Numerical Techniques

(a) Given the values :

x	f(x)
5	150
7	392
11	1452
13	2366
17	5202

Evaluate f(9) using Newton's divided difference formula.

(a) Explain the importance of numerical analysis in scientific computing.

(b) Perform 4 iteration of Bisection method to find out the root of function g(x)=2xcos(πx) - e^-x-1 = 0.

(b) Obtain Lagrange interpolation formula for equally spaced nodes.

(a) Explain the fitting of a straight line by least square method.

(b) What are the applications of curve fitting in physics and engineering ?

Evaluate : using.

1. Trapezoidal rule
2. Simpson's rule (1/3rd)
3. Simpson's rule (3/8th)

(a) Given dy/dx = y - x / y + x with initial condition y = 1 at x = 0, find y for x = 0.1 by Euler's method.

(b) Apply Runge-Kutta 4th order method to find approximate value of y for x = 0.2 in steps of 0.1 if dy/dx = x + y² given that y = 1 where x = 0.

Write down the errors in the approximation of : by the trapezoidal rule and Simpson's rule. Hence find the value of the constant C for which the trapezoidal rule gives the exact result for the calculation of ∫₀¹(x⁵ cx⁴)dx.

Explain why each of the following functions is continuous or discontinuous.

1. The temperature at a specific location as a function of time
2. The temperature at a specific time as a function of distance from a fixed point.

(a) Using Runge Kuttta method of order 4 find y(0.2) given that dy/dx = 3x + 1/2y, y(0) = 1 taking h = 0.1.

(b) Solve the following by Euler's modified method : dy/dx = log (x + y), y(0) - 2 at x = 1.2 and 1.4 with n = 0.2.

Use Gaussian elimination method (both with and without pivoting) to find the solution of the following systems :

1. 6x₁ + 2x₂ + 2x₃ = -2
2. 2x₁ + 0.6667x₂ + 0.3333x₃ = 1, x₁ + 2x₂ - x₃ = 0