Info Catalog octave: Evaluating Polynomials octave: Polynomial Manipulations octave: Products of Polynomials

octave: Finding Roots

 
 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.
 
DONTPRINTYET       See also: poly XREFpoly, compan XREFcompan, *noteDONTPRINTYET       See also: poly XREFpoly, compan XREFcompan, 
      fzero XREFfzero.
 
  -- : 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.
 
DONTPRINTYET       See also: eig XREFeig, eigs XREFeigs, *notecompan:
DONTPRINTYET       See also: eig XREFeig, eigs XREFeigs, 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.
 
DONTPRINTYET       See also: roots XREFroots, poly XREFpoly, *noteeig:
DONTPRINTYET       See also: roots XREFroots, poly XREFpoly, 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]
 
DONTPRINTYET       See also: residue XREFresidue, poly XREFpoly, *noteDONTPRINTYET       See also: residue XREFresidue, poly XREFpoly, 
      roots XREFroots, conv XREFconv, deconv XREFdeconv.
 

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