How spline interpolation differs with the Langrage's interpolation? Estimate the value of f(0) and f(4) using cubic spline interpolation from the following data.
xf(x)−1−101−2214386
[10]
2.
Fit the quadratic function for the data given below using least square method.
What is Gaussian integration formula? Evaluate the following integration using Gaussian integration three ordinate formula.
∫01xsinxdx
[5]
Solution of Nonlinear Equations
1.
How secant method can approximate the root of a non-linear equation? Explain with necessary derivation. Estimate a real root of following equation using secant method. Assume error precision of 0.01.
x3+2x−cos(x)=4
[10]
2.
Calculate a real root of the following function using bisection method correct upto 3 significant figures.
x2−ex=3
[5]
3.
What is fixed point iteration method? How can it converge to the root of a non-linear equation? Also explain the diverging cases with suitable examples.[5]
Solution of Ordinary Differential Equations
1.
Write down the program for solving ordinary differential equation using Heun's method.[5]
2.
Solve the following differential equation for 0≤x≤1 taking h=0.5 using Runge Kutta 4th order method.
y′(x)+y=3x with y(0)=2
[5]
3.
Solve the Poisson's equation ∇2f=3x2y over the square domain 0≤x≤3, 0≤y≤3 with f=0 on the boundary and h=1.[5]
Solving System of Linear Equations
1.
What is pivoting? Why is it necessary? Write an algorithm and program to solve the set of n linear equations using Gaussian elimination method.[10]
2.
Solve the following set of equations using Gauss Siedal method.