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What are inherent errors? Derive the Newton Raphson method for solving non-linear equation and using this method solve x²-5x+6=0, calculate upto 3 decimal places.
What are the limitations of direct methods for solving a system of linear equations? How Gauss Seidel method differs from Jacobi iteration? Solve the following system of linear equation using Jacobi iteration method.
2x-7y-10z=17
5x+y+3z=14
x+ 10y +9z=7
Write an algorithm and program to implement Lagrange interpolation method.
Attempt any Eight questions
[8x5=40]Consider the following data points estimate the f(0.6) using Newton's interpolation formula.
| x | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |
| f(x) | 2.68 | 3.04 | 3.38 | 3.69 | 3.97 |
What is regression analysis? Fit a second order polynomial for the following data values.
| x | 2 | 4 | 6 | 8 | 10 |
| y | 1.4 | 2.0 | 2.4 | 2.6 | 2.8 |
What is numerical differentiation? The table below gives the values of distance travelled by a vehicle at various time interval, estimate the velocity and acceleration at x=4.
| Time (x) | 1 | 2 | 4 | 8 | 10 |
| Distance (y) | 0 | 1 | 5 | 21 | 27 |
What is application of numerical integration? Find the value of integration for ∫₁² eˣ/x dx using Simpson's 3/8 rule with n=6.
Solve the following system of linear equations using Gauss-Jordan elimination method.
x+2y-3z=4
2x+4y-6z=8
x-2y+5z=4
Given the data points below
| X | 1.0 | 3.0 | 4.0 |
| f(x) | 1.5 | 4.5 | 9.0 |
Find cubic spline which belongs to 1<=x<=3 and estimate f(2) using cubic splines.
What is differential equation? Differentiate between ODE and PDE with example.
Solve dy/dx = x/y, y (0) =1, at x=0.4 using Runge-Kutta's 4ᵗʰ order method.
Solve the Poisson equation ∂²u/∂x² + ∂²u/∂y² =64xy, 0≤x≤1, 0≤y≤1 with boundary conditions:
u(0,y)=0, u(x,0)=0, u(1,y)=150, u(x,1)=150 and h=1/3.