Abstract
A wide variation in the magnitude of coefficients of polynomials may be a source of computational problems in root-finding algorithms, as the floating-point arithmetic operations on such coefficients may render floating-point overflow or underflow. This paper presents a new discrete method for (real) polynomial scaling—specifically, it determines a scale factor that minimizes variation in the magnitude of the coefficients of real polynomials. The method is conceptually simple and easy to implement. It is based on the phenomenon that the scale factor coincides with the intersection of certain monomials, defined by the respective magnitudes of the nonzero terms of the polynomial to be scaled. A constructive description of this phenomenon is presented. The method compares favorably with an existing general mathematical programming approach. Results on the effect of scaling polynomials on the numerical quality of their approximate roots are presented; these results show that the effect, if any, is insignificant.
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