Numerical Method TU Board 2075 question paper

What is non-linear equation? Derive the required expression to calculate the root of non-linear equation using secant method. Using this expression find a root of following equation.

$x^2+cos(x)–e^{-x}–2=0$

What is matrix factorization? Factorize the given matrix A into LU using Doolittle algorithm and also solve Ax = b for given b using L and U matrices.

A = $\left[\begin{array}{cccc}2& 4& -4& 0\\ 1& 5& -5& -3\\ 2& 3& 1& 3\\ 1& 4& -2& 2\end{array}\right]$ and b = $\left[\begin{array}{c}12\\ 18\\ 8\\ 8\end{array}\right]$

What is initial value problem and boundary value problem? Write an algorithm and program to solve the boundary value problem using shooting method.

Calculate a real negative root of following equation using Newton's method for polynomial.

$x^4+2x^3+3x^2+4x=5$

What is least squares approximation of fitting a function? How does it differ with polynomial interpolation? Explain with suitable example.

Find the lowest degree polynomial, which passes through the following points:

X
-2
-1
1
2
3
4
F(x)
-19
0
2
-3
-4
5

Using this polynomial estimate f(x) at x = 0

The fit function of type y = a + bx for the following points using the least square method.

X
-1
1.2
2
2.7
3.6
4
F(x)
1
20
27
33
41
45

Calculate the integral value of the function given below from x = 1.8 to x = 3.4 using Simpson's 1/3 rule.

X
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.4
F(x)
0.003
0.778
1.632
2.566
3.579
4.672
7.097
8.429

Evaluate the following integration using Romberg integration.

${\int }_0^1\frac{sinx}{x}dx$

Solve the following set of equations using Gauss Seidel method.

x + 2y + 3z = 4

6x – 4y + 5z = 10

5x + 2y + 2z = 25

From the following differential equation estimate y(1) using RK 4^th order method.

$\frac{dy}{dx}+2x^{2}y=4withy(0)=1$

[Take h = 0.5]

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.