Answer to Question #103594 in Quantitative Methods for Remi

Question

Perform 6 iteration to find zero of the equation: x+3=e^x using bisection algorithm. How many steps of the algorithm are needed to compute toot of the equation?

Answer 1

Solution. We write the equation in the form

$$x+3-e^x=0$$

Using for bisection algorithm for function f(x)=x+3-e^x.

Choose a and b such that f (a)> 0 and f (b) <0 (or vice versa). Find value c that equal to

$$c=\frac {a+b}{2}$$

Choose a and b such that f(a)> 0 and f (b) <0 (or vice versa). Find c

$$c=\frac {a+b}{2}$$

Find f(c). If f(c) =0 therefore c is root of equation f(x)=0. If f(c) ≠ 0 we check the sign of f(c).

If f(c) has the same sign as f(a) we replace a with c and we keep the same value for b.

If f(c) has the same sign as f(b), we replace b with c and we keep the same value for a.

The algorithm ends when the values of f(c) is less than a defined tolerance.

The number of iterations n needed to converge towards a root within the imposed tolerance Ɛ is calculated as:

$$n\eqslantgtr log_2(\frac{b-a}{Ɛ})$$

The equation x+3-e^x=0 has two roots.

Let's make two tables with 6 iterations for each case.