View source for Bernoulli method

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b0156301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

with the largest modulus (absolute value). The method was proposed by D. Bernoulli [[#References|[1]]] and is based on the following principle. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b0156302.png" /> be random numbers and let the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b0156303.png" /> be calculated by the following difference equation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b0156304.png" /></td> </tr></table>

In general, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b0156305.png" />, the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b0156306.png" /> tends to the value of the root of equation (*) with the largest modulus.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Bernoulli,   ''Comm. Acad. Sci. Imper. Petropolitanae'' , '''3''' , Petropolis  (1732)  pp. 62–69</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.T. Whittaker,   G. Robinson,   "Mathematische Bearbeitung von Resultaten der Observation" , Leningrad-Moscow  (1935)  (In Russian; translated from German)</TD></TR></table>



====Comments====
The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b0156307.png" /> 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. [[#References|[a1]]]). Let there be one (not necessarily real) root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b0156308.png" /> of largest modulus, with multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b0156309.png" />. Then the solution of the linear difference equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563010.png" /> is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563011.png" /></td> </tr></table>

as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563012.png" />. This expression is the solution of a linear difference equation with characteristic polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563013.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563014.png" /> satisfies the linear difference equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563017.png" />, then this equation can be solved explicitly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563018.png" />, otherwise (a1) is rewritten with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563019.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563020.png" /> and the resulting equations are solved in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563021.png" />.

If two roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563023.png" /> (of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563025.png" />) of the same maximum modulus occur, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563026.png" /> satisfies, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563027.png" />, a linear difference equation with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563028.png" /> as characteristic polynomial, and (a1) must be replaced by the corresponding difference equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015630/b01563029.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.B. Hildebrand,   "Introduction to numerical analysis" , McGraw-Hill  (1956)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.S. Householder,   "The numerical treatment of a single nonlinear equation" , McGraw-Hill  (1970)</TD></TR></table>