Bernoulli method

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A method for finding the real root of algebraic equations of the type


with the largest modulus (absolute value). The method was proposed by D. Bernoulli [1] and is based on the following principle. Let be random numbers and let the values of be calculated by the following difference equation:

In general, as , the expression tends to the value of the root of equation (*) with the largest modulus.


[1] D. Bernoulli, Comm. Acad. Sci. Imper. Petropolitanae , 3 , Petropolis (1732) pp. 62–69
[2] E.T. Whittaker, G. Robinson, "Mathematische Bearbeitung von Resultaten der Observation" , Leningrad-Moscow (1935) (In Russian; translated from German)


The expression tends to the root of largest modulus only if this root is simple and if no other roots of the same maximum modulus occur. If not, then one can proceed as follows (cf. [a1]). Let there be one (not necessarily real) root of largest modulus, with multiplicity . Then the solution of the linear difference equation for is

as . This expression is the solution of a linear difference equation with characteristic polynomial , so that satisfies the linear difference equation


as . If , then this equation can be solved explicitly in , otherwise (a1) is rewritten with replaced by and the resulting equations are solved in .

If two roots and (of multiplicity and ) of the same maximum modulus occur, then satisfies, as , a linear difference equation with as characteristic polynomial, and (a1) must be replaced by the corresponding difference equation of order .


[a1] F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1956)
[a2] A.S. Householder, "The numerical treatment of a single nonlinear equation" , McGraw-Hill (1970)
How to Cite This Entry:
Bernoulli method. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.N. Dovbysh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article