code: def residuez(b, a, tol=1e-3, 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`.
  
      Parameters
      ----------
      b : array_like
          Numerator polynomial coefficients.
      a : array_like
          Denominator polynomial coefficients.
  
      Returns
      -------
      r : ndarray
          Residues.
      p : ndarray
          Poles.
      k : ndarray
          Coefficients of the direct polynomial term.
  
      See Also
      --------
      invresz, residue, unique_roots
  
      """
      b, a = map(asarray, (b, a))
      gain = a[0]
      brev, arev = b[::-1], a[::-1]
      krev, brev = polydiv(brev, arev)
      if krev == []:
          k = []
      else:
          k = krev[::-1]
      b = brev[::-1]
      p = roots(a)
      r = p * 0.0
      pout, mult = unique_roots(p, tol=tol, rtype=rtype)
      p = []
      for n in range(len(pout)):
          p.extend([pout[n]] * mult[n])
      p = asarray(p)
      # Compute the residue from the general formula (for discrete-time)
      #  the polynomial is in z**(-1) and the multiplication is by terms
      #  like this (1-p[i] z**(-1))**mult[i].  After differentiation,
      #  we must divide by (-p[i])**(m-k) as well as (m-k)!
      indx = 0
      for n in range(len(pout)):
          bn = brev.copy()
          pn = []
          for l in range(len(pout)):
              if l != n:
                  pn.extend([pout[l]] * mult[l])
          an = atleast_1d(poly(pn))[::-1]
          # bn(z) / an(z) is (1-po[n] z**(-1))**Nn * b(z) / a(z) where Nn is
          # multiplicity of pole at po[n] and b(z) and a(z) are polynomials.
          sig = mult[n]
          for m in range(sig, 0, -1):
              if sig > m:
                  # compute next derivative of bn(s) / an(s)
                  term1 = polymul(polyder(bn, 1), an)
                  term2 = polymul(bn, polyder(an, 1))
                  bn = polysub(term1, term2)
                  an = polymul(an, an)
              r[indx + m - 1] = (
                  polyval(bn, 1.0 / pout[n])
                  / polyval(an, 1.0 / pout[n])
                  / factorial(sig - m)
                  / (-pout[n]) ** (sig - m)
              )
          indx += sig
      return r / gain, p, k
  
imports: from __future__ import division, print_function, absolute_import
  import operator
  import threading
  import sys
  import timeit
  from scipy.spatial import cKDTree
  from .None import sigtools, dlti
  from ._upfirdn import upfirdn, _output_len
  from scipy._lib.six import callable
  from scipy._lib._version import NumpyVersion
  from scipy import fftpack, linalg
  from scipy.fftpack.helper import _init_nd_shape_and_axes_sorted
  from numpy import allclose, angle, arange, argsort, array, asarray, atleast_1d, atleast_2d, cast, dot, exp, expand_dims, iscomplexobj, mean, ndarray, newaxis, ones, pi, poly, polyadd, polyder, polydiv, polymul, polysub, polyval, product, r_, ravel, real_if_close, reshape, roots, sort, take, transpose, unique, where, zeros, zeros_like
  import numpy
  import math
  from scipy.special import factorial
  from .windows import get_window
  from ._arraytools import axis_slice, axis_reverse, odd_ext, even_ext, const_ext
  from .filter_design import cheby1, _validate_sos
  from .fir_filter_design import firwin
label_desc: Use image node_id 30 for calling a global function with example usage: residuez(b, a, tol, rtype) and returns: unknown,  p,  k