(octave.info)Finding Roots

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28.2 Finding Roots
==================

Octave can find the roots of a given polynomial.  This is done by
computing the companion matrix of the polynomial (see the ‘compan’
function for a definition), and then finding its eigenvalues.

 -- : roots (C)

     Compute the roots of the polynomial C.

     For a vector C with N components, return the roots of the
     polynomial

          c(1) * x^(N-1) + ... + c(N-1) * x + c(N)

     As an example, the following code finds the roots of the quadratic
     polynomial

          p(x) = x^2 - 5.

          c = [1, 0, -5];
          roots (c)
          ⇒  2.2361
          ⇒ -2.2361

     Note that the true result is +/- sqrt(5) which is roughly +/-
     2.2361.

     See also: Note: poly, Note: compan, Note:
     fzero.

 -- : Z = polyeig (C0, C1, ..., CL)
 -- : [V, Z] = polyeig (C0, C1, ..., CL)

     Solve the polynomial eigenvalue problem of degree L.

     Given an N*N matrix polynomial

     ‘C(s) = C0 + C1 s + ... + CL s^l’

     ‘polyeig’ solves the eigenvalue problem

     ‘(C0 + C1 + ... + CL)v = 0’.

     Note that the eigenvalues Z are the zeros of the matrix polynomial.
     Z is a row vector with N*L elements.  V is a matrix (N x N*L) with
     columns that correspond to the eigenvectors.

     See also: Note: eig, Note: eigs, *note compan:
     XREFcompan.

 -- : compan (C)
     Compute the companion matrix corresponding to polynomial
     coefficient vector C.

     The companion matrix is

               _                                                        _
              |  -c(2)/c(1)   -c(3)/c(1)  ...  -c(N)/c(1)  -c(N+1)/c(1)  |
              |       1            0      ...       0             0      |
              |       0            1      ...       0             0      |
          A = |       .            .      .         .             .      |
              |       .            .       .        .             .      |
              |       .            .        .       .             .      |
              |_      0            0      ...       1             0     _|

     The eigenvalues of the companion matrix are equal to the roots of
     the polynomial.

     See also: Note: roots, Note: poly, *note eig:
     XREFeig.

 -- : [MULTP, IDXP] = mpoles (P)
 -- : [MULTP, IDXP] = mpoles (P, TOL)
 -- : [MULTP, IDXP] = mpoles (P, TOL, REORDER)
     Identify unique poles in P and their associated multiplicity.

     The output is ordered from largest pole to smallest pole.

     If the relative difference of two poles is less than TOL then they
     are considered to be multiples.  The default value for TOL is
     0.001.

     If the optional parameter REORDER is zero, poles are not sorted.

     The output MULTP is a vector specifying the multiplicity of the
     poles.  ‘MULTP(n)’ refers to the multiplicity of the Nth pole
     ‘P(IDXP(n))’.

     For example:

          p = [2 3 1 1 2];
          [m, n] = mpoles (p)
             ⇒ m = [1; 1; 2; 1; 2]
             ⇒ n = [2; 5; 1; 4; 3]
             ⇒ p(n) = [3, 2, 2, 1, 1]

     See also: Note: residue, Note: poly, Note:
     roots, Note: conv, Note: deconv.