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Explain different types of numerical errors with example. Find the root of the equation f(x) = x3 + 3x + 1 in the using Newton Raphson Method using initial value x₀ = 3 .
What is interpolation? Find the functional value at x = 0.5 from the following data using forward difference formula.
| X | 0 | 0.2 | 0.4 | 0.6 | 0.8 |
| f(x) | 0 | 0.24 | 0.55 | 0.92 | 1.63 |
Evaluate ∫₁² ex² dx using Simpson's rule with n=6. Also evaluate the same integral using Gaussian 3 -point formula and compare the result.
Attempt any Eight questions
[8x5=40]Solve the system of linear equation using Gauss elimination method.
3x + 6y + z = 16
2x + 4y + 3z = 13
x + 3y + 3z = 9
Solve the following differential equation dy/dx = 3x + y/2 with y(0) = 1 for x = 0.2 (h = 0.1) using Euler's Method.
Solve the Poisson's equation ∇²f = 2x²y² over the square domain 0≤ x ≤ 3 and 0 ≤ y ≤ 3 with f = 0 on the boundary and h=1.
What is initial value problem? Use RK fourth order method to estimate y(0.4) of the equation dy/dx = x² + y² with y(0)=0 (assume h=0.2).
Calculate the integral value of∫₀² 1/(x+4) dx by using Simpson's 3/8 rule with 9 segments.
Find the root of eˣ − 3x = 0, using bisection method correct to two decimal places.
Apply Gauss-Jordan elimination method to solve the following system of equations:
x + y + z = 9
2x-3y+4z=13
3x+4y+5z=40
Fit a second degree polynomial y=a+bx+cx² to the data (1, 1.1), (1.5, 1.3), (2.0, 1.6), (2.5, 2.0), (3.0, 2.7), (3.5, 3.4) and (4, 4.1).
Explain about least square linear regression and describe its application.