Fit a second order polynomial to the data in the table below:
XF(x)1226312420530
[5]
Numerical Differentiation and Integration
1.
Solve the double integration using Simpson's 1/3 rule. ∫22.6∫44.4xydxdy[5]
2.
Why Numerical Integration is required? Compute the integral: I=∫−11exdx using composite trapezoidal rule for n = 4.[5]
3.
Evaluate dxdy at x = 5 using Newton's forward interpolation formula using the following table.
Xy1−1.20312.805119.607472.8091302.80
[5]
Solution of Nonlinear Equations
1.
Write an algorithm and a C-Program to obtain roots of non-linear equation using Newton Raphson Method.[10]
2.
Use secant method to estimate the root of the equation x2−5x+6=0, with initial estimate x1=4 and x2=2 (EPS=0.05).[5]
3.
What are the sources of errors? Discuss various types of errors encounters in numerical computation.[5]
Solution of Ordinary Differential Equations
1.
Solve the following ordinary differential equation using shooting method. y′′+xy′−xy=2x with boundary conditions y(0)=1 and y(2)=10[10]
2.
Solve the following differential equation dxdy=3x+2y with y(0)=1 for x=0.2(h=0.1) using Euler's Method.[5]
Solution of Partial Differential Equations
1.
Solve the Poisson's equation ∂2f/∂x2+∂2f/∂y2=2x2y2 over the square domain 0<=x<=3 and 0<=y<=3 with f=0 on the boundary and h = 1.[5]
Solving System of Linear Equations
1.
Compare and contrast between Jacobi iterative methods and Gauss Seidal method? Solve the following equation using Gauss Seidal method. x+2y+3z=5,2x+8y+22z=6 and 3x+22y+82z=−10[10]
2.
Find the Eigen values and Eigen vectors of the Matrix: A=[31−11][5]