Quantitative Methods Answers

Q. Write and solve an example of Runge Kutta Fehlberg Method.

For the table of values f(x) =xe^x given by
x f(x)
1.8 10.8894
1.9 12.7032
2.0 14.7781
2.1 17.1489
2.2 19.8550
Find f"(2.0) using the central difference formula of 0(h^2) fir h=0.1 and h=0.2. Calculate T.E. and actual error.

The position f(x) of a particle moving in a line at various times xk is given in the following table. Estimate the velocity and acceleration of the particle at x=1.2
x f(x)
1.0 2.72
1.2 3.32
1.4 4.06
1.6 4.96
1.8 6.05
2.0 7.39
2.2 9.02

Consider the following data
x f(x)
1.0 0.7651977
1.3 0.6200860
1.9 0.2818186
2.2 0.1103623
Use Stirling's formula to approximate f(1.5) with x0= 1.6

Write down only one application of Runge Kutta Method to solve some real world problem

Write down Runge Kutta Fehlberg Method(Error Control)

using sin(0.1)= 0.09983 and sin(0.2)= 0.19867, find an approximate value of sin(0.15) by using Lagrange's interpolation. Obtain a bound on the truncation error.

The solution of the system of equations (1 2, 2 1)(x,y) =(4,-2) is attempt by the Gauss Jacobi and Gauss Seidel iteration schemes. Set up the two schemes in matrix form. Will the iteration schemes converge? Justify your answer.

Starting with x^(0)=[1 1 1]^T, find the dominant eigenvalue and corresponding eigenvector for the matrix A= [4 -1 1, 4 -8 1, -2 1 5] using the power mwthod

show that ux =c1eax +c2e-ax is the solution of the difference equation ux +1-2 ux Cosha+ux-1 =0