# scipy.signal.residuez

scipy.signal.residuez(b, a, tol=0.001, rtype='avg')

  Compute partial-fraction expansion of b(z) / a(z).

  If M is the degree of numerator b and N the degree of denominator a:

              b(z)     b[0] + b[1] z**(-1) + ... + b[M] z**(-M)
      H(z) = ------ = ------------------------------------------
              a(z)     a[0] + a[1] z**(-1) + ... + a[N] z**(-N)

  then the partial-fraction expansion H(z) is defined as:

              r[0]                   r[-1]
      = --------------- + ... + ---------------- + k[0] + k[1]z**(-1) ...
        (1-p[0]z**(-1))         (1-p[-1]z**(-1))

  If there are any repeated roots (closer than tol), then the partial fraction expansion has terms like:

           r[i]              r[i+1]                    r[i+n-1]
      -------------- + ------------------ + ... + ------------------
      (1-p[i]z**(-1))  (1-p[i]z**(-1))**2         (1-p[i]z**(-1))**n

  This function is used for polynomials in negative powers of z, such as digital filters in DSP. For positive powers, use `residue`.

  See Notes of `residue` for details about the algorithm.

  Parameters:
    b array_like

      Numerator polynomial coefficients.

    a array_like

      Denominator polynomial coefficients.

    tol float, optional

      The tolerance for two roots to be considered equal in terms of the distance between them. Default is 1e-3. See `unique_roots` for further details.

    rtype{‘avg’, ‘min’, ‘max’}, optional

      Method for computing a root to represent a group of identical roots. Default is ‘avg’. See `unique_roots` for further details.

  Returns:
    r ndarray

      Residues corresponding to the poles. For repeated poles, the residues are ordered to correspond to ascending by power fractions.

    p ndarray

      Poles ordered by magnitude in ascending order.

    k ndarray

      Coefficients of the direct polynomial term.

  See also

  `invresz`, `residue`, `unique_roots`