Sturm s theorem

The problem of separating the real roots of a numerical equation is, however, completely solved by what is known as Sturm s theorem. It is dear that if x assumes every possible value in succession from + oo to — oo, every change of sign will indicate the proximity of a real root. The total number of roots is known from the degree of the equation, therefore the number of imaginary roots can be determined by difference. 

Sturm s theorem enables these changes of sign to be readily detected. The process is as follows —... 

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... 

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