IGNOU MTE 10 2018 - The equation x2 + ax + b = 0 has two real roots p and q such that | p

Question

The equation x² + ax + b = 0 has two real roots p and q such that | p | < | q |. If we use the fixed point iteration $x_{k+1 = }\frac{-b}{x_{k} + a}$ , to find a root then to which root does it converge?

14 Feb 2018

Answer :

Word Count : 401

To solve this numerically, let's analyze the fixed-point iteration method provided: Given the equation $x^2 + ax + b = 0$ with roots $p$ and $q$ such that $ |p| < |q| $, the fixed-point iteration given is: \[ x_{k+1} = \frac{-b}{x_k + a} \] This iterative method is used to approximate the roots of the quadratic equation. The behavior of this iteration depends on the initial guess $x_0$. ### Step 1: Deriving Fixed Point Iteration for Convergence To determine to which root the iteration converges, we examine the iteration function: \[ f(x) = \frac{-b}{x + a} \] Now, we need to determine the fixed points of this

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x	0.1	1.2	2.4	3.9
f(x)	3.41	2.68	1.37	-1.48

x	1	2	3	4	5
$\sqrt\frac{{1+x^4}}{x^4}$	1.414	1.031	1.007	1.002	1.001