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Why is the study of errors important to a computational scientist? Differentiate between inherent and numerical errors.
Find the root of equation \( x^2 + 4x - 10 = 0 \) using bisection method where root lies between 5 and 6.
Find the square root of 3,5 using second order Lagrange interpolation polynomial using the following data table.
\[ \begin{array}{c|cccc} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 1 & 1.4142 & 1.7321 & 2 & 2.2361 \\ \end{array} \]
Write a program to calculate the integral using Trapezoidal Rule.
Solve the following set of equations using Gauss-Jordan Method.
\(3x-5y+2z=15\)
\(4xy+z=2\)
\(x-3y+7z=22\)
Use classical Runge-Kutta method to estimate \( y(0.2) \) when \( y'(x) = x^2 + y^2 \) with \( y(0) = 0 \) and \( h = 0.2 \).
Solve the Poisson equation \( \nabla^2 f = 2x^2 y^2 \) 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]a) Factorize the matrix \( \begin{array}{|c c c|} 1 & 2 & 3 \\ 2 & 3 & 22 \\ 3 & 22 & 82 \\ \end{array} \) Using Cholesky's method.
b) Locate the root of equation \( x^2 + x - 2 \) using the fixed point method.
a) Fit a straight line to the following set of data points.
| \(x\) | 1 | 3 | 4 | 6 | 8 | 9 |
| \(y\) | 1 | 3 | 4 | 4 | 5 | 7 |
b) Differentiate between ordinary and partial differential equations with their applications.
Given the data points:
| \(i\) | 0 | 1 | 2 |
| \(x_i\) | 2 | 3 | 4 |
| \(f_i\) | 4 | 9 | 16 |