Answer to Question #260165 in Quantitative Methods for Yuan

Question

Find the real root of the equation e^x=sinx correct to three decimal places by bisection method

Answer 1

Rewrite the equation in the form f(x) = 0:

$f(x) = e^x - \sin(x) = 0$

Let l denote left side of the interval contained a root, h right side, and m = (h + l)/ 2 the middle of the interval. As f(-4) = -0.73849 < 0, and f(-2) = 1.044633 > 0, get l = -4, and h = 2. Then fill the table. Here depending of the value signe in the next column we define which boundary should be shifted (we change left boundary if f(m) < 0, and right boundar if f(m) > 0)

$\begin{array}{|| c | c | c | c | c | c | c ||} h - l & l & f(l) & h & f(h) & m = (h+l)/2 & f(m) \\ 2.00000 & -4.00000 & -0.73849 & -2.00000 & 1.04463 & -3.00000 & 0.19091 \\ 1.00000 & -4.00000 & -0.73849 & -3.00000 & 0.19091 & -3.50000 & -0.32058 \\ 0.50000 & -3.50000 & -0.32059 & -3.00000 & 0.19091 & -3.25000 & -0.06942 \\ 0.25000 & -3.25000 & -0.06942 & -3.00000 & 0.19091 & -3.12500 & 0.06053 \\ 0.12500 & -3.25000 & -0.06942 & -3.12500 & 0.06053 & -3.18750 & -0.00462 \\ 0.06250 & -3.18750 & -0.00462 & -3.12500 & 0.06053 & -3.15625 & 0.02793 \\ 0.03125 & -3.18750 & -0.00462 & -3.15625 & 0.02793 & -3.17188 & 0.01165 \\ 0.01562 & -3.18750 & -0.00462 & -3.17188 & 0.01165 & -3.17969 & 0.00351 \\ 0.00781 & -3.18750 & -0.00462 & -3.17969 & 0.00351 & -3.18359 & -0.00055 \\ \end{array}$

So the root of the equation is -3.18 with 3 decimal digit precision.