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The negative Routh test and its application to the cases of vanishing leading elements and imaginary roots. (English) Zbl 0615.65051

The ordinary Routh test furnishes in a straightforward manner the number \(r_+\), of positive roots of a polynomial of degree n. A modification that allows one to obtain the number \(r_-\) of negative roots in the same way is presented. Thus, the number \(r_ 0\) of (purely) imaginary roots is immediately found as \(n-(r_++r_-)\). The modified test is especially useful when there is a first column vanishing element.

MSC:

65H05 Numerical computation of solutions to single equations
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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