scipy.signal.invres(r, p, k, tol=0.001, rtype='avg')[source]

Compute b(s) and a(s) from partial fraction expansion: r,p,k

If M = len(b) and N = len(a):

        b(s)     b[0] x**(M-1) + b[1] x**(M-2) + ... + b[M-1]
H(s) = ------ = ----------------------------------------------
        a(s)     a[0] x**(N-1) + a[1] x**(N-2) + ... + a[N-1]

         r[0]       r[1]             r[-1]
     = -------- + -------- + ... + --------- + k(s)
       (s-p[0])   (s-p[1])         (s-p[-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]
-------- + ----------- + ... + -----------
(s-p[i])  (s-p[i])**2          (s-p[i])**n
Parameters :

r : ndarray


p : ndarray


k : ndarray

Coefficients of the direct polynomial term.

tol : float, optional

The tolerance for two roots to be considered equal. Default is 1e-3.

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

How to determine the returned root if multiple roots are within tol of each other.

‘max’: pick the maximum of those roots.

‘min’: pick the minimum of those roots.

‘avg’: take the average of those roots.

See also

residue, unique_roots

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