A NOTE ON THE MULTIPLE ZEROS OF RANDOM POLYNOMINALS
Abstract
The solution of a differential or difference equation is expressible in terms of the roots of he charact ristic equation. The expression has one form if the roots of the characteristic equation are all distinct, and another, somewhat more complicated form if the characteristic equation has one or more multiple roots. It is shown that the second form need never be considered. The coefficients of the characteristic equation of a physical system are functions of the physical parameters of the system. However accurately an attempt is made to fix t ese parameters, they are still ran om variables with non-zero variance. The coefficients of the charac eristic equation will also be random variables with nonzero variance . Moreover, the nature of any macroscopic physical problem is such that the coefficients have true probability densities (no impulses), so that the probability of any coefficient having some fixed value is zero. The coefficients are statistically dependent, but are not subject to any purely deterministic relationships. Under these con itions t e probability that the characteristic equation has a multiple root is zero. ence, t e econd solution form of a differential or difference equation need never be used. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 31, 1962
- Accession Number
- AD0283777
Entities
People
- Joel Max
- John Fogarty
Organizations
- Massachusetts Institute of Technology