Question

Find a root of the equation 3x^3+10x^2+10x+7 = 0 which is close to −2.0, using
the Berge -Vieta method. Perform two iterations of the method.

Answer 1

At the first step of any iteration of the Birge-Vieta method, we write down the coefficients of the equation and put that

b₀=a₀;

c₀=a₀.

Also, we have the initial approximation p₀=-2.

Second step: find a coefficient

$$b_i=a_i+p_0⋅b_{i-1}.$$

Third step: find

$$c_i=b_i+p_0⋅c_{i-1}.$$

Fourth step: find

$$p_1=p_0-\frac{b_n}{c_{i-1}}.$$

Fifth step: when

$$|p_1-p_0|≤\epsilon,$$

we found the answer.

It is better to arrange the coefficients in the form of a table. Therefore, at the first iteration we have

and

$$p_1=p_0-\frac{b_3}{c_2}=-2-\frac{3}{6}=-2.5.$$

The second iteration gives

and

$$p_2=p_1-\frac{b_3}{c_2}=-2.5-\frac{-2.375}{16.25}=-2.3539.$$