Answer to Question #139397 in Quantitative Methods for elle

Question

find the roots using simple fixed point iteration. Stop until at 15th iteration or when approximate percent relative error is below 0.05%.

e^x+x=4

Answer 1

$$e^x+x=4$$

We know that there is a solution for the equation $e^x+x-4=0$ in $[1,2].$

$$x=\ln(4-x)$$

$$x_{i+1}=\ln(4-x_i)$$

$$\begin{matrix} i & x_i \\ 0 & 1\\ 1 & 1.098612 \\ 2 & 1.065189\\ 3 & 1.076643\\ 4 & 1.072733\\ 5 & 1.074069\\ 6 & 1.073613\\ 7 & 1.073769 \\ 8 & 1.073715\\ 9 & 1.073734\\ 10 & 1.073727\\ 11 & 1.073729 \\ 12 & 1.073729\\ \end{matrix}$$

$$\varepsilon =\big|\dfrac{n_{i+1}-n_i}{n_i}\big|\cdot 100\%$$

$$\varepsilon =\big|\dfrac{1.098612-1}{1}\big|\cdot 100\%=9.8612\%$$

$$\varepsilon =\big|\dfrac{1.065189-1.098612}{1.098612}\big|\cdot 100\%=3.0423\%$$

$$\varepsilon =\big|\dfrac{1.076643-1.065189}{1.065189}\big|\cdot 100\%=1.0753\%$$

$$\varepsilon =\big|\dfrac{1.072733-1.076643}{1.076643}\big|\cdot 100\%=0.3632\%$$

$$\varepsilon =\big|\dfrac{1.074069-1.072733}{1.072733}\big|\cdot 100\%=0.1245\%$$

$$\varepsilon =\big|\dfrac{1.073769-1.073713}{1.073713}\big|\cdot 100\%=0.005029\%$$

$$\varepsilon =\big|\dfrac{1.074069-1.072733}{1.072733}\big|\cdot 100\%=0.1245\%$$

$$\varepsilon =\big|\dfrac{1.073613-1.074069}{1.074069}\big|\cdot 100\%=0.0425\%$$

$$\varepsilon =\big|\dfrac{1.073769-1.073613}{1.073613}\big|\cdot 100\%=0.0145\%$$

$$\varepsilon =\big|\dfrac{1.073715-1.073769}{1.073769}\big|\cdot 100\%=0.0050\%$$

$x=1.073613$

The root is $1.074$