Navigation

Operators and Keywords

Function List:

C++ API

: [r, p, k, e] = residue (b, a)
: [b, a] = residue (r, p, k)
: [b, a] = residue (r, p, k, e)

The first calling form computes the partial fraction expansion for the quotient of the polynomials, b and a.

The quotient is defined as 

B(s)    M       r(m)        N
---- = SUM ------------- + SUM k(i)*s^(N-i)
A(s)   m=1 (s-p(m))^e(m)   i=1

where M is the number of poles (the length of the r, p, and e), the k vector is a polynomial of order N-1 representing the direct contribution, and the e vector specifies the multiplicity of the m-th residue’s pole.

For example, 

b = [1, 1, 1];
a = [1, -5, 8, -4];
[r, p, k, e] = residue (b, a)
   ⇒ r = [-2; 7; 3]
   ⇒ p = [2; 2; 1]
   ⇒ k = [](0x0)
   ⇒ e = [1; 2; 1]

which represents the following partial fraction expansion 

        s^2 + s + 1       -2        7        3
   ------------------- = ----- + ------- + -----
   s^3 - 5s^2 + 8s - 4   (s-2)   (s-2)^2   (s-1)

The second calling form performs the inverse operation and computes the reconstituted quotient of polynomials, b(s)/a(s), from the partial fraction expansion; represented by the residues, poles, and a direct polynomial specified by r, p and k, and the pole multiplicity e.

If the multiplicity, e, is not explicitly specified the multiplicity is determined by the function mpoles.

For example: 

r = [-2; 7; 3];
p = [2; 2; 1];
k = [1, 0];
[b, a] = residue (r, p, k)
   ⇒ b = [1, -5, 9, -3, 1]
   ⇒ a = [1, -5, 8, -4]

where mpoles is used to determine e = [1; 2; 1]

Alternatively the multiplicity may be defined explicitly, for example, 

r = [7; 3; -2];
p = [2; 1; 2];
k = [1, 0];
e = [2; 1; 1];
[b, a] = residue (r, p, k, e)
   ⇒ b = [1, -5, 9, -3, 1]
   ⇒ a = [1, -5, 8, -4]

which represents the following partial fraction expansion 

 -2        7        3         s^4 - 5s^3 + 9s^2 - 3s + 1
----- + ------- + ----- + s = --------------------------
(s-2)   (s-2)^2   (s-1)          s^3 - 5s^2 + 8s - 4

See also: mpoles, poly, roots, conv, deconv.

Package: octave