Answer to Question #143378 in Quantitative Methods for Usman

Question

Solve by iteration method:
(i) x^3 - 2 x^2 − 5=0

Answer 1

$$x^3-2x^2-5=0$$

$$x=g(x)=\sqrt[3]{2x^2+5}$$

$x_0=2.6$

$$x_{i+1}=g(x_i)=\sqrt[3]{2x_i^2+5}, i=0, 1,2,...$$

$$\def\arraystretch{1.5} \begin{array}{c:c:c} i & x_i & g(x_i) \\ \hline 0 & 2.6 & 2.64573897 \\ 1 & 2.64573897& 2.66839553 \\ 2 &2.66839553 & 2.67962112 \\ 3 & 2.67962112 & 2.68518353\\ 4 & 2.68518353 & 2.68793986 \\ 5 & 2.68793986 & 2.68930573\\ 6 & 2.68930573 & 2.68998257\\ 7 & 2.68998257 & 2.69031797\\ 8 & 2.69031797 & 2.69048418\\ 9 & 2.69048418 & 2.69056654\\ 10 & 2.69056654 & 2.69060735\\ 11 & 2.69060735 & 2.69062758\\ 12 & 2.69062758 & 2.69063760\\ 13 & 2.69063760 & 2.69064257\\ 14 & 2.69064257 & 2.69064503\\ 15 & 2.69064503 & 2.69064625\\ 16 & 2.69064625 & 2.69064685\\ 17 & 2.69064685 & 2.69064715\\ 18 & 2.69064715 & 2.69064730\\ 19 & 2.69064730 & 2.69064737\\ 20 & 2.69064737 & 2.69064741\\ 21 & 2.69064741 & 2.69064743\\ 22 & 2.69064743 & 2.69064744\\ \hdashline 23 &2.69064744 & 2.69064744\\ \end{array}$$

$x\approx2.69064744$