Polynomials With Real Roots
It is possible to determine the bounds of the roots of a polynomial using Samuelson's inequality. This method is due to a paper by Laguerre.
Let
be a polynomial with all real roots. The roots are located in the interval with endpoints
- .
Example: The polynomial
has four real roots -3, -2, -1 and 1. The formula gives
- ,
its roots are contained in
- I = .
If the polynomial f has real simple roots the Hessian H(f) evaluated on the interval is always ≥ 0. In symbols
H(f) = (n − 1)2 f' 2 − n(n − 1) f f' ≥ 0
where f' is the derivative of f with respect to x.
When n > 1 this simplifies to
f'(x) ≥ n f(x)
This relation applied to polynomials with complex roots is known as Bernstein's inequality.
Read more about this topic: Properties Of Polynomial Roots
Famous quotes containing the words real and/or roots:
“It is a war against the pines, the only real Aroostook or Penobscot war.”
—Henry David Thoreau (1817–1862)
“Einstein is not ... merely an artist in his moments of leisure and play, as a great statesman may play golf or a great soldier grow orchids. He retains the same attitude in the whole of his work. He traces science to its roots in emotion, which is exactly where art is also rooted.”
—Havelock Ellis (1859–1939)