Horners method for approximating to the real roots of numerical equations

When the first significant digit or digits of a root have been obtained, by, say, Sturm s theorem, so that one root may be 

In outline, the method is as follows Find by means of Sturm s theorem, or otherwise, the integral part of a root, and transform the equation into another whose roots are less than those of the original equation by the number so found. Suppose we start with the equation 

The first significant figure of the root of this equation is 0 05. Lower 

The next significant figure of the root, deduced from (3), is 006. 

We could have continued in this way until the root had been obtained of any desired degree of accuracy. 

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