(0) = 2, when

1. h = 0.5
2. h = 0.25

Write down the program for solving ordinary differential equation using Heun's method.

Find the missing term in the following using Newton's divided difference formula.

x
0
1
2
3
4
y
1
3
9
….
81

Using a method of least square find the relation of the form y = ax+b

x
0.301
0.4771
0.6021
0.6990
y
1.4440
1.7931
2.0414
2.2068

Find the largest eigen value and the corresponding eigen vector of the following matrix :

$\left[\begin{array}{ccc}1& 2& 0\\ 2& 1& 0\\ 0& 0& -1\end{array}\right]$.

Factorize the matrix using Cholesky's decomposition.

$\left[\begin{array}{ccc}1& 2& 3\\ 2& 8& 22\\ 3& 22& 82\end{array}\right]$.

Use Romberg estimate to evaluate R(2,2)

$$\int \frac{1}{1+x}$$ from 0 to 2.

Calculate the integral value of the following tabulated function from x = 0 to x = 1.6 using simpson's 3/8 rule.

x
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
f(x)
0
0.24
0.55
0.92
1.63
1.84
2.37
2.95
3.56

Solve the Poison's equation over the square domain 0 ≤ x ≤ 1.5, 0 ≤ y ≤ 1.5 with f = 0 on the boundary and h = 0.5.

Estimate a real root of the following non-linear equation using bisection method correct upto three significant figure x^2 – e^-x = 3.