Answer to Question #143377 in Quantitative Methods for Usman

Question

Solve by iteration method:
(i) sinx = x+1/x-1

Answer 1

$$(x-1)\sin x=x+1$$

$$x=g(x)=-\dfrac{1+\sin x}{1-\sin x}$$

$x_0=-0.4$

$$x_{i+1}=g(x_i)=-\dfrac{1+\sin x_i}{1-\sin x_i}, i=0, 1,2,...$$

$$\def\arraystretch{1.5} \begin{array}{c:c:c} i & x_i & g(x_i) \\ \hline 0 & -0.4 & -0.43405335 \\ 1 &-0.43405335 & -0.40790368 \\ 2 &-0.40790368 & -0.43196121 \\ 3 & -0.43196121 & -0.40978834\\ 4 & -0.40978834 & -0.43019040 \\ 5 & -0.43019040 & -0.41138892\\ 6 & -0.41138892 & -0.42869108\\ 7 & -0.42869108 & -0.41274796\\ 8 & -0.41274796 & -0.42742130\\ 9 & -0.42742130 & -0.41390172\\ 10 & -0.41390172 & -0.42634568\\ 11 & -0.42634568 & -0.41488107\\ 12 & -0.41488107 & -0.42543436\\ 13 & -0.42543436 & -0.41571225\\ 14 & -0.41571225 & -0.42466214\\ 15 & -0.42466214 & -0.41641760\\ 16 & -0.41641760 & -0.42400769\\ 17 & -0.42400769 & -0.41701612\\ 18 & -0.41701612 & -0.42345300\\ 19 & -0.42345300 & -0.41752394\\ 20 & -0.41752394 & -0.42298282\\ 21 & -0.41752394 & -0.42298282\\ 22 & -0.42298282 & -0.41795478\\ 23 &-0.41795478 & -0.42258425\\ 24 & -0.42258425 & -0.41832028\\ 25 & -0.41832028 & -0.42224635\\ 26 & -0.42224635 & -0.41863034\\ 27 & -0.41863034 & -0.42195987\\ 28 & -0.42195987 & -0.41889336\\ 29 & -0.41889336 & -0.42171698\\ 30 & -0.42171698 & -0.41911646\\ 31 & -0.41911646 & -0.42151104\\ 32 & -0.42151104 & -0.41930569\\ 33 & -0.41930569 & -0.42133643\\ 34& -0.42133643 & -0.41946619\\ 35 & -0.41946619 & -0.42118838\\ 36 & -0.42118838 & -0.41960232\\ 37 & -0.41960232 & -0.42106283\\ 38 & -0.42106283 & -0.41971778\\ 39 & -0.41971778 & -0.42095638\\ 40 & -0.42095638 & -0.41981570\\ 41 &-0.41981570 & -0.42086611\\ 42 &-0.42086611 & -0.41989875\\ 43 & -0.41989875 & -0.42078956\\ 44 & -0.42078956 & -0.41996919\\ 45 & -0.41996919 & -0.42072364\\ 46 & -0.42072364 & -0.42002985\\ 47 & -0.42002985 & -0.42066875\\ 48 & -0.42066875 & -0.42008037\\ 49 & -0.42008037 & -0.42062220\\ 50 &-0.42062220 & -0.42012322\\ 51 & -0.42012322 & -0.42058272\\ 52 &-0.42058272 & -0.42015956\\ 53 &-0.42015956 & -0.42054924\\ 54 & -0.42054924 & -0.42019038\\ 55 &-0.42019038 & -0.42052085\\ 56 &-0.42052085 & -0.42021652\\ 57 &-0.42021652 & -0.42049677\\ 58 &-0.42049677 & -0.42023869\\ 59 &-0.42023869 & -0.42047635\\ \hdashline 60 &-0.42047635 & -0.42025749\\ \end{array}$$

$x\approx-0.4204$