`residue`

Partial fraction expansion of a ratio of polynomials.

Syntax

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

Expands b(s)/a(s) into residues r, poles p and the direct polynomial term k.

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

Converts back: combines the partial fractions into numerator b and denominator a.

Description

The decomposition is b(s)/a(s) = r(1)/(s - p(1)) + ... + r(n)/(s - p(n)) + k(s). A repeated pole p appears several times in p, and its residues correspond to the terms 1/(s-p), 1/(s-p)^2, and so on. It is the standard tool for inverse Laplace and Z transforms, where each term is inverted separately.

Close or repeated poles make the result sensitive to rounding, because they are found with roots.

Input arguments

b, a: Coefficient vectors of the numerator and denominator, highest power first.
r, p, k: Residues, poles and direct term, as returned by the first form.

Output arguments

r, p: Column vectors of residues and poles of the same length.
k: Coefficients of the polynomial part; empty when the degree of b is less than that of a.

Example

Expand and rebuild

[r, p, k] = residue([1 0 0 1], [1 3 2])
[b, a] = residue(r, p, k)

Output

r = 2×1

     7
     0
p = 2×1

    -2
    -1
k = 1×2

     1    -3
b = 1×4

     1     0     0     1
a = 1×3

     1     3     2

residue