Bsc CSIT Semester 3 – Numerical Method – Unit 1: Solution of Nonlinear Equations (8 Hrs.)

What are inherent errors? Derive the Newton Raphson method for solving non-linear equation and using this method solve $x2-5x+6=0$. Calculate upto 3 decimal places.

How secant methods differs from Newton Rhapson method? Derive the formula for Secant Method. Solve the equation $Cosx + 2Sinx - x² = 0$ using Secant method. Assume error precision as 0.01. Discuss the drawbacks of the Newton Rhapson method.

Define the terms approximate error and relative approximate error? Discuss the working of Half Interval method for finding the roots of non-linear equation.

How secant method can approximate the root of a non-linear equation? Explain with necessary derivation. Estimate a real root of following equation using secant method. Assume error precisionn of 0.01.

$x3 + 2x - cos(x) = 4$

Calculate a real root of the following function using bisection method correct upto 3 significant figures.

What is fixed point iteration method? How can it converge to the root of a non-linear equation? Also explain the diverging cases with suitable examples.

How can Horner’s rule be used to evaluate the $f(x)$ and $f(x)$ of a polynomial at a given point? Explain. Write an algorithm and program to calculate a real root of a polynomial using Horner’s rule.

Derive the formula for Newton Raphson Method. Solve the equation $x^2 + 4x - 9 = 0$ using Newton Raphson method. Assume error precision is $0.01.$ Discuss drawbacks of the Newton Raphson method.

How the half-interval method can be estimate a root of a non-linear equation? Find a real root of the following equation using the half-interval method to correct up to two decimal places.

$x^2 - e^{-x} - x =$1

Calculate the real root of the given equation using fixed point iteration correct up to 3 significant figures.