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What do you mean by true error? Find a positive root of equation \(x^2 = 1\), which lies between 0 and 1 using bisection method.
What is interpolation? Given the following set of data points, obtained the table of divided difference. Use the table to estimate the value of \(f(1.5)\).
| \(i\) | 0 | 1 | 2 | 3 | 4 |
| \(x_i\) | 1 | 2 | 3 | 4 | 5 |
| \(f(x_i)\) | 0 | 7 | 26 | 63 | 124 |
Solve the following system of equations by Gauss elimination method.
\(2x + y + 4z = 12\)
\(8x - 3y + 2z = 23\)
\(4x + 11y - z = 33\)
The distance of motorbike at interval 2 minutes are given below:
| Time(minutes) | 0 | 2 | 4 | 6 |
| Distance(Km) | 0 | 1.5 | 3.8 | 6.7 |
Evaluate the velocity and acceleration of motorbike at time 2.37 and 4.57 minutes.
Write a program to compute value of \(y\) when \(x = 2\) such that \(\frac{dy}{dx} = 3x^2 + 1\) with \(y(1) = 2\) and \(h = 0.25\) using Euler's method.
Find the largest Eigen value and corresponding Eigen vector of the following matrix using power method.
\[\begin{bmatrix} 1 & 2 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}\]
Solve the Poisson equation \(\nabla^2 f = 2x^2y^2\) over the square domain \(0 \leq x \leq 3\) and \(0 \leq y \leq 3\) with \(f = 0\) on the boundary and \(h = 1\).
Attempt any TWO questions
[2x10=20]What do you mean by ill condition? Solve the following system using Jacobi iteration method.
\(2a + b + c = 5\)
\(3a + 5b + 2c = 15\)
\(2a + b + 4c = 8\)
How interpolation is differ from regression? Write down algorithm and program for Lagrange interpolation polynomial.
What is fixed point iteration method? How it converge the root of non-linear equation? Find the root of equation \(f(x) = x^2 - 3x + 2\) using N-R method.