Question

The equation 0 2 5
3 2
x + x − = has a positive root in the interval [2,1] . Write a fixed
point iteration method and show that it converges. Starting with initial approximation
5.1 x0 = find the root of the equation. Perform two iterations.

Answer 1

Answer on Question #46682 – Math – Algorithms | Quantitative Methods

The equation 0 2 5

3 2

$x + x - =$ has a positive root in the interval [2,1]. Write a fixed

point iteration method and show that it converges. Starting with initial approximation

5.1 x0 = find the root of the equation. Perform two iterations.

Solution.

$$x ^ {3} + 2 x ^ {2} - 5 = 0 \rightarrow x = \sqrt [ 3 ]{5 - 2 x ^ {2}}$$

For the equation $x = g(x)$ fixed point iteration method is:

$$x _ {n + 1} = g (x _ {n}), n = 0, 1, \dots$$

If $g(x)$ and $g'(x)$ are continuous on an interval $J$ about their root $s$ of the equation $x = g(x)$ and $|g'(x)| < 1$ for all $x$ in the interval $J$ then the fixed point iterative process $x_{n+1} = g(x_n), n = 0, 1, \ldots$ will converge to the root $x = s$ for any initial approximation $x_0$ belongs to the interval $J$ .

In our case: $g(x) = \sqrt[3]{5 - 2x^2}$ , $g'(x) = \frac{1}{3} (5 - 2x)^{-\frac{2}{3}}$ , $g(x)$ and $g'(x)$ are continuous on the interval $(1, 2)$ , $|g'(x)| < \frac{1}{3}$ on $(1, 2)$ , thus the iteration method converges.

$$x _ {0} = 1. 5,$$ $$x _ {1} = \sqrt [ 3 ]{5 - 2 * 1. 5 ^ {2}} = 1. 4 0$$ $$x _ {2} = \sqrt [ 3 ]{5 - 2 * 1. 4 ^ {2}} = 1. 4 5$$ $$x _ {3} = \sqrt [ 3 ]{5 - 2 * 1. 4 5 ^ {2}} = 1. 4 2 6.$$

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