Partial Fraction Expansions

residue finds the partial fraction expansion of the ratio of two polynomials. This is particularly useful for applications that represent systems in transfer function form. For polynomials b and a, if there are no multiple roots,

b(s)a(s)=r1sp1+r2sp2++rnspn+ks

where r is a column vector of residues, p is a column vector of pole locations, and k is a row vector of direct terms. Consider the transfer function

4s+8s2+6s+8

b = [-4 8];
a = [1 6 8];
[r,p,k] = residue(b,a)

r =
   -12
     8

p =
    -4
    -2

k =
     []

Given three input arguments (r, p, and k), residue converts back to polynomial form:

[b2,a2] = residue(r,p,k)

b2 =
    -4     8
a2 =
     1     6     8
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