BIT Numerical Methods

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.

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.

Construct the divided difference table for the following data value and find first and second order derivative at \(x=2\).

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.

What is least squares method of fitting a function? Fit the second order polynomial for the following data values.

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 \(dy/dx = 1-2xy\), given that \(y = 0\) when \(x = 0\) with \(h=0.2\) using Heun's method.

Consider a steel plate of size 24cm x 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 x 8cm.

Write an algorithm for Honer's method. Evaluate the polynomial \(f(x) = x^4 + 3x^3 + 5x^2 + 7^x + 9\) at \(x = 2\) by using Honer's method.