Define true error and relative error. Derive the bisection method for solving non-linear equation and using this method solve \(2x^3 - 2x - 5\) with initial \(x_0 = 1\) and \(x_1 = 2\). Calculate upto \(10^{th}\) iteration.
What are the applications of interpolation? Differentiate between interpolation and regression. Consider the following data points estimate the \(f(10)\) using Lagrange's interpolation.
| x | 5 | 6 | 9 | 11 |
| y | 13 | 14 | 15 | 16 |
What do you mean by numerical integration? Write any one application of numerical integration. Write an algorithm and c program to implement multi-segment trapezoidal rule.
Attempt any Eight questions
[8×5=40]Construct the divided difference table for the following data value and find first and second order derivative at \(x = 2\).
| x | 1 | 2 | 4 | 8 | 10 |
| y | 0 | 1 | 5 | 21 | 27 |
Why it is better to use composite Simpson's \(\frac{3}{8}\) rule instead of Simpson's \(\frac{3}{8}\) rule? Find the value of integration for following data set using Simpson's \(\frac{3}{8}\) rule.
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| y | 1 | 0.5 | 0.2 | 0.1 | 0.0588 | 0.0385 | 0.027 |
What is least squares method of fitting a function? Fit the second order polynomial for the following data values.
| x | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| y | 2 | 6 | 7 | 8 | 10 | 12 | 15 |
Solve the following system of linear equations using Gaussian elimination method.
\(2x + 2y + z = 12\)
\(3x + 2y + 2z = 8\)
\(5x + 10y - 8z = 10\)
Solve the following system of linear equations using Gauss-Seidal method.
\(10x + y + z = 12\)
\(2x + 10y + z = 13\)
\(2x + 2y + 10z = 14\)
Define eigen value and eigen vector. Explain how shooting method is used to solve boundary value problem.
Find the approximate value of \(y\) when \(x = 0.6\) of \(\frac{dy}{dx} = 1 - 2xy\), given that \(y = 0\) when \(x = 0\) with \(h = 0.2\) using Heun's method.
Consider a steel plate of size 24cm × 24cm. If two of the opposite sides are held at 100 degree Celsius and the other two opposite sides at 0 degree Celsius, find the steady state temperatures of interior points, assuming a grid size of 8cm × 8cm.
Write an algorithm for Horner's method. Evaluate the polynomial \(f(x) = x^4 + 3x^3 + 5x^2 + 7x + 9\) at \(x = 2\) by using Horner's method.