Lobachevskii's Method

Lobachevskii’s Method 

a method of approximate (numerical) solution of algebraic equations that was discovered independently by the Belgian mathematician G. Dandelin, the Russian mathematician N. I. Lobachevskii (in 1834 in its most complete form), and the Swiss mathematician K. Graffe.

The essence of Lobachevskii’s method consists in the construction of the equation f₁(x)= 0, whose roots are the squares of the roots of the initial equation f[x) = 0. Then we construct the equation f₂(x) = 0, whose roots are the squares of the roots of the equation f₁(x) = 0. By repeating this process several times we obtain an equation whose roots are far apart. If all roots of the initial equation are real and different in absolute value, then there exist simple computational schemes in Lobachevskii’s method for approximating the roots. In the case of roots with equal absolute values and in the case of complex roots, the computational schemes of Lobachevskii’s method prove to be very involved.

REFERENCES

Lobachevskii, N. I. “Algebra ili vychisleniia konechnykh.” Poln. sobr. soch., vol. 4. Moscow-Leningrad, 1948.
Berezin, I. S., and N. P. Zhidkov. Metody vychislenii, 2nd ed., vol. 2. Moscow, 1962.