Algebraic & Transcendental Equations
We will cover following topics
PYQs
Regula-Falsi Method
1) Develop an algorithm for Regula-Falsi method to find a root of \(f(x)=0\) starting with two initial iterates \(x_{0}\) and \(x_{1}\) to the root such that sign \(\left(f\left(x_{0}\right)\right) \neq \operatorname{sign}\left(f\left(x_{1}\right)\right)\). Take \(n\) as the maximum number of iterations allowed and epsilon be the prescribed error.
[2009, 30M]
2) Find the smallest positive root of equation \(x e^{x}-\cos x=0\) using Regula-Falsi method. Do three iterations.
[2008, 12M]
3) Use the method of false position to find a real root of \(x^{3}-5 x-7=0\) lying between 2 and 3 and correct to 3 places of decimals.
[2007, 12M]
4) Find a real root of the equation \(f(x)=x^{3}-2 x-5=0\) by the method of false position.
[2002, 12M]
Newton-Raphson Method
1) Apply Newton-Raphson method, to find a real root of transcendental equation \(xlog_{10} x=1.2\), correct to three decimal places.
[2019, 10M]
2) Write an algorithm in the form of a flow chart for Newton-Raphson method. Describe the cases of failure of this method.
[2017, 15M]
3) Apply Newton-Raphson method to determine a root of the equation \(\cos x-x e^{x}=0\) correct up to four decimal places.
[2014, 10M]
4) Develop an algorithm for Newton-Raphson method to solve \(f(x)=0\) starting with initial iterate \(x_{0}\), \(n\) be the number of iterations allowed, epsilon be the prescribed relative error and delta be the prescribed lower bound for \(f^{\prime}(x)\).
[2013, 20M]
5) Use Newton-Raphson method to find the real root of the equation \(3 x=\cos x+1\) correct to four decimal places.
[2012, 12M]
6) Find the positive root of the equation \(10 x e^{-x^{2}}-1=0\) correct up to 6 decimal places by using Newton-Raphson method. Carry out computations only for three iterations.
[2010, 12M]
7) Draw a flow chart for solving equation \(F(x)=0\) correct to five decimal places by Newton-Raphson method.
[2008, 30M]
8) How many positive and negative roots of the equation \(e^{x}-5 \sin x=0\) exist? Find the smallest positive root correct to 3 decimals, using Newton-Raphson method.
[2004, 10M]
9) Find the positive root of the equation \(2 e^{-x}=\dfrac{1}{x+2}+\dfrac{1}{x+1}\) using Newton-Raphson method correct to four decimal places. Also show that the following scheme has error of second order:
\(x_{n+1}=\dfrac{1}{2} x_{n}\left(1+\dfrac{a}{x_{n}^{2}}\right)\).
[2003, 30M]