Re: bound on eigenvalues...


Robert Low wrote:
> José Carlos Santos wrote:
> > The eigenvalues are the roots of the characteristic polynomial, which is
> > P(x) = -x^3 + a x^2 + b x + c, with a = 0.501, b = -0.024089, and
> > c = 0.0000524046. Since it's a third degree polynomial, it can't have
> > more than three roots. Now, since P(0) = 0.0000524046, P(0.05) =
> > -0.0000245449, P(0.1) = 0.00165351, P(0.4) = 0.00657681, and P(0.5) =
> > -0.0117421, there must be a root between 0 and 0.1, a second one between
> > 0.1 and 0.4, and a third one between 0.4 and 0.5. This proves that all
> > roots are smaller than 0.5.
>
> But I think he *meant* to write that there must be a root
> between 0 and 0.05, a second one between 0.05 and 0.1,
> and a third one between 0.4 and 0.5, since that's how the
> sign changes go.

But these approaches look like numerical... it is perfectly OK for
practice ... but here we need analytical methods...

.

Relevant Pages

  • Re: Searching zeros of complex function
    ... >> polynomial of the order of A, n say, and its n roots are the ... >> A. IMSL has routines to determine the eigenvalues of general complex A. ... In the current post to the OP I gave him a hint and ... in a followup post I gave him the answer to his homework assignment. ...
    (comp.lang.fortran)
  • Re: Finding eigenvalues
    ... Arturo Magidin wrote/skrev/kaita/popisal/schreibt: ... were able to write an upper case W. ... different eigenvalues, ... the two other cubic roots of 1 in the complex numbers. ...
    (sci.math)
  • Re: Symbolic solution of quadratic matrix equations
    ... degree-2n polynomial will work (assuming the roots are distinct). ... roots correspond to matrices whose eigenvalues involve no reciprocal pairs ... according to the number of reciprocal pairs of eigenvalues for X. ...
    (sci.math)
  • Re: eigenvalues
    ... The question about eigenvalues will have to be at least as hard as ... the corresponding question about roots of polynomials (every polynomial ... But even for polynomials the question is not easy; ... best answer there is probably the use of Sturm sequences, ...
    (sci.math.num-analysis)
  • Re: Nth root of a matrix (with some constraints)
    ... eigenvalues with distinct magnitudes and having eigenvectors V such that ... roots of those of PN which must therefore also have distinct magnitudes, ... probabilistic type solution for each N. This greatly narrows down the ...
    (comp.soft-sys.matlab)