# scipy.signal.residue

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

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

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

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

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

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

  If there are any repeated roots (closer together than tol), then H(s) has terms like:

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

  This function is used for polynomials in positive powers of s or z, such as analog filters or digital filters in controls engineering. For negative powers of z (typical for digital filters in DSP), use `residuez`.

  See Notes 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

  `invres`, `residuez`, `numpy.poly`, `unique_roots`

  Notes

  The “deflation through subtraction” algorithm is used for computations — method 6 in [1].

  The form of partial fraction expansion depends on poles multiplicity in the exact mathematical sense. However there is no way to exactly determine multiplicity of roots of a polynomial in numerical computing. Thus you should think of the result of `residue` with given tol as partial fraction expansion computed for the denominator composed of the computed poles with empirically determined multiplicity. The choice of tol can drastically change the result if there are close poles.

  References

  [1]

  J. F. Mahoney, B. D. Sivazlian, “Partial fractions expansion: a review of computational methodology and efficiency”, Journal of Computational and Applied Mathematics, Vol. 9, 1983.